View Full Version : Highest Possible Crit Range?

lvl 1 sharnian

2009-11-25, 11:51 AM

In an attempt to try and find the highest crit range I can get, I came up with this:

An Aptitude Rapier has 15-20 crit range

Street Fighter Barbarian 7 gives +1 to Threat Range on a charge or flat-footed target = 14-20

Disciple of Dispater 8, gives x3 to crit range, which ends up as 9-20 -1 for Barb7 = 8-20 Crit Range

Throw in Arcane Duelist adds in a false keen, = 6-20 -1 for Barb7 = 5-20

Add OA Weapon Master for -2 = 3-20 Crit Range

Throw the Psychic Weapon Master for -2 again = 1-20 Crit Range

Combined with Lightning Maces for Infinite Attacks.

Using beneficial stacking rules, is there a way to make this legal? Probably impossible pre-epic (I know, there are worse things in Epic than critical hits) , but if playing Gestalt, that might be possible pre-epic with Barbarian//PsychicWarrior.

Also, if this works, can you threaten a critical hit on a natural 1?

Sleepingbear

2009-11-25, 11:54 AM

Crit range does not affect ability to hit.

So you do not threaten a crit on a natural one as a natural one always misses.

As for the rest of it, I have no idea.

drengnikrafe

2009-11-25, 11:55 AM

I'm afraid I cannot poke any holes in your logic, but only because I haven't actually investiaged whether or not those stack. As for the last question... There is a statement somewhere that says regardless of your threat range, you only automatically hit on a natural 20 (which can be changed with variant rules). IIRC, a natural 1 is an automatic miss, and thus... would miss.

EDIT: Ninja'd!

Tyndmyr

2009-11-25, 11:59 AM

The spell Dolorous Blade doubles crit range, and causes crit threats to autoconfirm. Now, it doesn't stack with everything(keen being a listed exception), but if you have a decent crit range already, it does make crits vastly more likely.

Sleepingbear

2009-11-25, 12:01 PM

I'm afraid I cannot poke any holes in your logic, but only because I haven't actually investiaged whether or not those stack. As for the last question... There is a statement somewhere that says regardless of your threat range, you only automatically hit on a natural 20 (which can be changed with variant rules). IIRC, a natural 1 is an automatic miss, and thus... would miss.

EDIT: Ninja'd!

Sweet! I'm the worlds first Ninja Bear!

Ostien

2009-11-25, 12:02 PM

An immediate flaw I see, RAWwise you don't provoke a critical every time you roll in a critical threat. You have to roll a critical threat and hit in order for it to provoke. So say you roll a 3 and that is in your critical threat range but a 3 + attack does not hit. Then you just miss.

Edit: Only a natural 20 is an auto-hit that gets around this, so that even if your 20+attack does not hit the AC numbers wise you still hit. You'll probably miss on the confirm but still hit. Though you could roll two 20's in a row.

lvl 1 sharnian

2009-11-25, 12:09 PM

Not actually important to hit, as Lightning Maces triggers an extra attack everytime it threatens a critical so what I'm trying to do is cause an infinite amount of attacks which should include an infinite amount of natural 20s so that unless they have enough DR to overcome a possible max damage critical, they're dead.

Blood in the Water stance is also nice

Tyndmyr

2009-11-25, 12:15 PM

You do have to hit to threaten, though, right?

Now, you can optimize to-hit, but 1s will always break the chain unless you use a method such as the rogue option(feat, I believe) to treat a roll of a 1 as a natural 20. Or use rerolls granted by skill tricks, etc.

lvl 1 sharnian

2009-11-25, 12:18 PM

Ahh, forgot that important part, but if you can't hit on a natural one, you can get rid of some of those barbarian levels and use Dextrious Attack from Arcane Duelist with a +5 Rapier for +10 to hit, or just punk them out with Wraithstrike as all the extra attacks are still in the same round.

Any way to get enough divine ranks to not fail on a natural one?

ericgrau

2009-11-25, 12:19 PM

There's got to be an ability somewhere that lets you reroll 1's, that'd get this going.

drengnikrafe

2009-11-25, 12:21 PM

There's got to be an ability somewhere that lets you reroll 1's, that'd get this going.

The world's first, and only, use for luck feats. How nice. :smallbiggrin:

lvl 1 sharnian

2009-11-25, 12:22 PM

Well, to be infinite I'd need that, but at the level where I can finish off the build, I'd have atleast 11 BAB, so at 3 attacks, with a 95% chance of threatening, that would have some insane amount of attacks. I need to find a way to add in the Two Weapon Fighting chain.

RagnaroksChosen

2009-11-25, 12:34 PM

After reading this thread I wonder what the biggest crit range is using only 3.5 raw only, no 3.0 stuff. Best i could do was 13-20.

which was psi-warrior/psychic weapon master.

Smeggedoff

2009-11-25, 01:08 PM

Forgive my lack of knowledge but what the heck is an Aptitude Rapier?

lvl 1 sharnian

2009-11-25, 01:11 PM

Aptitude is a weapon enhancement from Tome of Battle that lets you use that weapon as the required weapon for other feats such as shadow blade or in this case, Lightning Maces to get near infinite attacks

Eldariel

2009-11-25, 01:20 PM

That's the broken reading of Aptitude, mind. It's speculated that its real purpose is allowing using your Weapon Focus-line and such with another weapon type, but then people decided to combine it with Weapon Style-feats and do something busted. Like Lightning Maces which is balanced by the base weapon being 20/x2.

Also, just pump up To Hit and the Crit won't be a problem. As a Barbarian, that shouldn't be hard. Apply Wraithstrike as desired. Blood in the Water should quickly erase any miss chance.

Heliomance

2009-11-25, 03:08 PM

In an attempt to try and find the highest crit range I can get, I came up with this:

An Aptitude Rapier has 15-20 crit range

Actually, I'm pretty sure it has an 18-20 crit range.

Street Fighter Barbarian 7 gives +1 to Threat Range on a charge or flat-footed target = 14-20

I don't know the Street Fighter variant, so I can't comment on this.

Disciple of Dispater 8, gives x3 to crit range, which ends up as 9-20 -1 for Barb7 = 8-20 Crit Range

Alright so far, though I would point out that the Disciple of Dispater abilities only work with iron or steel weapons.

Throw in Arcane Duelist adds in a false keen, = 6-20 -1 for Barb7 = 5-20

Again, I don't know the Arcane Duelist, so I can't comment. I've never seen it in similar crit-focused builds though - double check the wording to se if it stacks.

Add OA Weapon Master for -2 = 3-20 Crit Range

Assuming the OA Weapon Master is the same as the Sword and Fist Weapon master, this is valid.

Throw the Psychic Weapon Master for -2 again = 1-20 Crit Range

Psychic Weapon master only works with crystal weapons. Disciple of Dispater only works with iron or steel weapons. You can't get both. The largest crit range I've sen the CO boards come up with is 9-20, though I can't remember how. Incidentally, you forgot the simple Improved Critical.

Combined with Lightning Maces for Infinite Attacks.

Using beneficial stacking rules, is there a way to make this legal? Probably impossible pre-epic (I know, there are worse things in Epic than critical hits) , but if playing Gestalt, that might be possible pre-epic with Barbarian//PsychicWarrior.

Also, if this works, can you threaten a critical hit on a natural 1?

I don't believe so.

Kudaku

2009-11-25, 03:24 PM

Don't mean to throw a spanner in your plans here, and I don't have access to my books atm so this is purely based on my google-fu, but:

Doesn't Lightning Maces require you to use two Light Maces for it to work?

Edited for spelling.

Curmudgeon

2009-11-25, 03:29 PM

Doesn't Lightning Maces require you to use a Light Mace for it to work?

The trick here is the Aptitude weapon special ability from Tome of Battle, which lets you apply weapon feat benefits for some other type of weapon to that Aptitude weapon.

Eldariel

2009-11-25, 04:09 PM

Actually, I'm pretty sure it has an 18-20 crit range.

It's Keen/Imp. Crit.

I don't know the Street Fighter variant, so I can't comment on this.

See here (http://www.wizards.com/default.asp?x=dnd/we/20070228a).

Again, I don't know the Arcane Duelist, so I can't comment. I've never seen it in similar crit-focused builds though - double check the wording to se if it stacks.

See here (http://www.wizards.com/default.asp?x=dnd/re/20030224a).

Psychic Weapon master only works with crystal weapons. Disciple of Dispater only works with iron or steel weapons. You can't get both. The largest crit range I've sen the CO boards come up with is 9-20, though I can't remember how. Incidentally, you forgot the simple Improved Critical.

Wouldn't Aptitude get past this? And Improved Crit was already included in the original weapon. Also, I recall 3-20 crit range on Char Ops.

Curmudgeon

2009-11-25, 04:55 PM

Arcane Duelist's False Keen specifically does not stack with either Improved Critical or keen.

Keshay

2009-11-25, 04:55 PM

Originally Posted by Kudaku

Doesn't Lightning Maces require you to use a Light Mace for it to work?

The trick here is the Aptitude weapon special ability from Tome of Battle, which lets you apply weapon feat benefits for some other type of weapon to that Aptitude weapon.

Even using an Aptitiude weapon, would you not still have to be dual wielding? I'm not sure what the actual verbatim on the mod is...

Curmudgeon

2009-11-25, 04:58 PM

Even using an Aptitiude weapon, would you not still have to be dual wielding? I'm not sure what the actual verbatim on the mod is...

Yes, you would. You'd need two Aptitude weapons. They just don't have to be light maces.

lvl 1 sharnian

2009-11-25, 05:11 PM

That puts a damper on things, right now without a way of getting past the crystal weapon and the Arcane Duelist, that's 6-20 on a charge or flat-footed.

term1nally s1ck

2009-11-25, 05:13 PM

If you're using Aptitude weapons, Use Roundabout kick.

it's a feat in Complete Warrior that gives you a free unarmed strike each time you confirm a critical. But you have aptitude, so that's another Rapier attack...meaning any confirmed crits actually give you 2 extra attacks, and you can actually go truly infinite.

Of course you want the Lion totem Barbarian to get a full attack on the charge to make sure you go infinite.

I think with Improved critical, and DoD, you get to 3-20 anyway? or is it 9-20? Never sure how to stack that kind of multiplier...I think since it explicitly says you double it, then triple it, you get to 3-20. If I'm wrong, then 9-20 is still pretty good...but unless you're hitting on a 7, you won't go infinite.

If, however, you can use a spell like Dolorous Blow, or Critical Strike (and if it stacks, not sure), you could get it infinite quite easily. (all you need is the ability to hit and threaten a critical on an 8, one more point is all...but more is always good)

Signmaker

2009-11-25, 05:21 PM

If you're using Aptitude weapons, Use Roundabout kick.

it's a feat in Complete Warrior that gives you a free unarmed strike each time you confirm a critical. But you have aptitude, so that's another Rapier attack...meaning any confirmed crits actually give you 2 extra attacks, and you can actually go truly infinite.

NI!=Infinite

Of course you want the Lion totem Barbarian to get a full attack on the charge to make sure you go infinite.

I think with Improved critical, and DoD, you get to 3-20 anyway? or is it 9-20? Never sure how to stack that kind of multiplier...I think since it explicitly says you double it, then triple it, you get to 3-20. If I'm wrong, then 9-20 is still pretty good...but unless you're hitting on a 7, you won't go infinite.

As above.

If, however, you can use a spell like Dolorous Blow, or Critical Strike (and if it stacks, not sure), you could get it infinite quite easily. (all you need is the ability to hit and threaten a critical on an 8, one more point is all...but more is always good)

As above.

Once a large enough string of ones is hit, the attack routine stops. You need to be able to circumvent ones.

term1nally s1ck

2009-11-25, 05:48 PM

NI!=Infinite

As above.

As above.

Once a large enough string of ones is hit, the attack routine stops. You need to be able to circumvent ones.

If I find a monster with sufficient damage reduction, I think I can make infinitely many attacks quite easily.

Well, not if you use roundabout kick. If every attack you make spawns 1.1 attacks on average, then even if some give you none, you'll usually end up with more than you had before, because it's actually more likely for you to get two attacks off any given attack than it is for you to miss. If I crit and confirm on 6 or higher, then there's only a 1/4 chance of missing, while there's a greater than half chance of confirming a crit, giving you two extra attacks. Yes, some attacks get nothing, but more attacks actually INCREASE the number of attacks you have to use.

I just had a bad bad bad idea. Isn't there an ability that lets you take 10 on an attack? If we can work that into the build, it IS infinite if you hit on a 10.

Signmaker

2009-11-25, 05:52 PM

Well, not if you use roundabout kick. If every attack you make spawns 1.1 attacks on average, then even if some give you none, you'll usually end up with more than you had before, because it's actually more likely for you to get two attacks off any given attack than it is for you to miss. If I crit and confirm on 6 or higher, then there's only a 1/4 chance of missing, while there's a greater than half chance of confirming a crit, giving you two extra attacks. Yes, some attacks get nothing, but more attacks actually INCREASE the number of attacks you have to use.

You're misapplying averages to predict long-term relationships. A positive attack increase does NOT guarantee an infinite string unless you can remove the miss chance from a natural one.

Zincorium

2009-11-25, 05:54 PM

Psychic weapon master and disciple of dispater can't ever stack by the books- one requires a crystal weapon, and the other an iron/steel one.

You're also using multiple 3.0 sources where stacking of critical hit effects from different sources was allowed, and it is not the case in 3.5. So like DR 30/+1, it's effectively errata'd to not function the way it did. You can ignore it for theoretical optimization purposes, but I think this has been done before.

Also, if 3.0 stuff is allowed, I believe that bladed gauntlets exist which have a base crit range of 17-20. Double due to Improved Critical, add disciple of Dispater which stacks explicitly with imp. critical to make it x4, and use OA weapon master to add +2 for a RAW legal total of 3-20. I don't think you can stack anything with that, and it's very fragile considering the 3.0 stuff.

term1nally s1ck

2009-11-25, 06:30 PM

You're misapplying averages to predict long-term relationships. A positive attack increase does NOT guarantee an infinite string unless you can remove the miss chance from a natural one.

No, it doesn't guarantee it, but you have an good chance of it happening. In the case I mentioned earlier (hit and crit on a 6), the chance of an infinite string of attacks is 5/9, off one attack. If you can get a second attack at the same stats (Haste comes to mind, among others), then you have a 65/81 chance of going infinite, and even if you don't go infinite, the likelihood is that you make a lot of attacks anyway.

I'm wondering how the splitting property works with this....Could we find a way to get a Great Crossbow to work? 18-20 still, but there's the whole reloading thing...fairly sure there's a way around that though...and the splitting would make every critical threat give you two more attacks. (Since I have no memory, do you roll two attacks, or just one?)

BLOOD IN THE WATER.

It stacks, it keeps building up as you crit, and you can charge it up with a bag of rats or something....(note that this build can do the same thing as the great cleave sweeping strike cheese, if you drop a bag of rats out where you can hit them)

If you get to crit on a 7(easy in this build), and hit on a 2 (crit lots of rats), then there's a 60/133 chance of going infinite. Nearly half, and achievable easily.

SparkMandriller

2009-11-25, 06:32 PM

Will you please learn what the word infinite means?

term1nally s1ck

2009-11-25, 06:40 PM

Infinity (symbolically represented by ∞) is a concept in mathematics and philosophy that refers to a quantity without bound or end.

The attacks have a chance of having no end.

OracleofWuffing

2009-11-25, 06:47 PM

Psychic weapon master and disciple of dispater can't ever stack by the books- one requires a crystal weapon, and the other an iron/steel one.

Well... Iron crystals do exist in catgirl-killing land... I'd imagine such a weapon wouldn't last that long, though.

Signmaker

2009-11-25, 06:50 PM

Infinity (symbolically represented by ∞) is a concept in mathematics and philosophy that refers to a quantity without bound or end.

The attacks have a chance of having no end.

Put it this way:

For any given interval of attacks, there is a probability associated with it of the attack stream ending right then and there. Maybe you have it absurdedly low.

As the limit tends to an infinite number of attacks, the probability of a long enough stream of '1's tends to 1. Which means that it is, in fact, NOT an infinite amount of attacks. Just an arbitrarily high amount. Therein lies the difference between finite and infinite.

Claudius Maximus

2009-11-25, 07:00 PM

Also, if 3.0 stuff is allowed, I believe that bladed gauntlets exist which have a base crit range of 17-20.

An Erratum changed this to 19-20.

term1nally s1ck

2009-11-25, 07:28 PM

Put it this way:

For any given interval of attacks, there is a probability associated with it of the attack stream ending right then and there. Maybe you have it absurdedly low.

As the limit tends to an infinite number of attacks, the probability of a long enough stream of '1's tends to 1. Which means that it is, in fact, NOT an infinite amount of attacks. Just an arbitrarily high amount. Therein lies the difference between finite and infinite.

There's a chance that I hit with exactly 0 attacks.

There's a chance that I hit with exactly 1 attack.

There's a chance that I hit with exactly 2 attacks.

There's a chance that I hit with exactly 5 attacks.

There's a chance that I hit with exactly 34676456 attacks.

If you add all those chances together for every number that exists, you get a number which is LESS than 1, for the best builds. Meaning there is a positive chance for me to attack a number of times that is not any writeable number, and exceeds any number you care to name. What do you call that, except an infinite number of attacks?

Signmaker

2009-11-25, 07:41 PM

There's a chance that I hit with exactly 0 attacks.

There's a chance that I hit with exactly 1 attack.

There's a chance that I hit with exactly 2 attacks.

There's a chance that I hit with exactly 5 attacks.

There's a chance that I hit with exactly 34676456 attacks.

If you add all those chances together for every number that exists, you get a number which is LESS than 1, for the best builds. Meaning there is a positive chance for me to attack a number of times that is not any writeable number, and exceeds any number you care to name. What do you call that, except an infinite number of attacks?

Partially incorrect. For the natural numbers, the sum of the probabilities adds to one if there exists a chance of failure (fumbling), and zero if there does not exist a chance of failure (removing the ability to fumble, while recieving more attacks = an attack stream that doesn't stop).

SparkMandriller

2009-11-25, 07:43 PM

A very high number, but still not infinite. If there's even the smallest chance that your attacks will stop at some point, then your attacks will stop at some point. Guaranteed.

Fiery Diamond

2009-11-25, 08:02 PM

Remember, for example, that 1/3 = .33333333333333333333333333333333333...,

2/3 = .66666666666666666666666666666666666666...,

and 3/3 = .99999999999999999999999999999999999999999..., which means that since 3/3 = 1, .99999999999999999999999999999999999999999... = 1. This is why if there is a chance of failure the probability of failure will reach 1 as the number of times reaches infinity - because math is confusing like that.

quiet1mi

2009-11-25, 08:13 PM

all that work... then the Dm throws nothing but crit immune zombies.... this would not be funny except that it is true that this would happen... :smallfrown:

Zeful

2009-11-25, 08:15 PM

What do you call that, except an infinite number of attacks?

Arbitrarily high. There are legitimate inifinite loops in D&D, but since the game deals with real numbers, the loop has to be non-terminating. Pun-pun, the greatest rules abuse in the game, does not have infinite ability scores.

Even if you have only a 5% (20/100=5) chance of any attack not generating another attack, eventually you will run out of attacks because 1/20 attacks will fail statistically. Eventually a long string of 1s will occur, whether this is part of the third iteration or after Graham's Number (http://en.wikipedia.org/wiki/Graham%27s_number) of iterations is irrelevant. Statistically you will roll 1 enough times to end the sequence, eventually. Therefore it's not infinite.

Eldariel

2009-11-25, 08:17 PM

all that work... then the Dm throws nothing but crit immune zombies.... this would not be funny except that it is true that this would happen... :smallfrown:

With insane numbers of attacks (that doesn't depend on opps being Crittable), it doesn't really matter what opponents the DM throws.

Saintheart

2009-11-25, 08:53 PM

With insane numbers of attacks (that doesn't depend on opps being Crittable), it doesn't really matter what opponents the DM throws.

It does if the DM gives your enemy appallingly high Concealment chances, á la the way a lot of optimised wizards create their armor. I'd understood (and someone can correct me if I've got it wrong) that Concealment chance cuts across everything bar a natural 20, including critical hit damage. Assuming the insane number of attacks depends on a continual chain of guaranteed hits, I would've thought most decent Concealment applications have a 40% chance of shutting down this build.

term1nally s1ck

2009-11-25, 08:54 PM

I'm going to avoid typing out the full calculation I used, because it's hellish. If you want to continue arguing, I'll type it up, but tbh, you do not want to try and wrap you're brain around it. I'm in my second year studying maths at Cambridge uni, and it took a lot of tricks I've learned and a coupla hours of calculation. (Yes, I've seen the build before, it's on WotC optimisation board, and there's one on this board too...both use TWF kukris though.)

Here's a slightly simpler way of explaining it.

Every time you make an attack, if there's a 3/4 chance of critting and hitting, (hit+crit on a 6) you (on average) spawn 21/16 attacks.

Then, when you make your 21/16 attacks, you multiply the expected number of attacks by 21/16 to get the new number.

The expected number of attacks on your nth 'round' of attacks is (21/16)^n

And the chance of all of them missing is (1/4)^((21/16)^n)

We have to multiply by the probability that the solution hasn't failed yet

And then the sum of those is equal to just under 1/2, nowhere near 1.

The full solution relies on no less than 4 partial sums and 1 sum to infinity. You REALLY don't want to see this. I can provide it if you wish, but I suggest you don't.

EDIT: @Zeful,

The build I'm considering not only continues to generate attacks, but generates MORE attacks as time goes by. It becomes less and less probable for the sequence to end faster than we gain in chances due to sheer numbers of attacks.

SartheKobold

2009-11-25, 09:04 PM

With insane numbers of attacks (that doesn't depend on opps being Crittable), it doesn't really matter what opponents the DM throws.

He's got a good point. You can still Crit against things immune to Critical Hits/Percision Damage. You still Crit, and can still confirm that it Crits, it just doesn't take the extra damage...

I seem to recall, probabbly from 3.0, of the psionic powers Embed Weapon which would let you graft two weapons onto your hands. Then manifest Duodimensional Hands to cheese the Crit-Range of whatever you're using... Forgive me if it goes pear-shaped, but I remember it working as such. No idea how it stacks...

quiet1mi

2009-11-25, 09:09 PM

So the build is defeated by a smokestick?

term1nally s1ck

2009-11-25, 09:22 PM

Yeah, concealment tends to shut down the build, but it'll still get a fair few atttacks off against anything with concealment.

Interesting question....if you crit against a mirror image (as the spell), would you get an extra attack?

And since nobody answered, I'll ask again, do you roll once or twice to attack with a weapon with the splitting property?

Jack Mann

2009-11-25, 09:28 PM

There's actually another reason it doesn't give you truly infinite attacks. Let's say you average 15 points of damage per hit (not real high, but you've sunk your resources into an absurdly large crit range). You're fighting a Cauchemar. The Cauchemar has 172 hit points. After around twelve hits, you're probably going to stop.

Unless you want to keep beating a dead horse, of course.

sofawall

2009-11-25, 09:30 PM

Wouldn't Aptitude get past this?

No, actually. Aptitude is feats, and those are classes.

I know it was a page back, but many folk don't seem to know that. Just wanted to get it out there.

term1nally s1ck

2009-11-25, 09:40 PM

There's actually another reason it doesn't give you truly infinite attacks. Let's say you average 15 points of damage per hit (not real high, but you've sunk your resources into an absurdly large crit range). You're fighting a Cauchemar. The Cauchemar has 172 hit points. After around twelve hits, you're probably going to stop.

Unless you want to keep beating a dead horse, of course.

Well, I can think of a couple of scenarios it would be effective.

You have a weapon which deals negative levels somehow and want to buff an undead to ridiculous levels. (you need neg energy damage, or the undead to have DR high enough to not take much damage from normal attacks.)

The Tarrasque. This build can literally put the Tarrasque down, to the extent that it won't get back up for BILLIONS of years.

Can you crit when attacking an object? If so, this build can break the earth with enough time :smalltongue:

sofawall

2009-11-25, 09:44 PM

Well, I can think of a couple of scenarios it would be effective.

You have a weapon which deals negative levels somehow and want to buff an undead to ridiculous levels. (you need neg energy damage, or the undead to have DR high enough to not take much damage from normal attacks.)

The Tarrasque. This build can literally put the Tarrasque down, to the extent that it won't get back up for BILLIONS of years.

Can you crit when attacking an object? If so, this build can break the earth with enough time :smalltongue:

Negative levels don't do anything to undead. Enervation just so happens to add temporary hitpoints.

ex cathedra

2009-11-25, 09:45 PM

It does if the DM gives your enemy appallingly high Concealment chances, á la the way a lot of optimised wizards create their armor. I'd understood (and someone can correct me if I've got it wrong) that Concealment chance cuts across everything bar a natural 20, including critical hit damage. Assuming the insane number of attacks depends on a continual chain of guaranteed hits, I would've thought most decent Concealment applications have a 40% chance of shutting down this build.

I'm fairly sure that concealment negates natural 20s. Additionally, I'm not sure if concealment stops you from rolling a critical threat. This build relies on rolling crit threats, not hitting or critting. It may be entirely possible to effectively ignore concealment, if it's only negating 40% of your nigh-infinite number of attacks.

sofawall

2009-11-25, 09:50 PM

I'm fairly sure that concealment negates natural 20s. Additionally, I'm not sure if concealment stops you from rolling a critical threat. This build relies on rolling crit threats, not hitting or critting. It may be entirely possible to effectively ignore concealment, if it's only negating 40% of your nigh-infinite number of attacks.

It still makes it far more likely to end abruptly.

Roderick_BR

2009-11-25, 09:51 PM

The paladin spell Bless Weapon makes all threats become auto-crits, but doesn't work with weapons that already have some critical enhancing ability.

But I think that one of the abilities of the Fist of Raziel PrC is to gain this auto confirm against evil creatures when using a smite. Unlike the Bless Weapon spell, it does stack with others critical-enhancing abilities. So, if you can afford a level of paladin and a level of Fist of Raziel, you could at least have an auto crit against some creatures once or twice a day.

edit:

Can you crit when attacking an object? If so, this build can break the earth with enough time :smalltongue:

No, because the earth (or whatever planet) is an object, therefore immune to critical hits. There are better ways to destroy the world in D&D, though...:smallwink:

term1nally s1ck

2009-11-25, 10:16 PM

Just being immune to crits isn't enough, they still threaten and confirm, just don't do extra damage, so they do proc extra attacks. Personally, I think destroying the planet as a standard action at mid-teens ( I think....we don't need too much warblade to get the stance we want and the feats...how much weaponmaster do we need?) in level seems pretty good tbh.

I don't think our weapons have any crit enhancing ability....Improved critical, DoD and weaponmaster levels are all that are used.

@sofawall: Concealment does indeed cause a problem, but if you have enough + to hits from blood on the water, and if you have a good crit threat range, the build can still go infinite vs concealment. As usual, though, true seeing would be very nice to negate some of that miss chance.

I'm sad I can't have the mega-undead...it looked so fun :smalltongue: but guess I forgot to check that one properly, I just went by memory, since enervation is the only time I usually come across neg levels in most games.

ex cathedra

2009-11-25, 10:21 PM

Just being immune to crits isn't enough, they still threaten and confirm, just don't do extra damage, so they do proc extra attacks. Personally, I think destroying the planet as a standard action at mid-teens ( I think....we don't need too much warblade to get the stance we want and the feats...how much weaponmaster do we need?) in level seems pretty good tbh.

Strongheart Halfling Crusaders with feycraft daggers and a reroll device do it at eleven, if memory serves.

sofawall

2009-11-25, 10:27 PM

@sofawall: Concealment does indeed cause a problem, but if you have enough + to hits from blood on the water, and if you have a good crit threat range, the build can still go infinite vs concealment. As usual, though, true seeing would be very nice to negate some of that miss chance.

This build cannot go infinite even with no concealment, unless I;m missing something.

Is there a way to get infinite rerolls or ways to treat nat 1s as something else, even if just for one round?

Also, why do you even bring up to-hit in a discussion about concealment? +10065 to-hit against -10 AC will still miss 50% of the time if the person with -10 AC is blinking.

Lord of Syntax

2009-11-25, 10:39 PM

There is a feat from one of the completes that lets rogues treat nat 1s as nat 20s :smallamused:

sofawall

2009-11-25, 10:40 PM

There is a feat from one of the completes that lets rogues treat nat 1s as nat 20s :smallamused:

Is that Better Lucky Than Good? Because I don't think that works.

term1nally s1ck

2009-11-25, 10:41 PM

Because if I only hit on 11s, and you're blinking, I only hit 1/4 of the time.

And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.

@aethernox is that the d2 trick? I love that trick, it's just funny.

Halfling: I poke him with my tiny dagger (basically a letter opener

DM: The impact completely destroys him, and nothing is left.

sofawall

2009-11-25, 10:46 PM

Because if I only hit on 11s, and you're blinking, I only hit 1/4 of the time.

And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.

@aethernox is that the d2 trick? I love that trick, it's just funny.

Halfling: I poke him with my tiny dagger (basically a letter opener

DM: The impact completely destroys him, and nothing is left.

You do not understand math. EDIT:Or at least that math.

As people have said, if there is any chance, any chance of not going infinite, if there is any chance of rolling enough natural ones to break the chain, as you get closer and closer to infinity, the chance of you failing to hit infinity approaches one. Therefore, as you hit infinity, you screw up and miss, thus resulting in a non-infinite result.

Yay high-school math.

SparkMandriller

2009-11-25, 10:55 PM

Watch out dude, he's gonna use his Cambridge maths powers on you. They take hours to power up, they must be hugely destructive. Like Goku.

AirGuitarGod32

2009-11-25, 10:55 PM

Has anyone thought of using a Disciple of Dispater with an Iron Aptitude Kopesh?

ghashxx

2009-11-25, 10:55 PM

Because if I only hit on 11s, and you're blinking, I only hit 1/4 of the time.

And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.

@aethernox is that the d2 trick? I love that trick, it's just funny.

Halfling: I poke him with my tiny dagger (basically a letter opener

DM: The impact completely destroys him, and nothing is left.

Unless there's something terribly going wrong here in the real world then you're eventually going to roll a one, missing completely and not even threatening to crit, thereby ending the chain. If there is even the ittiest bittiest tiniest chance of ever failing then it's not infinite. What you're claiming is like saying that every time I successfully throw a basketball through a hoop that my chances of succeeding next time increases exponentially. But so long as I ever at any point in time eternal have a chance of not hitting, then eventually I will miss meaning that my chain of hits will end.

ex cathedra

2009-11-25, 10:56 PM

I think the part that people are having trouble grasping is that, as the chain goes on, it progressively becomes more likely to generate attacks to what seems like an infinite amount. However, any amount that is infinite must go through an infinite set of numbers. This includes natural 1s. This loop is not literally infinite, but it's near enough as to not matter.

SparkMandriller

2009-11-25, 10:57 PM

I don't think that anyone's arguing that. They're just arguing that it's not infinite. Because it's not.

ex cathedra

2009-11-25, 10:57 PM

@aethernox is that the d2 trick? I love that trick, it's just funny.

Halfling: I poke him with my tiny dagger (basically a letter opener

DM: The impact completely destroys him, and nothing is left.

Indeed it is. :smallsmile:

Any 1d2 weapon + Aura of Chaos + something like Imbued Healing: Luck.

So the build is defeated by a smokestick?

Or DR equal to your weapons max critical damage.

Ping, pah-ping-ping, pingity-ping!

ex cathedra

2009-11-25, 11:09 PM

Or DR equal to your weapons max critical damage.

Ping, pah-ping-ping, pingity-ping!

Blood in the Water has already been mentioned; it grants you a cumulative +1 attack and damage each time you crit. So your weapon's base damage quickly becomes moot.

lvl 1 sharnian

2009-11-25, 11:10 PM

Doesn't Dragon Ascendant from Draconomicon give 1 divine rank?

Just play a Dragonwrought Kobold and at epic levels, they'll be able to get infinite

Zeful

2009-11-25, 11:14 PM

Indeed it is. :smallsmile:

Any 1d2 weapon + Aura of Chaos + something like Imbued Healing: Luck.

Does Imbued Healing: Luck work on attack rolls or just damage rolls*? Cause if it's possible to get at least a 2-20 critical, then it's possible to go infinite by simply ignoring 1s.

*I'm pretty sure it works only on damage rolls, but it's nice to be sure.

Jack Mann

2009-11-25, 11:18 PM

@aethernox is that the d2 trick? I love that trick, it's just funny.

Halfling: I poke him with my tiny dagger (basically a letter opener

DM: The impact completely destroys him, and nothing is left.

Essentially, you can now make a tonberry in D&D.

"DOINK!"

ex cathedra

2009-11-25, 11:19 PM

I do not believe so, no.

term1nally s1ck

2009-11-25, 11:21 PM

Aether has it.

The basic reason the build tends to go infinite is that it generates attacks faster than basic probability can generate nat 1s (or merely non-crits) to stop the trick. The chances that the build stops are actually higher EARLIER in the attack chain, just because by the 9th-10th attack, you're likely to have 3-4 attacks to use, and as you get further and further through the attack sequence, you'll have more and more attacks.

As an example, lets use the simpler crits and hits on 6+ to have a look.

If I want to get 10 nat 1s in a row, the expected number of rolls I need to make are 20^10, or 1.02400 * 10^13. Meanwhile, after that many attacks, I expect to have 4.16737342 * 10^23 attacks....which is so far above 10 I can't even explain that. Nearly all the chances for the attack loop to stop happen early on....there's a 1/4 chance of the first attack missing, but only about a .2 chance of it stopping ever if the first attack crits, more than halving the miss chance.

Of course, having thought about it, I found a very pretty and simple way to calculate the probability of a given build going infinite, but it requires you to know that it's infinite in the first place. I rigorously proved this by looking at the chances of failure, but you seem to not want to accept this.

How about this one.

The infinite sum 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ........ is precisely equal to 1. It is a geometric series with a constant factor of 1/2.

The series of 'miss chances' I calculated earlier ((1/4)^((21/16)^n))

starts at 0 with 1/4. When n increases by 1, you multiply the 'miss chance' by (1/4)^(21/16), which is smaller than 1/2.

The miss chances at each step then, not only start smaller than the first infinite sum, term-by-term, but get smaller faster. The infinite sum of the 'miss chances', therefore, is AT LEAST 1/4 below 1. (The first two terms are 1/4 apart)

I'm taking miss chances here to mean if I use every attack I have currently generated, and miss with every single one.

This means that there is a finite chance that the number of attacks I make is NOT A FINITE NUMBER. Non-finite is pretty much the definition of infinity mathematically.

EDIT: Oh, and Sofawall...High school math teaches you precisely NOTHING about dealing with infinities. I couldn't do it til I took Further Maths at A-Level, and even then I was only looking at very specific cases, and without any real understanding. Yes the amount of numbers and chains of nat 1s I go through gets arbitrarily high as you approach infinity, but the number of attacks I make gets MUCH MUCH higher, much much faster, so I can weather them easily.

Here's another trick that might help....If I get to the point where it's likely for me to roll n 1s in a row, it's also equally likely I roll the same number of nat 20s in a row, which would completely cancel the 1s out.

sofawall

2009-11-26, 12:13 AM

Infinity does not work like normal numbers, though...

If something can happen, in an infinite series of probabilities, it will. Is it possible to roll enough ones to end the chain? Yes. It will, therefore, happen.

True, it becomes less and less likely it will happen as time goes on. However, all that does is make the chain more and more likely to be really really mind-bogglingly large before it ends.

Watch out dude, he's gonna use his Cambridge maths powers on you. They take hours to power up, they must be hugely destructive. Like Goku.

I bow to Paul Muad'dib.

term1nally s1ck

2009-11-26, 12:20 AM

No, it doesn't, which is why it's so hard to deal with. In fact, you cannot consider what happens at infinity, but what happens as things approach infinity. In my case, I've proven that through all the numbers on the number line, there is only a finite chance of me attacking a certain number of times.

I'm going to ask you a question now. If I add up the probability of attacking n times, for all n that exist on the number line (the infinite sum), I get 4/9 for the total. I'd like you to tell me now, if the build will always finish it's attacking at some point, how many attacks do I make for the remaining 5/9? Every natural number from 1 to 45762342454687564645, (And 0), and every number higher still is included in the 4/9, so your answer cannot be any of those, so how many attacks am I making?

lvl 1 sharnian

2009-11-26, 01:57 AM

Theorectical optimization turns to theorectical mathmatics? I'm dreadin college now

tcrudisi

2009-11-26, 02:36 AM

Theorectical optimization turns to theorectical mathmatics? I'm dreadin college now

Meh, just avoid Calculus and Advanced Mythology and you should be fine.

Heliomance

2009-11-26, 03:17 AM

In simple terms, the reason why this has a chance to go infinite is that it's not a fixed number of natural 1s needed to end the chain. As the number of attacks you make tends to infinity, the number of consecutive natural 1s you need to roll to end it also tends to infinity. Yes, on your way to infinity you will roll every conceivable number of consecutive natural 1s at some point, but the number you need to roll is ever increasing, and so there's a pretty good chance that however many you roll in a row, at the point at which you roll it it won't be enough.

Fortuna

2009-11-26, 03:31 AM

Hmmmm. If I'm right (and I am never right), then you need to be able to show a conceptual point of some kind (not neccesarily a point on the number line, just a point) at which the odds of your string of natural ones are infinitesimal. Is that correct? If you can show such a point, then doesn't the whole thing end at infinity (specifically, at aleph-0) when such a ratio loses meaning and the two are equal? And, assuming that we ignore the gross stupidity that this post has no doubt shown so far, does that not mean that it is a terminating sequence which nevertheless sums to infinity?

Arcane Duelist's False Keen specifically does not stack with either Improved Critical or keen.

Are we reading the same Arcane Duelist?

False Keenness (Ex): To create the idea that she is more effective than she really is, the arcane duelist can make her chosen melee weapon keen once per day per arcane duelist level. To activate the keen effect, she must subtract the weapon's normal threat range for critical hits from her attack bonus. For example, if the arcane duelist wields a rapier, she must subtract 3 from her attack bonus, since the rapier has a threat range of 18-20, to make it keen. The keen effect lasts for 1 round per arcane duelist level, and it does not stack with the keen edge spell or other magical effects that make a weapon keen. The ability does stack with the increased threat range granted by the Improved Critical feat and by a keen magic weapon. If the arcane duelist uses this power in conjunction with the dexterous attack power, the damage dealt by a critical hit is reduced by the weapon's critical multiplier. Thus, the arcane duelist wielding a rapier and using dexterous attack to reduce the damage by 4 on her rapier has it reduced by 12 if she scores a critical hit using this power. The weapon must still do a minimum 1 point of damage.

Emphasis added. It doesn't stack with keen edge, but it DOES stack with Improved Crit and an actual Keen weapon. Confusing, but that's WotC for you.

lvl 1 sharnian

2009-11-26, 11:26 AM

Thank you my friend, I believe it's now 3-20 Crit Range for either Psionic or regular version

Curmudgeon

2009-11-26, 11:48 AM

Are we reading the same Arcane Duelist?

Yes, we're reading the same class description. What you're not reading are

the date of the Arcane Duelist (02/24/2003)

the Primary Sources errata rule

Errata Rule: Primary Sources

When you find a disagreement between two D&D® rules sources, unless an official errata file says otherwise, the primary source is correct. One example of a primary/secondary source is text taking precedence over a table entry. An individual spell description takes precedence when the short description in the beginning of the spells chapter disagrees.

Another example of primary vs. secondary sources involves book and topic precedence. The Player's Handbook, for example, gives all the rules for playing the game, for playing PC races, and for using base class descriptions. If you find something on one of those topics from the Dungeon Master's Guide or the Monster Manual that disagrees with the Player's Handbook, you should assume the Player's Handbook is the primary source. The Dungeon Master's Guide is the primary source for topics such as magic item descriptions, special material construction rules, and so on. The Monster Manual is the primary source for monster descriptions, templates, and supernatural, extraordinary, and spell-like abilities.

The primary source for the Improved Critical feat is the 3.5 Player's Handbook, which came out in 2003 July, after the Arcane Duelist (a 3.0 prestige class).

This effect doesn’t stack with any other effect that expands the threat range of a weapon. The primary source for the keen weapon special ability is the 3.5 Dungeon Master's Guide, which also came out in 2003 July.

This benefit doesn’t stack with any other effect that expands the threat range of a weapon (such as the keen edge spell or the Improved Critical feat). WotC's official rule is that, since both of these disagree with the Arcane Duelist's False Keenness, and they're from primary sources, they're both right. The Arcane Duelist's False Keen specifically does not stack with either Improved Critical or keen.

Eldariel

2009-11-26, 11:57 AM

Except specific trumps general, Arcane Duelist specifically stacks with both so it works. Improved Critical and Keen have just general line "stacks with nothing", so the "except"-abilities function just fine.

It doesn't matter that the said limitations were added in the 3.5 revision, no errata has been applied to Arcane Duelist so its ability still functions as written and thus trumps the limitation on Imp. Crit and Keen (even though there was none by the time of the printing of the class).

Emmerask

2009-11-26, 12:37 PM

Not actually important to hit, as Lightning Maces triggers an extra attack everytime it threatens a critical

Sorry but you are wrong there

from the srd:

Critical Hits

Sometimes your threat range is greater than 20. That is, you can score a threat on a lower number. In such cases, a roll of lower than 20 is not an automatic hit. Any attack roll that doesn’t result in a hit is not a threat

So you have to actually hit to threaten a critical... it is important to hit ;)

lvl 1 sharnian

2009-11-26, 12:37 PM

Don't even bring in Disciple of Dispater or Weapon Master or Psychic Weapon or Streetfighter Barbarian or anything like that (all 3.0) which all state that they stack with improved critical

Curmudgeon

2009-11-26, 12:39 PM

Except specific trumps general ...

It doesn't matter that the said limitations were added in the 3.5 revision

The game is D&D, version 3.5. Wizards of the Coast has set up rules for how material that differs from the 3.5 primary sources is handled. Page 4 of the 3.5 Dungeon Master's Guide says:

This is an upgrade of the d20 System, not a new edition of the game. This revision is compatible with all existing products, and those products can be used with the revision with only minor adjustments. The core rules say that there will need to be minor adjustments to incorporate 3.0 material. The primary sources errata rule specifies how to make those minor adjustments in the case where there's a disagreement.

This isn't a case of there being a general 3.5 rule and a specific exception to that rule. This is a case where the general rules used to be different and then were changed. (That is, Improved Critical and keen used to stack in 3.0.) Most 3.0 content doesn't have specific update information, so you follow the required 3.5 rules for how to make the necessary adjustments to that older material.

Eldariel

2009-11-26, 12:48 PM

The game is D&D, version 3.5. Wizards of the Coast has set up rules for how material that differs from the 3.5 primary sources is handled. Page 4 of the 3.5 Dungeon Master's Guide says: The core rules say that there will need to be minor adjustments to incorporate 3.0 material. The primary sources errata rule specifies how to make those minor adjustments in the case where there's a disagreement.

This isn't a case of there being a general 3.5 rule and a specific exception to that rule. This is a case where the general rules used to be different and then were changed. (That is, Improved Critical and keen used to stack in 3.0.) Most 3.0 content doesn't have specific update information, so you follow the required 3.5 rules for how to make the necessary adjustments to that older material.

Indeed, but since the only necessary adjustments are changing obsolete skills and DR system, Arcane Duelist remains untouched without an update or errata. The critical feats themselves changed, but that system is still identical to the 3.0 system and as such, 3.0 classes utilizing the system work as they used to. Same goes for Weapon Master, Disciple of Dispater and so on. They all have specific exceptions (now with the 3.5 rules) that enables them to function.

Curmudgeon

2009-11-26, 01:05 PM

Indeed, but since the only necessary adjustments are changing obsolete skills and DR system

That argument doesn't hold water. It's also necessary to change a bunch of other things to work with 3.5 rules. One example is 3.0 weapon sizes, where the rules are quite different in 3.5. Another example is that things that increase threat range don't stack any more. Those minor adjustment are necessary, too.

The Random NPC

2009-11-26, 01:10 PM

Do not argue RAW with Curmudgeon, he is always right.

Eldariel

2009-11-26, 01:17 PM

That argument doesn't hold water. It's also necessary to change a bunch of other things to work with 3.5 rules. One example is 3.0 weapon sizes, where the rules are quite different in 3.5. Another example is that things that increase threat range don't stack any more. Those minor adjustment are necessary, too.

I don't see how you can argue that an ability which functions perfectly fine as written under 3.5 rules needs to be spontaneously adjusted when playing 3.5. There is no precedent for that and it's not a procedure mentioned anywhere. If anything, it's houseruling.

Talbot

2009-11-26, 01:35 PM

With all the arguing about infinity, I lost track of what the build actually is right now... can someone post both the 'theoretical' build (using the 3.0 stuff) and the 'RAW legal' build (3.5 only stuff) for the sake of reference?

Curmudgeon

2009-11-26, 01:38 PM

I don't see how you can argue that an ability which functions perfectly fine as written under 3.5 rules

I've already pointed out the specific 3.5 rule (primary sources) which dictates that the 3.5 Improved Critical and keen restrictions win their disagreements with the Arcane Duelist's class ability.

If anything, it's houseruling.

Please, let's try to keep this civil. That accusation is baseless, as I've referenced the official WotC rules supporting my position. Your attempt to claim material which requires update -- strictly according to the 3.5 rules -- instead "functions perfectly fine" without following those rules, is the actual house rule.

Heliomance

2009-11-26, 01:43 PM

Most of the time when Curmudgeon weighs in on an argument, his points are unpalatable but correct. In this instance, though, I believe him to be wrong. I think the "specific trumps general" rule is in effect here, and the "This stacks with Improved Critical" line in the various classes is more specific than the "This doesn't stack with anything" line in Improved Critical.

Eldariel

2009-11-26, 01:47 PM

I've already pointed out the specific 3.5 rule (primary sources) which dictates that the 3.5 Improved Critical and keen restrictions win their disagreements with the Arcane Duelist's class ability.

Please, let's try to keep this civil. That accusation is baseless, as I've referenced the official WotC rules supporting my position. Your attempt to claim material which requires update -- strictly according to the 3.5 rules -- instead "functions perfectly fine" without following those rules, is the actual house rule.

None of what you quoted has any implication on this matter though. We aren't talking about a printing of the Improved Critical feat, which did change. We aren't talking about a printing of the Keen enhancement, which did change. For those, primary source is indeed 3.5 PHB. And yes, they changed. But that class does not use Improved Critical. It merely refers it, and very specifically says that they stack.

Regardless of changes to Improved Critical, that class feature's wording remains the same and so they stack. "Minor revisions" is far too nebulous to actually draw any RAW changes upon. Indeed, the only changes you can argue must be made in 3.5 upgrade are changes to things that no longer exist; given there's no guideline saying critical effects must be updated to not stack or such, they remain the same.

You have not quoted anything in the contrary. The primary source-rules determine what happens in the case of a clash, which doesn't exist here, and the "minor revisions" is merely a guideline to DMs on how to use the compatible products and doesn't include any actual guidelines for said revisions.

term1nally s1ck

2009-11-26, 03:05 PM

Never mind, just read the above post.

Curmudgeon

2009-11-26, 03:29 PM

The primary source-rules determine what happens in the case of a clash, which doesn't exist here

How is there not a clash? False Keen says that it stacks with Improved Critical and keen, and both Improved Critical and keen say they don't "stack with any other effect that expands the threat range of a weapon". There's a prima facie disagreement. Thus we follow the primary source rule and there's no stacking.

False Keen can't be a case of specific trumps general rules under 3.5 (actually an exception to specific rules, to use the Rules Compendium terminology), because the 3.5 specific rules about critical threat ranges weren't yet written when False Keen was created. It's just a case of a rule disagreement with the primary sources. The primary sources win all such disagreements.

sofawall

2009-11-26, 03:51 PM

How is there not a clash? False Keen says that it stacks with Improved Critical and keen, and both Improved Critical and keen say they don't "stack with any other effect that expands the threat range of a weapon". There's a prima facie disagreement. Thus we follow the primary source rule and there's no stacking.

False Keen can't be a case of specific trumps general rules under 3.5 (actually an exception to specific rules, to use the Rules Compendium terminology), because the 3.5 specific rules about critical threat ranges weren't yet written when False Keen was created. It's just a case of a rule disagreement with the primary sources. The primary sources win all such disagreements.

So when is it primary vs. other and when is it specific vs. general?

For example, sundering normally cannot be done with a bow. Complete Warrior has a feat that lets you do that. Players Handbook is the Primary Source on sundering, therefore the feat does nothing?

Also, would you mind linking me or giving me a page cite to somewhere that says primary source rule beats general vs. specific rule?

Curmudgeon

2009-11-26, 04:29 PM

For example, sundering normally cannot be done with a bow. Complete Warrior has a feat that lets you do that. Players Handbook is the Primary Source on sundering, therefore the feat does nothing? Nope. That feat provides an exception to a specific rule. The Normal section of Ranged Sunder explains the specific rule, and the Benefit section of the feat describes the exception.

Also, would you mind linking me or giving me a page cite to somewhere that says primary source rule beats general vs. specific rule?

There isn't one. The primary source rule is absolute, so when there's a disagreement you follow the primary source. To not fall under the primary source rule you have to show that there is no disagreement in the rules, and that the conflicting rule is instead providing an exception to a specific rule -- as Ranged Sunder clearly does. It's not possible to make that claim for False Keen, because the specific 3.5 rule it disagrees with didn't exist yet.

Eldariel

2009-11-26, 05:27 PM

Class features and feats are written differently. Class features don't have a "benefits"-section at all; it's simply the class feature listed with all relevant considerations. Consider it equivalent to a feat that in "Benefits"-section lists the exception to this rule.

Curmudgeon

2009-11-26, 05:53 PM

Consider it equivalent to a feat that in "Benefits"-section lists the exception to this rule.

You can't create an exception to a rule that doesn't yet exist. Thus the primary source rule treats it as a disagreement, and False Keen loses.

Wings of Peace

2009-11-26, 06:17 PM

You can't create an exception to a rule that doesn't yet exist. Thus the primary source rule treats it as a disagreement, and False Keen loses.

Explain your logic with the primary source rule to me. I am not criticizing it I am just not understanding it. It seems you're arguing that it doesn't't count because the rule didn't exist yet but that doesn't seem relevant to whether or not the class feature overrides the rule that exists now if we look at things purely from their current forms and what is written.

Talbot

2009-11-26, 06:27 PM

Regardless of what was written when, D&D has always worked on a "specific trumps general" rule. If the creators didn't want it to, they could easily have either errata'd it, or put under the Keen/IC entries some sort of a "without exception" clause, but they did neither. Since the feature explicitly states that the features stack, they stack. Until something is written which specifically says they don't.

Curmudgeon

2009-11-26, 06:42 PM

Since the feature explicitly states that the features stack, they stack. Until something is written which specifically says they don't. That something has long since been written.

The critical threat range expanders stacked under 3.0 rules, which is when this prestige class was created. Then WotC changed the rules for 3.5, and said if there are disagreements, you obey the rules in the primary sources. It's the rules in those primary sources, which specifically state that such features don't stack, which we are required to follow.

Talbot

2009-11-26, 08:26 PM

That something has long since been written.

The critical threat range expanders stacked under 3.0 rules, which is when this prestige class was created. Then WotC changed the rules for 3.5, and said if there are disagreements, you obey the rules in the primary sources. It's the rules in those primary sources, which specifically state that such features don't stack, which we are required to follow.

No, what it says is "This effect doesn’t stack with any other effect that expands the threat range of a weapon (such as the keen edge spell)." That would be the general rule. Since the Arcane Duelist class explicitly points out that False Keen DOES stack with Improved Critical, which is a more specific statement, they DO stack until Errata or other says differently. It's not an issue of what was published most recently; if things were that simple, there'd be no reason for the "Specific vs. General" policy in the first place.

Of course, this is the sort of thing that's up to a specific DM. A DM who subscribes to the traditional "explicit trumps general" idea would allow it, a DM who's more beholden to the "sanctity" or sourcebooks, or who doesn't allow 3.0 material, wouldn't.

But, in the absence of a DM, I think the specific/general setup is a much better policy than core vs. splatbooks or 3.5 vs. 3.0 (except, obviously, in the cases where there are both 3.0 and 3.5 versions of the book/item in question).

Kelb_Panthera

2009-11-26, 10:04 PM

Just wanted to weigh in on the issue of a weapon having to be crystal for the psychic weapons master, or steel for the disciple of dispater..... crysteel weapon? It's eberron specific, but it's effectively both steel and crystal.

Jastermereel

2009-11-27, 09:27 AM

In simple terms, the reason why this has a chance to go infinite is that it's not a fixed number of natural 1s needed to end the chain. As the number of attacks you make tends to infinity, the number of consecutive natural 1s you need to roll to end it also tends to infinity. Yes, on your way to infinity you will roll every conceivable number of consecutive natural 1s at some point, but the number you need to roll is ever increasing, and so there's a pretty good chance that however many you roll in a row, at the point at which you roll it it won't be enough.

In the wise words of Yoda, "This is why you fail."

Well that's part of it anyway.

Infinity cares not for your "there's a pretty good chance." It's infinite. It doesn't matter if you need to hit a run of bad luck so long that the metaphor would break down. If you're going to infinity and beyond you'll find it somewhere between here and there. That's the problem with mixing probability and infinity. If something can happen along the way that will mess up the "nigh infinite damage", even if it's almost impossible for it to happen, until that probability of failure reaches absolute zero (not just really really really close) it will do just that.

Also, even if the math did work, it won't deal infinite damage for the same reason the 1d2 crusader won't. You can enter an endless (or not) loop of rolling, but that's limited by your fellow players patience.

In the end, the only damage done would be from your fellow players to you.

Even if it did work, which it doesn't.

Signmaker

2009-11-27, 11:47 AM

What he's trying to say is that certain infinities grow faster than other infinities. Because I'm too lazy to confirm, I can't tell if the growth of attacks outruns the loss of attacks, but yes, as time passes the odds of the stream ending actually decreases, rather than increases. The assumption being made by the "always" finite crew is a twist on the Gambler's Fallacy, in that it's believed that there will be a long enough stream of fumbles to counter the growth of attacks as you tend to infinity.

I'm reminded of a riddle which may or may not help (most certainly not, but hey). An impish pixie has a bucket, and outside the bucket an infinite number of balls (the natural number set, to be precise). At time 1, he puts in ball 1 and 2, takes out the lowest numbered ball. At time 2, balls 3 and 4 are added in, and the lowest (this time 2) is removed. He does this for an infinite number of time. At any finite amount of time, the number of balls in the bucket is the number of intervals progressed. However, the argument at infinity is that there are no balls in the bucket, because every natural number has been removed at its respective time interval. Infinities can do weird things, as this riddle implies.

What am I getting at? At any finite time, there is a distinct probability that the set of attacks will terminate. If we have a setup where you fumble on ones, generate 1 attack on 2-19(self-sustain), and generate, say, 3 attacks on 20, we have a decreasing probability at interval. However, it becomes hazy to determine what happens at 'infinity'. Is it infinite, by expected growth patterns? Or is it not? All we can hope for is that some math graduate student can shed some light, because I'm sure as hell not qualified.

sofawall

2009-11-27, 03:21 PM

That something has long since been written.

The critical threat range expanders stacked under 3.0 rules, which is when this prestige class was created. Then WotC changed the rules for 3.5, and said if there are disagreements, you obey the rules in the primary sources. It's the rules in those primary sources, which specifically state that such features don't stack, which we are required to follow.

But WotC also has a rule that says everything written in 3.o works as written except when what it references doesn't exist anymore (such as some skills and feats) or if there is an update. Since Imp. Crit. still exists, it was just changed, the 3.0 class has a reference point. Imp. Crit. provides a general rule (nothing stacks with it) and the class provides a specific exception (it stacks with Imp. Crit.).

So now what is the hierarchy of rules? Since the "Primary Source rule" (apparently) trumps the Specific vs. General rule, where does the 3.0=3.5 rule fall into this?

ocdscale

2009-11-27, 05:40 PM

If something can happen, in an infinite series of probabilities, it will. Is it possible to roll enough ones to end the chain? Yes. It will, therefore, happen.

Quite a few posters have repeated this assertion as proof that the proposed idea isn't capable of infinite attacks. But it seems they're misapplying (to put it gently).

Here's my attempt to explain it as plainly as possible.

An infinite series of rolls will result in every conceivable sequence of results. Eventually, this character will roll ten thousand 1s in a row, or a hundred thousand, or 10^100 or whatever.

However, you cannot simply assume that an arbitrarily high 1s in a row will stop the attack string because the build might generate additional attacks on non-1 rolls (or whatever the range is) to buffer against the misses. So you need the math (as other posters have said)

I'm going to make up numbers just to prove a point (and because I'm not familiar enough to use the actual numbers) Lets suppose the build is such that it misses on a 1. On anything else, it grants you two extra attacks. and it starts with one attack.

What is the probability that this attack sequence ends on the first iteration? Easy, 5%

What is the probability that this attack sequence ends on the second iteration? Still pretty easy. You have two attacks on the second iteration, and both have to miss. 0.25% (Note, at this step you never have 0 attacks, because that probability is accounted for in the previous step: failing on the first iteration)

What is the probability that this attack sequence ends on the third iteration?

A bit more complicated because there is some probability you have four attacks and some probability you have 2. Approximately 0.0243%

Now, what is the probability you fail sometime before the fourth step? Easy, you simply add the above probabilities. There's about a 5.25243% chance of not surviving to make the fourth iteration.

Someone with a decent grasp of mathematics will note that the failure summation is always going to be less than 6% (or even 5.2525% for that matter). The probability this sequence fails before the 10^100th step? Less than 6%. The probability this sequence fails before the 10^100^100 step? Less than 6%. The probability this sequence generates an infinite series? Over 94%. (If someone pipes up with "But over an infinite period of time, eventually all failure the probabilities you add will add up to more than 6%," I'm going to get an aneurysm.)

Can you still say "Well, in a long enough series of attacks, eventually you're going to roll enough 1s to end the series." Of course you can say that, and look, it's true approximately 5.25% of the time. The rest of the time this particular sequence generates an infinite amount of attacks.

Jastermereel

2009-11-27, 07:37 PM

In short, if you flip a coin you CAN get an infinite number of heads in a row and there's a clear possibility that you will do just that.

The build allows for an infinite number of attacks, but the 1d2 crusader delivers them.

Sure, you'll probably kill a plane's worth of Tarrasque, but, then again, you might not. It's a cute loop, but might not start right and it's not a sound follow through. If the RAW crew approves (I think the debate is on rules about what rules apply to other rules on rules (talk about infinite regressions!)) it'll be a neat new exploit, but darned if there aren't easier ways to get better results.

The Crit Panther!

60% of the time it works everytime!

:smalltongue:

Myrmex

2009-11-27, 07:39 PM

In short, if you flip a coin you CAN get an infinite number of heads in a row and there's a clear possibility that you will do just that.

The probability of that happening is zero, isn't it?

The limit of (1/2)^infinity = 0.

Jastermereel

2009-11-27, 07:55 PM

The probability of that happening is zero, isn't it?

The limit of (1/2)^infinity = 0.

On a long enough timeline the survival rate for every critical hit string reaches zero.

Well it approaches zero, but it could happen just that once against the Lich if I'm lucky. :smalltongue:

Mind, I don't mean to be extra snarky, I just enjoy arguing.

Myrmex

2009-11-27, 07:56 PM

On a long enough timeline the survival rate for every critical hit string reaches zero.

Well it approaches zero, but it could happen just that once against the Lich if I'm lucky. :smalltongue:

Mind, I don't mean to be extra snarky, I just enjoy arguing.

If the Lich has a truly infinite number of HP, then being able to an arbitrarily large amount of damage won't help.

ocdscale

2009-11-27, 08:07 PM

The probability of that happening is zero, isn't it?

The limit of (1/2)^infinity = 0.

Yes, in that coin example. That coin example is an example of what you could call a 'sustaining' infinite series. Meeting the requirement (getting heads) only lets you continue what you've been doing (flipping one coin). Any success rate under 100% is going to result in that series eventually ending.

The proposed build is more of what you'd call a 'multiplicative' infinite series. A success results in a quadratic (or is it exponential?) improvement in your future success rates, which mean that given enough success, the probability of failing approaches 0 because one failure is not enough, you need enough failures to make up for all the successes you've had up until then.

A good way to conceptualize the two is imagine rolling a d100.

The first: If you roll 2-100, continue rolling. Stop rolling on a 1.

The second: If you roll 2-100, grab another d100, continue rolling with all your dice. If a die comes up 1, stop rolling that die.

In the first, despite it having a 99% success rate, the probability of it reaching infinite rolls approaches 0.

In the second, the probability of it achieving infinite rolls is (approximating) near 98%.

term1nally s1ck

2009-11-27, 08:57 PM

People....people who can do Maths :smallbiggrin:

I was starting to worry that it was gonna be me trying to convince the whole Giantitp community that this worked.

Yeah, this stuff is hard to do, since it requres some odd prestige classes.

The main issue is that we need 15 levels in the two prestige classes, so we HAVE to enter at 5. DoD needs BAB+6, so is out, so we need to fulfil Wep master by LV5. This needs: 4 ranks in intimidate, Dodge, Mobility, Combat Reflexes, Expertise, weapon Focus, Whirlwind attack. Whirlwind attack needs Spring Attack.

Be Human, take 2 flaws, and 4 levels in fighter and 1 in another full BAB, and all of the above can be done. You then Take Disciple of darkness and Power attack as your 6 and 9 feats, to qualify ffor DoD.

We then need to get Lightning mace, Imp critical, and Roundabout kick to make it work.

Lightning Mace needs TWF, and Roundabout Kick needs Improved Unarmed Strike....but we only have 3 more feats to get. Simple to solve, just take Barbarian 1 (City Brawler variant). It gives you TWF (for unarmed strikes only, but we don't need it to actuallly work, just be present (and Aptitude means later on you don't care), and Imp Unarmed strike.

The build is then:

Human (2 Flaws), 15+, 15+, X, 8+, X, X

1: Barb (City brawler) (Imp unarmed strike, TWF (Unarmed strike), Weapon Focus (light mace), Combat Reflexes, Dodge, Combat Expertise)

2: Fighter (Mobility)

3: Fighter (Spring Attack, Whirlwind Attack)

4: Fighter +1 to a stat.

5: Fighter (Free feat or possibly a replacement level of something worth dipping...Not sure what I'd use though. Free feat is useful, gives it another trick if needed. Power attack if you're playing it through all these levels, and use the later levels for Imp TWF or something like that.)

6-12 Weapon master (Sword and Fist) (Disciple of darkness, Power attack, Improved Critical)

13-20 Disciple of Dispater (The Book of Vile darkness) (Lightning Mace, Roundabout Kick.)

Achieved pre-epic, with some minor cheese...I could easily find a way to get Imp Unarmed or the real TWF as a 1-dip, I'm sure, since we have a feat spare. This is the only build that I think can achieve this by 20, and I couldn't get warblade in, unfortunately. if anyone can find me a way to get a level of warblade in, that'd be great. Alternatively, can I use Tiger Claw Bracers to get a maneuver for my character, qualifying him for the Martial Stance feat for Blood on the Water?

To recap, Human (2 Flaws) Barbarian (City Brawler) 1, Fighter 4, Weapon Master 7, Disciple of Dispater 8 can get this combo working.

Oh, and another proof that the average hits of one of these builds is infinite.

If the average hits was, for example, N, where N is an absolutely monstrous, finite number. Consider the probability of the sequence ending (as calculated above) before 10*N hits. You'll be left with about 73/133, as before. This then means that there's at least a 60/133 chance that the sequence generates more than 10*N attacks. This means that the average damage is at least (60/133)*10*N, or about (4 3/4)*N, which is a contradiction.

We have used perfect logic, not mentioning infinities at any point, and yet we have reached a contradiction. Using Reductio ad Absurdum, this means that we have made an incorrect assumption that the average no. of hits was a finite number. Since it is not finite, it must be infinite.

EDIT: If the Bracers+Feat trick works, take Claw at the Moon as your maneuver. Synergy :D

term1nally s1ck

2009-11-27, 09:11 PM

In short, if you flip a coin you CAN get an infinite number of heads in a row and there's a clear possibility that you will do just that.

The build allows for an infinite number of attacks, but the 1d2 crusader delivers them.

Sure, you'll probably kill a plane's worth of Tarrasque, but, then again, you might not. It's a cute loop, but might not start right and it's not a sound follow through. If the RAW crew approves (I think the debate is on rules about what rules apply to other rules on rules (talk about infinite regressions!)) it'll be a neat new exploit, but darned if there aren't easier ways to get better results.

The Crit Panther!

60% of the time it works everytime!

:smalltongue:

Yes, the crusader is very much more powerful...but this one is more flavorful, and more importantly is playable by RAW. The d2 crusader will never stop rolling for damage, so cannot be used in a game (and is disallowed for this reason in most games). The Chuck Norris (Infinite Roundabout(Roundhouse) Kicks and Lightning Maces(Fists)) can stop killing something when it's dead, and move on to start hitting the next guy, meaning that provided your DM doesn't make some quick infinite sums, he'll never know that you're not just lucky with those crits, and it's a practical, if OP build to use in a game.

OracleofWuffing

2009-11-27, 09:28 PM

Yes, the crusader is very much more powerful...but this one is more flavorful, and more importantly is playable by RAW. The d2 crusader will never stop rolling for damage, so cannot be used in a game (and is disallowed for this reason in most games).

Actually, the wording for the d2 Crusader is that you "can" roll the dice, not that you must, so it's your choice when you want to stop.

Curmudgeon

2009-11-27, 09:46 PM

But WotC also has a rule that says everything written in 3.o works as written except when what it references doesn't exist anymore (such as some skills and feats) or if there is an update.

I've never heard of a rule like that. What I do know of is the following:

This is an upgrade of the d20 System, not a new edition of the game. This revision is compatible with all existing products, and those products can be used with the revision with only minor adjustments.

The Nature of the Core Rulebooks’ Changes

The purpose of this booklet is not to provide a comprehensive list of everything that has changed with the 3.5 revision. The changes are too large in number and varied in scope to be able to provide an all-inclusive inventory. Instead, what we want to do is to show you a broad picture of what has changed and to provide you with support for updating our most popular product.

...

Player’s Handbook

The changes throughout the book range from very minor to quite substantial.

...

Feats

Likewise, feats underwent revision. Many new feats entered the book. Standardized pairs of skills grant +2 bonuses from feats like Negotiator and Stealthy. Many old feats changed significantly.

...

Dungeon Master’s Guide

This book experienced a dramatic reorganization. The focus on the revision was on user-friendly and useful text. A great deal of new material was added. Other areas of the book underwent review and revision.

So, in summary:

The 3.5 DMG says minor adjustments will be necessary to use 3.0 material.

The 3.5 update booklet details some of those adjustments, including significantly changed feats in the PH and revisions in the DMG.

The Improved Critical feat and keen weapon special ability are included in those revisions.

The changes to Improved Critical and keen specifically disallow stacking any threat range expanders under 3.5 rules.

Jastermereel

2009-11-27, 10:11 PM

Yes, the crusader is very much more powerful...but this one is more flavorful, and more importantly is playable by RAW. The d2 crusader will never stop rolling for damage, so cannot be used in a game (and is disallowed for this reason in most games). The Chuck Norris (Infinite Roundabout(Roundhouse) Kicks and Lightning Maces(Fists)) can stop killing something when it's dead, and move on to start hitting the next guy, meaning that provided your DM doesn't make some quick infinite sums, he'll never know that you're not just lucky with those crits, and it's a practical, if OP build to use in a game.

At this level of cheese, claiming one is more flavorful is like arguing about who has a better back story for PunPun. We can argue about infinity, we can argue about RAW, but I don't think anyone is going to argue that this is a build any DM would want you to bring to a table.

Also, I think "Crit Panther" is a more practical name than "The Chuck Norris (Infinite Roundabout(Roundhouse) Kicks and Lightning Maces(Fists))" even at the cost of some dignity.

flabort

2009-11-28, 12:44 AM

Just found this, real intresting. I think I'll break this down page by page...

Psychic weapon master and disciple of dispater can't ever stack by the books- one requires a crystal weapon, and the other an iron/steel one.

I'm screaming inside: this is from page 1, and haven't gotten to page 2 yet.

he needs two weapons to use lightning mace, and two different types of weapons for these two feats, so using TWO aptitude rapiers made from each of the materials, or if they are a special material by being aptitude, counting as each of the materials due to APTITUDE rapier, would qualify for all three!

So the build is defeated by a smokestick?

Page 2, most interesting quote i found. it would appear to be true, by that point. interesting to see were the build is heading by this point. and also the conversation. concerning the conversation about ∞, I can hardly wrap my head around tesseracts and other 4 dimensional objects, or other n-dimensional objects, and related speres, let alone infinite and related spheres! although, i do know n∞=∞^n=∞/n=∞+n=∞-n=(∞+n)^n=∞... basicly, anything you do to ∞ results in ∞, even ∞/∞=∞. i also do know that 1+1/2+1/4+1/8+1/16... ect., halving the value each time, never quite reaches 2, but by the time it gets to the point were it might as well be, there's ∞ terms...

$%@#, could you imagine a ∞-dimensional object?

Interesting question....if you crit against a mirror image (as the spell), would you get an extra attack?

And since nobody answered, I'll ask again, do you roll once or twice to attack with a weapon with the splitting property?

ooh, cool. still page 2, I love mirror image abuse cheese. fill a room with yourself, ect.

Personally, i'd let the crit to the illusion count for another attack, so as not to tip the player off that it's an illusion. And, i've never heard of the splitting property. guess i'll find out later in the thread, maybe.

No, actually. Aptitude is feats, and those are classes.

I know it was a page back, but many folk don't seem to know that. Just wanted to get it out there.

Ooh, sucker punch to my comment on page 1. Ok, but if the weapons CAN be made of different materials still, that's still cool.

There is a feat from one of the completes that lets rogues treat nat 1s as nat 20s :smallamused:

Ok, page 2 is a GOLD MINE of good material. let's just pack that on....

reaching page three, i'm seeing some people are talking as if one miss will end the whole chain, and others talking about the attacks like a tree rather than a chain, and using different math. are we using permutations, combanations, or niether here? arg. i hate math 30 pre-probability.

Blood in the Water has already been mentioned; it grants you a cumulative +1 attack and damage each time you crit. So your weapon's base damage quickly becomes moot.

Finally a page 3 quote. so, that's what that does. and how are we packing so many feats into this build?! well, by the 27th hit, we're hitting +27 dammage, or more, due to this one feat, and are hitting on everything but a 1. hunng. and we're getting, like, 2.98337243294982749832734398273489x10^∞ hits.... or more. Mozza-swiss with a little cheddar, much?

Do not argue RAW with Curmudgeon, he is always right.

LOL. that is all from that quote.

-Arguing with Curmudgeon-

LOL. again.

Ok, so to recap, We're arguing about the chances and odds that this is indeed dealing infinite hits, i'm questioning trying to pass high school, we've got two Aptitude Rapiers with splitting property, one made of crystal and one of iron, quallifying for the Lightning mace feat, and the Psycic Weapon master and Disiple of dissasture classes, we've got some rouge with a feat so 1's act as 20's, we've got Blood in Water so no damage reduction can stand in our way, we've got some barbarian variant (street fighter?!), and a couple others, so that we hit with crits from 1-20, and as a result we kill if we're allowed one hit, and a simple smokestick foils all of our planning. and this is all pre-epic, and we can STILL kill a lvl. 100. harsh.

Limburger and monteray Jack indeed, even if I missed something!

term1nally s1ck

2009-11-28, 02:51 AM

Oh, I'm not using the Psychic weaponmaster in my build (Which I will call the Chuck Norris if I want...the brackets were to explain why..), jeez, I can barely fit in the levels I've got...

To be honest, this build has exactly the same issue as any ubercharger....if your DM throws a monster at you that you can't attack, or that has miss chances/huge AC, you're gonna struggle to get the hit off, even if that one hit would kill it. Discounting the AC due to ToB cheese, miss chances and anything that can keep out of melee range can't be killed...well, you COULD one-shot the earth, taking it down with it, but that's a last resort, surely :P

(although, the idea of having a quest to traverse a mountain range, as has happened to me before, and responding with 'I hit the mountain. The mountain dies.' would be most definitely follow the rule of cool.)

The whole miss chances thing was why I wanted Splitting...because then I could probably get seeking on a pile of arrows, too, and annihilate any miss chances. Splitting is an enhancement for a bow/crossbow, where it duplicates each arrow you fire. I just can't recall if you would roll the attacks seperately or just roll once,or roll once and count it twice...I think you'd get two chances to proc extra attacks with them, which would nearly guarantee that you kill whatever you attack. Plus, y'know....range...

I take no responsibility for anyone blowing all their WBL on arrows within the first 6 seconds of the first battle in a campaign. You could literally blot the sun out, provided you could carry enough arrows. It's like having a heavy machinegun to use.

Eldariel

2009-11-28, 03:02 AM

Oh, I'm not using the Psychic weaponmaster in my build (Which I will call the Chuck Norris if I want...the brackets were to explain why..), jeez, I can barely fit in the levels I've got...

Since the name Chuck (http://community.wizards.com/go/thread/view/75882/19861942/Chuck_E_Cheese?post_id=338123498#338123498) is already taken, you may wish to reconsider that lest you want a roundhouse in the face Tornado Throw to the other side of the continent.

Fortuna

2009-11-28, 03:27 AM

Sorry to those who have been trying to explain advanced mathematics to everyone, I am just curious about the little things (I once had an interesting discussion about philosophers, thought for an hour, and then pointed out very succinctly that Bishop Berkely was a moron and why). At any rate, I would like a clarification of this please.

We have a system defined below. This is meant to be a simplified version which should be infinite if the system which we are working with is infinite (note that I have probably used more axioms than neccessary).

Axiom 1: A system is entirely defined by these axioms and two numbers p and m.

Axiom 2: A system begins containing one instance.

Axiom 3: A system's population is equal to the cardinality of the set of instances in the system.

Axiom 4: A system develops in discrete iterations.

Axiom 5: At any iteration, there is a probability that the system will fail equal to 1/p^population.

Axiom 6: If a system does not fail at a given iteration, its population is multiplied by m.

Axiom 7: If a system fails, no further iterations occur.

Hypothesis: That a system with finite m and 0<p<1 will never satisfy population=aleph-0.

Conceptual Proof: At an iteration i, the chance that the system will fail is equal to 1/p^population, and population=m^(i-1). The probability that the system fails is therefore equal to 1/p^(m^(i-1)).

The probability that a system exists at iteration i+1 is equal to the product of (1-1/p^(m^(x-1))) for all integer values of x from 1 to i inclusive. Let i equal an arbitrarily high finite number. Then the probability that the system exists is the product of an arbitrary number of values less than 1 but greater than zero. Therefore as i increases, the product of (1-1/p^(m^(x-1))) for x from 0 to i is strictly decreasing.

Now, this is as far as I can get. From here, i believe that the clincher is the limit of the product of (1-1/p^(m^(x-1))) for x from 0 to i as i increases without bound. Please show me how you calculate this limit, or at least outline your method, unless of course I have stuffed up my reasoning in which case please point out how.

term1nally s1ck

2009-11-28, 04:13 AM

You won't be able to make this proof, I'm afraid.

if M = (say) 2, and p = 10

The product is equal to:

1 - SUM(over x)(1/p^(m^(x-1))) + SUM (over X and Y)((1/p^(m^(x-1)))(1/p^(m^(y-1)))

- .......

But that second sum is equal to (SUM(over x)(1/p^(m^(x-1))))^2, and the same is true as you go down the line, making the entire series equal to 1-S+(S^2)-(S^3).....which, since S is trivially provable to be of absolute value less than 1, converges to a finite value. In this case, i can't tell you what that value is, because I can't write down a nice number that's equal to that sum, but the final value is between 1 and 1-S (a bit under 0.9)

we know for certain that S<0.2 though, so [1, 1-S] does not include 0

we can get what the sum is by a nice trick though.

Let P be the Probability that the sequence starting with population = 1 ends.

Looking at the first generation, we get that P = 1/10 (extinct after 1 try) + (9/10)*(P^2) (pop = 2 now). Solving quadratically, we get that P = 1 or 1/9

we have already established above that (1-P) =/= 0, so P =/= 1, therefore the probability in this case is 8/9 that it reaches aleph-0

Bad luck, you had a tougher job there. I had to only find an exception, you wanted to prove a general rule.

If I want to now find the set of values that can lead to infinities, I need to find the values of m and p that give me S between -1 and 1.

Zincorium

2009-11-28, 06:35 AM

I'm screaming inside: this is from page 1, and haven't gotten to page 2 yet.

he needs two weapons to use lightning mace, and two different types of weapons for these two feats, so using TWO aptitude rapiers made from each of the materials, or if they are a special material by being aptitude, counting as each of the materials due to APTITUDE rapier, would qualify for all three!

Fail. Specifically, failure to read the Aptitude property. Actually, failure to even skim the description in the tome of battle.

Aptitude counts for feats, and only feats. It's not even a bit misleading- feats are all that are ever mentioned. Class features that are not bonus feats are not feats at all. So it still *must* be made primarily of iron for the class feature of the Disciple of Dispater to increase it's crit range. Aptitude does not change this. The psychic weapon master requires a *standard* crystal weapon- one without any special property- in order to use the class ability gained. Aptitude, again, does not change this.

Sorry for the rant, but calling someone out without even bothering to research it is MY schtick. Don't be a copycat.

(actually, I feel really guilty when I do it, but that rarely prevents me from flying off the handle the next time)

ocdscale

2009-11-28, 01:06 PM

The probability that a system exists at iteration i+1 is equal to the product of (1-1/p^(m^(x-1))) for all integer values of x from 1 to i inclusive. Let i equal an arbitrarily high finite number. Then the probability that the system exists is the product of an arbitrary number of values less than 1 but greater than zero. Therefore as i increases, the product of (1-1/p^(m^(x-1))) for x from 0 to i is strictly decreasing.

Forgive me if I misunderstand your post, and forgive me if this post is somewhat superfluous given the terminally sick's calculations above.

I wanted to just explain why the section in bold doesn't quite support your hypothesis.

The fact that the probability of existence inexorably decreases does not necessarily mean the probability of existence converges to 0, only that it converges to some point.

As a simple proof of concept, imagine a system that has a probability of existence of:

0.75/(t+1) + 0.25

Where t is the time in rounds. Clearly the probability of existence decreases over time, but equally clear that it has a healthy 25% chance of existence even as t approaches infinity.

Fortuna

2009-11-28, 03:08 PM

I am aware of this. As stated below that, that is where my reasoning breaks down. What I am asking for is a simple demonstration of the limit to infinity of that product. I notice that there have been many mentions of sums to infinity in this thread, but I am fairly sure that this is a case for products to infinity, and I have little to no experience with those.

lvl 1 sharnian

2009-11-28, 04:51 PM

Fail. Specifically, failure to read the Aptitude property. Actually, failure to even skim the description in the tome of battle.

Aptitude counts for feats, and only feats. It's not even a bit misleading- feats are all that are ever mentioned. Class features that are not bonus feats are not feats at all. So it still *must* be made primarily of iron for the class feature of the Disciple of Dispater to increase it's crit range. Aptitude does not change this. The psychic weapon master requires a *standard* crystal weapon- one without any special property- in order to use the class ability gained. Aptitude, again, does not change this.

Sorry for the rant, but calling someone out without even bothering to research it is MY schtick. Don't be a copycat.

(actually, I feel really guilty when I do it, but that rarely prevents me from flying off the handle the next time)

Well, the main point of his post wasn't that Aptitude would bypass the restriction, just an addition. His main point was that one could wield a Crystal Aptitude Rapier in one hand and a Steel Aptitude Rapier in the other so as to gain the benefit of both.

However, that wouldn't be worth it, as it would cause differing threat ranges that don't apply.

Someone mentioned a crysteel material from Ebberon that functioned as both steel and crystal. Source?

Edit: I can't recall, does nonlethal damage do anything besides knock you out? Because if not, just Non-lethal damage to keep it infinite. Also, if you coup-de-grace, then you automatically hit, no need for a roll, would that also apply for the extra hits?

term1nally s1ck

2009-11-28, 05:10 PM

I am aware of this. As stated below that, that is where my reasoning breaks down. What I am asking for is a simple demonstration of the limit to infinity of that product. I notice that there have been many mentions of sums to infinity in this thread, but I am fairly sure that this is a case for products to infinity, and I have little to no experience with those.

Take a look at my most recent post, I found that product in terms of a sum of sums, and proved that the limit of the product is finite (and in a certain case, 8/9) Your product is of a set of functions with a fairly difficult set of conditions...In general when you're looking at products, you re-arrange it into sums of sums, or take logs. I think logs would work here, but I'm not sure...

@Sharnian

If you don't roll, I don't think you can crit.

sofawall

2009-11-28, 05:13 PM

Well, the main point of his post wasn't that Aptitude would bypass the restriction, just an addition. His main point was that one could wield a Crystal Aptitude Rapier in one hand and a Steel Aptitude Rapier in the other so as to gain the benefit of both.

However, that wouldn't be worth it, as it would cause differing threat ranges that don't apply.

Someone mentioned a crysteel material from Ebberon that functioned as both steel and crystal. Source?

Edit: I can't recall, does nonlethal damage do anything besides knock you out? Because if not, just Non-lethal damage to keep it infinite. Also, if you coup-de-grace, then you automatically hit, no need for a roll, would that also apply for the extra hits?

I think it needs to be iron :P

term1nally s1ck

2009-11-28, 05:21 PM

But if you get Crysteel, you can give it the Metalline property to convert the metal part to cold iron.

OracleofWuffing

2009-11-28, 05:28 PM

@Sharnian

If you don't roll, I don't think you can crit.

Coup de grace scores a crit without rolling.

Coup de Grace

As a full-round action, you can use a melee weapon to deliver a coup de grace to a helpless opponent. You can also use a bow or crossbow, provided you are adjacent to the target.

You automatically hit and score a critical hit. If the defender survives the damage, he must make a Fortitude save (DC 10 + damage dealt) or die. A rogue also gets her extra sneak attack damage against a helpless opponent when delivering a coup de grace.

term1nally s1ck

2009-11-28, 05:50 PM

Hmmmmmm I think that works for Roundabout Kick, but I'm not sure about lightning mace..you have to 'roll a threat on an attack roll' which is not happening imo.

Roundabout kick is whenever you hit with a critical though, so does work.

Fortuna

2009-11-28, 05:56 PM

Ah well. It is always good to try to do something that you've been told is impossible, if only so that you convince yourself of that. Thank you very much for that clarification, and my apologies for wasting your time.

Kalirren

2009-11-28, 06:20 PM

Sorry to those who have been trying to explain advanced mathematics to everyone, I am just curious about the little things (I once had an interesting discussion about philosophers, thought for an hour, and then pointed out very succinctly that Bishop Berkely was a moron and why). At any rate, I would like a clarification of this please.

We have a system defined below. This is meant to be a simplified version which should be infinite if the system which we are working with is infinite (note that I have probably used more axioms than neccessary).

Axiom 1: A system is entirely defined by these axioms and two numbers p and m.

Axiom 2: A system begins containing one instance.

Axiom 3: A system's population is equal to the cardinality of the set of instances in the system.

Axiom 4: A system develops in discrete iterations.

Axiom 5: At any iteration, there is a probability that the system will fail equal to 1/p^population.

Axiom 6: If a system does not fail at a given iteration, its population is multiplied by m.

Axiom 7: If a system fails, no further iterations occur.

Hypothesis: That a system with finite m and 0<p<1 will never satisfy population=aleph-0.

Conceptual Proof: At an iteration i, the chance that the system will fail is equal to p^population, and population=m^(i-1). The probability that the system fails is therefore equal to p^(m^(i-1)).

The probability that a system exists at iteration i+1 is equal to the product of (1-p^(m^(x-1))) for all integer values of x from 1 to i inclusive. Let i equal an arbitrarily high finite number. Then the probability that the system exists is the product of an arbitrary number of values less than 1 but greater than zero. Therefore as i increases, the product of (1-p^(m^(x-1))) for x from 1 to i is strictly decreasing.

Now, this is as far as I can get. From here, i believe that the clincher is the limit of the product of (1-p^(m^(x))) for x from 0 to i as i increases without bound. Please show me how you calculate this limit, or at least outline your method, unless of course I have stuffed up my reasoning in which case please point out how.

Corrected a few errors.

Well, you're looking for limit to infinity of P(x) = PRODUCT_i [1-p^(m^i)] from i = 0 to infinity. Consider ln P(x)= SUM_i [ln(1-p^(m^i))]. Since m>1 and 0<p<1, p^(m^i) is very small for large i, and the approximation

ln(1+u) ~ u - 1/2 u^2 + O(x^3) < u

is valid. Thus SUM_i [ln(1-p^(m^i))] converges iff SUM_i [-(p^(m^i))] converges, by direct comparison. (The negative sign on that last actually doesn't matter, so we'll consider SUM_i [p^(m^i)] instead.)

But since i is very large and m>1, m*i < m^i, which means that SUM_i [p^(m^i)] < SUM_i [p^(m*i)] (since 0<p<1). But SUM_i [p^mi] is just the sum of a geometric series with r=p^m, which is finite since 0<p<1 and m>0, so SUM_i [p^mi] converges. Thus SUM_i [ln(1-p^(m^i))] also converges to a finite negative number by direct comparison, which means that our original PRODUCT_i [1-p^(m^i)] must also converge to a finite positive number greater than 0.

term1nally s1ck

2009-11-28, 07:54 PM

^^Yeah, that's muuuch easier. Should have done logs to start, but wasn't sure it'd sum nicely, using the approximation is very clever (though you really should use inequalities for proofs like this, and the < means it still works)

lvl 1 sharnian

2009-11-28, 10:33 PM

I think it needs to be iron :P

Not exactly...

Iron Power (Ex): When using an iron or steel weapon, a

4th-level disciple of Dispater gains a +1 insight bonus on

attack and damage rolls. Furthermore, his threat range is

doubled as if he were using a keen weapon. At 8th level, the

insight bonus improves to +2, and the threat range triples.

This ability does not stack with the keen weapon quality, but

it does stack with the Improved Critical feat.

SensFan

2009-11-28, 10:46 PM

Because if I only hit on 11s, and you're blinking, I only hit 1/4 of the time.

And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.

You seem to not understand the concept of infinite.

Does your build keep attacking even if you roll one hundred billion 1s in a row? If not, you won't go infinite.

In fact, if you roll long enough, you will have n rolls, and will then roll 9999n 1s in a row.

Haven't read past this post, so if you learnt what infinite means in the meantime, great.

Kalirren

2009-11-28, 10:56 PM

^^Yeah, that's muuuch easier. Should have done logs to start, but wasn't sure it'd sum nicely, using the approximation is very clever (though you really should use inequalities for proofs like this, and the < means it still works)

Yeah, you're right, I should have made the error of the approximation more explicit...as you said, the next-higher order term in the approximation is

ln(1+u) = u - 1/2 u^2 + O(u^3) < u

so it happens to work. I was relying on the fact that as u-> 0, it's really just u +/- O(u^2), but I guess that's not really sound, is it?

sofawall

2009-11-28, 11:11 PM

Not exactly...

Probably only allows steel because it contains iron... Stupid Dispater, not sticking to his guns...

Kalirren

2009-11-28, 11:12 PM

You seem to not understand the concept of infinite.

Does your build keep attacking even if you roll one hundred billion 1s in a row? If not, you won't go infinite.

In fact, if you roll long enough, you will have n rolls, and will then roll 9999n 1s in a row.

Haven't read past this post, so if you learnt what infinite means in the meantime, great.

On the contrary, -you- seem not to understand the concept of infinite. Specifically. you're confusing "finite" with "non-deterministically infinite".

As it stands, the sequence of partial sums representing the probability that the attack sequence terminates converges to a number less than 1. This build therefore has a finite, nonzero probability of an attack sequence not terminating. It has a -chance- of going infinite in the sense that it does not terminate. It -also- has a chance of terminating. This is thus "non-deterministically infinite", and the expected value of the number of attacks you make is undefined.

Suppose that a revised version of this build had a luck feat or something attached to it that allowed the rerolling of natural 1s, and one was attacking on a 2 to hit. Then the probability that the attack sequence does not terminate is 1. This is -also- infinite: it differs from the first case in that it is deterministically infinite.

OracleofWuffing

2009-11-28, 11:29 PM

Not exactly...

Hm... Any way we can abuse the wording on this, given that the text says you need to be "using" the weapon, as opposed to "wielding" or "attacking"? I guess you could set up a rube machine mouse trap back at your home that uses an iron or steel weapon, thus freeing up a hand...

"... Since I'm using the steel weapon as a lever back home, my crystal weapon's threat range is increased, and I'll just two-handed power attack for full..." :smalltongue:

(Except not power attacking, since we're already taking TWF anyways.)

*Gets Curmudgeoned*

AgentPaper

2009-11-29, 12:12 AM

On the contrary, -you- seem not to understand the concept of infinite. Specifically. you're confusing "finite" with "non-deterministically infinite".

As it stands, the sequence of partial sums representing the probability that the attack sequence terminates converges to a number less than 1. This build therefore has a finite, nonzero probability of an attack sequence not terminating. It has a -chance- of going infinite in the sense that it does not terminate. It -also- has a chance of terminating. This is thus "non-deterministically infinite", and the expected value of the number of attacks you make is undefined.

Suppose that a revised version of this build had a luck feat or something attached to it that allowed the rerolling of natural 1s, and one was attacking on a 2 to hit. Then the probability that the attack sequence does not terminate is 1. This is -also- infinite: it differs from the first case in that it is deterministically infinite.

I'm not sure what you're talking about here. To make sure we're talking the same case, we're talking about a situation where every time you make an attack, you have a 40% chance of getting two additional attacks, a 55% chance of getting one additional attack, and a 5% chance of getting no additional attacks. This essentially creates a "chain" of attacks, rolling a certain amount of attacks per step, adding an attack to the next step if you get two attacks, and removing an attack from the next step if you get no attacks.

So, first attack, you have a 5% chance of the chain ending, a 55% chance of continuing with no change, and a 40% chance of adding another attack to the chain. Now, let's say you got average results for the next few hundred rolls, and so for each step of the chain, you are now rolling 200 dice. The chance for each of those rolls to terminate is still 5%. The chance for ALL of them to terminate right now and end the chain is about 6.22x10^-261. That's ridiculously small, but it's still there, so it CAN happen.

Now continue another few hundred thousand steps in this chain. Let's say now you're rolling a few million dice. Each of those rolls STILL has a 5% chance of terminating, so there is STILL a chance that all of them will do so and the chain will suddenly end. That chance is even more ridiculously small, but it IS there, and no matter how high your numbers get, it will always be above 0. For such a chain to result in infinite damage, it would need to continue for an infinite amount of steps. And since each of those steps will always have some chance greater than 0 to stop right there, you have to multiply that chance by infinity. And since even 1x10^-googleplex multiplied by infinity is still infinity, you have a 100% chance of the chain terminating before you reach infinity.

Yes, you can and likely will reach a number so absurdly high that the number of atoms in the universe looks laughable next to it, but that's NOT infinity, and it never will be.

SilverSheriff

2009-11-29, 12:14 AM

Don't know if anyone else has covered this or if it is relevent to the discussion, but the 20th level fighter ability 'Weapon Mastery' in Pathfinder gives an auto-confirm on criticals.

Gpope

2009-11-29, 01:10 AM

That chance is even more ridiculously small, but it IS there, and no matter how high your numbers get, it will always be above 0. For such a chain to result in infinite damage, it would need to continue for an infinite amount of steps. And since each of those steps will always have some chance greater than 0 to stop right there, you have to multiply that chance by infinity. And since even 1x10^-googleplex multiplied by infinity is still infinity, you have a 100% chance of the chain terminating before you reach infinity.

The problem is that you're trying to multiply an arbitrarily small number by infinity, when the actual situation involves multiplying an infinitely small number by infinity. You're not rolling an infinite number of attempts at 1 in 10^(10^100) odds, you're rolling e.g. an attempt at 1 in 10^(10^101), then an attempt at 1 in 10^(10^102), then at 10^(10^103), and so on. Obviously that's not the exact math, but it illustrates the point.

Intuitively, you might say "So what? There's still always a chance to fail." But because that chance to fail is dropping at such a rapid rate, if you add up the chance to fail for all attacks from 1 to infinity, you end up with a total chance of failure that is less than 100%, which means there is a chance that the chain will truly never end.

AgentPaper

2009-11-29, 01:57 AM

The problem is that you're trying to multiply an arbitrarily small number by infinity, when the actual situation involves multiplying an infinitely small number by infinity. You're not rolling an infinite number of attempts at 1 in 10^(10^100) odds, you're rolling e.g. an attempt at 1 in 10^(10^101), then an attempt at 1 in 10^(10^102), then at 10^(10^103), and so on. Obviously that's not the exact math, but it illustrates the point.

Intuitively, you might say "So what? There's still always a chance to fail." But because that chance to fail is dropping at such a rapid rate, if you add up the chance to fail for all attacks from 1 to infinity, you end up with a total chance of failure that is less than 100%, which means there is a chance that the chain will truly never end.

Except it's NOT an infinitely small number. That would be 0, IE no chance of failing. If there's any chance of it failing, it cannot reach infinity. The fact that the chance of failing is decreasing at an exponential rate is irrelevant. There is a chance to fail, so it WILL fail eventually. The only way for it to have a chance at reaching infinity would be for that chance to fail to reach 0%, which it never does.

Gpope

2009-11-29, 02:11 AM

Except it's NOT an infinitely small number.

Yes it is, unless you care to show that the chance of X attacks in a row failing has a lower bound other than 0.

Gan The Grey

2009-11-29, 02:24 AM

Okay, I'm going to say this about infinity without dropping math equation bombs all over the place.

1. The basic premise of infinity is something that never ends. If there is a problem with this definition, webster it.

2. Infinity is NOT a specific number. It is an idea, and sometimes a state.

3. In order to achieve the ability to continue on for an infinite amount of time, there must be a 100% chance or greater to do so.

4. Infinity either IS or IS NOT. There is no middle ground. Either you have a 100% chance of being infinite, or you are finite.

Fancy math doesn't change this. The only way for this build to gain infinite attacks, is if it never suffers a miss on a natural 1. Even with luck rerolls, this is not possible. It doesn't matter how many attacks it generates, there is still a chance that you will roll 1 freakin godmillion 1's in a row and end your series. As long as that chance exists, you are not infinite.

AgentPaper

2009-11-29, 02:28 AM

Yes it is, unless you care to show that the chance of X attacks in a row failing has a lower bound other than 0.

Yes, it approaches 0. It WOULD hit 0, AFTER reaching infinity, except it doesn't reach infinity, because it fails an infinite amount of times before reaching infinity.

Mooch

2009-11-29, 02:30 AM

would casting single crystal blades out of iron be a bit too cheesy? (in order to qualify for both weapon properties)

Fortuna

2009-11-29, 02:40 AM

Okay, I'm going to say this about infinity without dropping math equation bombs all over the place.

The problem is that you need "math equation bombs" to talk about infinity in any kind of rigorous fashion.

1. The basic premise of infinity is something that never ends. If there is a problem with this definition, webster it.

2. Infinity is NOT a specific number. It is an idea, and sometimes a state.

3. In order to achieve the ability to continue on for an infinite amount of time, there must be a 100% chance or greater to do so.

4. Infinity either IS or IS NOT. There is no middle ground. Either you have a 100% chance of being infinite, or you are finite.

1. Fine

2. Fine

3. Wrong. By means of "math equation bombs", it has been demonstrated that the build has a chance to achieve a number of attacks which is not finite, that is, which is infinite. Specifically, it has been shown that as the number of attacks which has been made increases arbitrarily, the likelihood that the system has collapsed DOES NOT become arbitrarily close to zero. It instead becomes arbitrarily close to something very close to 5/9. Therefore, it is possible to plug in the highest natural number, that is aleph-0, as the number of attacks which has been made and recieve a probability that the system has not collapsed a non-zero number. Which means that there is a chance that the system will achieve an infinite amount of attacks, specifically about 5 times in nine starting attacks (an interesting side effect is that each attack which is spawned has the same chance to go infinite, but that is beside the point).

4. See 3.

Gan The Grey

2009-11-29, 02:48 AM

The problem is that you need "math equation bombs" to talk about infinity in any kind of rigorous fashion.

3. Wrong. By means of "math equation bombs", it has been demonstrated that the build has a chance to achieve a number of attacks which is not finite, that is, which is infinite. Specifically, it has been shown that as the number of attacks which has been made increases arbitrarily, the likelihood that the system has collapsed DOES NOT become arbitrarily close to zero. It instead becomes arbitrarily close to something very close to 5/9. Therefore, it is possible to plug in the highest natural number, that is aleph-0, as the number of attacks which has been made and recieve a probability that the system has not collapsed a non-zero number. Which means that there is a chance that the system will achieve an infinite amount of attacks, specifically about 5 times in nine starting attacks (an interesting side effect is that each attack which is spawned has the same chance to go infinite, but that is beside the point).

Wrong. There is no 'chance'. It either is, or is not. The failure chance gets smaller the further and further in you go, but that chance is ALWAYS there. There is no getting rid of it. And as long as it is there, the possibility of going on forever is 0.

You are treating infinity as a specific, though massively large, number. Something that infinity IS NOT. Infinity is not 100. Nor is it 10^10000^10000000. Nor is it aleph-0. It's not even a number. It's an idea.

I'm not arguing that you will get a lot of attacks. But alot does not equal infinite. Either their is no chance of failure, or there is.

AgentPaper

2009-11-29, 03:01 AM

I think what you're saying is that, as the number you're trying to reach increases, the chance you will get to that number approaches 5/9. Right? Unfortunately, this has no correspondence to infinity, since infinity isn't actually a number.

Zeful

2009-11-29, 03:05 AM

Okay the way I found to treat infinities in situations like this is to state the following:

Assertion 1: Infinity is not a number.

Assertion 2: D&D works with numbers.

Therefore: Nothing in D&D is truly infinite.

Therefore: Infinity will instead represent all permutations of chance simultaneously.

Therefore: If there is a permutation that would end the supposed chain, no matter how mathematically improbable, comparing it to infinity means that it will occur.

Therefore: An Infinite loop must show that all possible permutations of an event have no chance of not seeding the next permutation (i.e, you must prove that the build will continue under the most statistically unfavorable of conditions).

Gan The Grey

2009-11-29, 03:08 AM

I think what you're saying is that, as the number you're trying to reach increases, the chance you will get to that number approaches 5/9. Right? Unfortunately, this has no correspondence to infinity, since infinity isn't actually a number.

Thanks for the backup. :smallbiggrin:

Zeful, I like your style, but I think redefining words will just get us into more trouble.

Superglucose

2009-11-29, 03:09 AM

It's possible that there's a point where it tends to be infinite.

If we let the number of attacks be N, then the number of 1s that must be rolled in a row to break the chain will be f(N). That is, if we roll 1 attack, it will take 1.1 1s in a row to break the chain (assuming the average attack nets us 1.1 attacks afterwords). The probability of rolling f(N) 1s in a row is easy, it's (1/20)^(f(N)) or 1/(20^f(N)).

Now we can examine end behavior simply:

if the limit as f(N) approaches infinity is infinity, then the loop is infinite in that as you get more and more attacks it will be less and less likely to break out of the loop. If you get 1.1 attacks back for every attack you get in, then f(N) is 1.1^N, and as N approaches infinity, 1.1^N approaches infinity, so the loop is infinite in the sense that its limit is infinite.

more importantly: because the probability of rolling what you need to roll decreases as N gets large, it's not as simple as "infinite monkeys" scenario, because the "infinite monkeys" are making the proverbial works of shakespear even less likely. The probability that ALL monkeys on typewriters type the works of shakespear decreases as you get more and more monkeys.

Gan The Grey

2009-11-29, 03:15 AM

It's possible that there's a point where it tends to be infinite.

If we let the number of attacks be N, then the number of 1s that must be rolled in a row to break the chain will be f(N). That is, if we roll 1 attack, it will take 1.1 1s in a row to break the chain (assuming the average attack nets us 1.1 attacks afterwords). The probability of rolling f(N) 1s in a row is easy, it's (1/20)^(f(N)) or 1/(20^f(N)).

Now we can examine end behavior simply:

if the limit as f(N) approaches infinity is infinity, then the loop is infinite in that as you get more and more attacks it will be less and less likely to break out of the loop. If you get 1.1 attacks back for every attack you get in, then f(N) is 1.1^N, and as N approaches infinity, 1.1^N approaches infinity, so the loop is infinite in the sense that its limit is infinite.

more importantly: because the probability of rolling what you need to roll decreases as N gets large, it's not as simple as "infinite monkeys" scenario, because the "infinite monkeys" are making the proverbial works of shakespear even less likely. The probability that ALL monkeys on typewriters type the works of shakespear decreases as you get more and more monkeys.

The problem here is you are talking about certainties in a realm of possiblities. It is not certain that you will always get 1.1 attacks back. That's an average. Averages don't take into account the ten times out of twelve I rolled less than a five on a d20. If we went by that....

If winning the lottery was a 1 in a million chance, and I bought 1 million tickets, I would win. But it doesn't work like that. (Lets not get caught up on this example. I think you know what I mean.)

If you can tell me that there is NO chance of rolling more 1's than any of the other numbers combined, then you are right. Unfortunately, we both know that isn't true.

Fortuna

2009-11-29, 03:39 AM

As I see it, there are two arguments going on here. One is based on a conceptual argument, and one on a rigorous mathematical proof. I humbly request, sir, that you attempt to state your argument mathematically as I tried to (and failed), by directly countering any step in the proof that this does in fact produce an infinite number of attacks some of the time.

And incidentally, aleph-0 is not infinity. I beg your pardon. Aleph-0 is a transfinite number, and if you insist then I will go back and edit every single one of my posts to make it clear that I am talking only about a number which emulates infinity in EVERY WAY IMAGINABLE.

Gan The Grey

2009-11-29, 03:50 AM

As I see it, there are two arguments going on here. One is based on a conceptual argument, and one on a rigorous mathematical proof. I humbly request, sir, that you attempt to state your argument mathematically as I tried to (and failed), by directly countering any step in the proof that this does in fact produce an infinite number of attacks some of the time.

The problem is not with your math. I applaud your math, sir. Your math freakin rocks.

The problem is a problem of logic. Something cannot 'possibly' be infinite. It's a true or false, not a percentage.

Here's what little math I can muster to prove that infinity and percentages cannot function together:

100% of infinity = infinity

99% of infinity = infinity

infinity = infinity K that works.

99% = 100% that doesn't work.

Hell, 1% of infinity is still infinity, except that percentages are a measurement of numbers, and infinity is NOT a number, so frankly I can't even say that.

Either something goes on forever 100% of the time (100% = true) or 0% of the time (0% = false). Anything in between doesn't fit the boundaries of the equation. For a true/false, there is no in between.

And incidentally, aleph-0 is not infinity. I beg your pardon. Aleph-0 is a transfinite number, and if you insist then I will go back and edit every single one of my posts to make it clear that I am talking only about a number which emulates infinity in EVERY WAY IMAGINABLE.

I'm not sure what's with your tone here, and frankly I'm not even sure what you are trying to say to me. Sarcasm (or whatever this is) doesn't have much of a place in an intelligent discussion, especially a text-based intelligent discussion.

To quote the Wikipedia:

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.

We're talking about infinity here, more so towards the 'increases without bound'. Aleph-0, while similar, is not the same, and for the purposes of this conversation, not appropriate.

Fortuna

2009-11-29, 03:57 AM

I just accepted that we could not work with the truly infinite. There, you win. Working with the truly infinite is, as far as I am aware, at the boundaries of modern mathematics. Therefore, let's see whether we can achieve Cantor's poor imitation thereof. You say that the problem is not with math, but with logic. Your argument in that post says nothing to me except that dividing by infinity gives nonsensical answers. Where did we divide by infinity? If we did not, then I do not think that your argument holds water. And also, all that your last statement says is that any average value that we extract will also be nonsensical.

Gpope

2009-11-29, 04:10 AM

I just accepted that we could not work with the truly infinite.

This is the crux of the issue, because no, you can't "really" get infinite attacks. Mathematically you have an infinite series; while for all intents and purposes you can treat this as being an infinite number, it does not actually exist as infinity.

For the numbers term1nally s1ck mentioned earlier, the odds of getting X number of attacks are going to be approximately 5/9. This is not literally the same thing as an infinite number of attacks, but it does mean that any time you declare you're going to be rolling an attack, you have a 5/9 chance of having however many attacks as you want.

The odds that you will have a thousand attacks to work with? 5/9.

The odds that you will have a million attacks to work with? 5/9.

The odds that you will have a hundred billion attacks to work with? 5/9.

The odds that you will have more attacks than the number of atoms that exist in the universe? 5/9.

AgentPaper

2009-11-29, 04:11 AM

Well, something *can* be possibly infinity. Just for example, say you got some slew of feats that made it so, with each attack, if you rolled a 10 or more, you get to attack again, with a cumulative +1 bonus. And let's say the designers were silly, and this +1 bonus counted for getting above 10, so if you were on your fifth roll, and rolled a 7, you'd get 7+5=12, giving you another roll and another +1 bonus. There's a certain % chance that you will do this 10 times, giving you a +10 bonus, at which point you will make an infinite amount of attacks. Thus, you can say that each attack has an X% chance of going infinite.

However, that's not what's happening here. For the above to happen, the chance of each attack terminating has to become 0% at some point, which it does not. Instead, this would generate an arbitrarily higher number, so high that it may as well be infinite (maybe this is the aleph-0 you're talking about? might be nice to know what that even is) but it is NOT infinite. It is in fact exactly as far away from infinity as the number 0, or a gaboojillion*, or even negative gaboojillion*.

*A Gaboojillion being X^Xth power X number of times, with X being the number of atoms in the universe

Fortuna

2009-11-29, 04:17 AM

Aleph-0 is, as far as I am aware, the lowest transfinite number, being defined as the cardinality of the set of natural numbers. It is the largest number which you can name, including by means of this process, with an arbitrary increase in size. It is about as close to infinity as you get while still being vaguely able to work with it sanely.

Gan The Grey

2009-11-29, 04:22 AM

Well, something *can* be possibly infinity. Just for example, say you got some slew of feats that made it so, with each attack, if you rolled a 10 or more, you get to attack again, with a cumulative +1 bonus. And let's say the designers were silly, and this +1 bonus counted for getting above 10, so if you were on your fifth roll, and rolled a 7, you'd get 7+5=12, giving you another roll and another +1 bonus. There's a certain % chance that you will do this 10 times, giving you a +10 bonus, at which point you will make an infinite amount of attacks. Thus, you can say that each attack has an X% chance of going infinite.

However, that's not what's happening here. For the above to happen, the chance of each attack terminating has to become 0% at some point, which it does not. Instead, this would generate an arbitrarily higher number, so high that it may as well be infinite (maybe this is the aleph-0 you're talking about? might be nice to know what that even is) but it is NOT infinite. It is in fact exactly as far away from infinity as the number 0, or a gaboojillion*, or even negative gaboojillion*.

*A Gaboojillion being X^Xth power X number of times, with X being the number of atoms in the universe

That was sexy. I like being shown holes in my argument. *cookie for you* :smallbiggrin: Very interesting, though I think you step-outside the boundaries of mathematics when you start houseruling lol. So the only way you have the ability of possibly achieving the state of infinity is if you can somehow completely eliminate all chance of failure. I dig that.

And Random, I was providing proof that infinity is not a number, but a state. Numbers can function with percentages, states cannot. That's all I was saying with that example. Like I said, your math rocks. I was in no way criticizing or even attempting to understand the sexiness that was those 'math equation bombs'. I was trying to show that infinity was being used wrong, something which didn't really require math to show. That's all.

Oh, and I quoted the differences between Aleph-0 and infinity a few posts back if you are curious.

Fortuna

2009-11-29, 04:24 AM

Good. I am glad that we have reached an understanding. You be happy with your proof that it is not quite infinity, and I'll be happy with our proof that it is not quite finite either.

Gpope

2009-11-29, 04:24 AM

However, that's not what's happening here. For the above to happen, the chance of each attack terminating has to become 0% at some point, which it does not. Instead, this would generate an arbitrarily higher number, so high that it may as well be infinite (maybe this is the aleph-0 you're talking about? might be nice to know what that even is) but it is NOT infinite. It is in fact exactly as far away from infinity as the number 0, or a gaboojillion*, or even negative gaboojillion*.

There will always be a chance for each attack to terminate, but the chance for all attacks to terminate becomes infinitely close to 0 as the number of attacks increases. While you never get a literal infinity of attacks, the number of attacks generated in a successful chain is mathematically equal to infinity.

AgentPaper

2009-11-29, 04:27 AM

Aleph-0 is, as far as I am aware, the lowest transfinite number, being defined as the cardinality of the set of natural numbers. It is the largest number which you can name, including by means of this process, with an arbitrary increase in size. It is about as close to infinity as you get while still being vaguely able to work with it sanely.

See, that I can agree with. I have no problem with this having a chance of reaching an arbitrarily high number. I even said it would do that. It's just when you start confusing an arbitrarily high number with infinity that the problems crop up.

Gan The Grey

2009-11-29, 04:29 AM

Good. I am glad that we have reached an understanding. You be happy with your proof that it is not quite infinity, and I'll be happy with our proof that it is not quite finite either.

Lol alrighty then. Though if you've got a math equation to explain how something can't be either of only two possibilities, I'm up for some learning. :smallsmile:

Wings of Peace

2009-11-29, 04:42 AM

Okay the way I found to treat infinities in situations like this is to state the following:

Assertion 1: Infinity is not a number.

Assertion 2: D&D works with numbers.

Therefore: Nothing in D&D is truly infinite.

Therefore: Infinity will instead represent all permutations of chance simultaneously.

Therefore: If there is a permutation that would end the supposed chain, no matter how mathematically improbable, comparing it to infinity means that it will occur.

Therefore: An Infinite loop must show that all possible permutations of an event have no chance of not seeding the next permutation (i.e, you must prove that the build will continue under the most statistically unfavorable of conditions).

This is mostly nitpicking but there are some cases in D&D such as the Omnificer which achieve actually infinite values.

Edit: Possible never mind. May have misread your last point.

Gan The Grey

2009-11-29, 05:10 AM

This is mostly nitpicking but there are some cases in D&D such as the Omnificer which achieve actually infinite values.

Edit: Possible never mind. May have misread your last point.

Yeah, I just re-re-reread Zeful's post, and I think I finally understand what he's trying to say. And I like it.

term1nally s1ck

2009-11-29, 05:13 AM

Yeah, you're right, I should have made the error of the approximation more explicit...as you said, the next-higher order term in the approximation is

ln(1+u) = u - 1/2 u^2 + O(u^3) < u

so it happens to work. I was relying on the fact that as u-> 0, it's really just u +/- O(u^2), but I guess that's not really sound, is it?

Not quite, when you're dealing with a proof like this...you need a bound, not an approximation, because we don't know how good the approximation is without the bound. In general, if you have an approximation, you can always have it work, because for small/large/specific enough values of your numbers in the equations, you have a bound which satisfies the conditions, but you have to specify what bound you use, because there are a few times when you're dealing with something REALLY close to your approximation that it won't quite work.

You seem to not understand the concept of infinite.

Does your build keep attacking even if you roll one hundred billion 1s in a row? If not, you won't go infinite.

In fact, if you roll long enough, you will have n rolls, and will then roll 9999n 1s in a row.

Haven't read past this post, so if you learnt what infinite means in the meantime, great.

By the time it is remotely likely for me to roll that many 1s, I have more attacks than that.

On the contrary, -you- seem not to understand the concept of infinite. Specifically. you're confusing "finite" with "non-deterministically infinite".

As it stands, the sequence of partial sums representing the probability that the attack sequence terminates converges to a number less than 1. This build therefore has a finite, nonzero probability of an attack sequence not terminating. It has a -chance- of going infinite in the sense that it does not terminate. It -also- has a chance of terminating. This is thus "non-deterministically infinite", and the expected value of the number of attacks you make is undefined.

Suppose that a revised version of this build had a luck feat or something attached to it that allowed the rerolling of natural 1s, and one was attacking on a 2 to hit. Then the probability that the attack sequence does not terminate is 1. This is -also- infinite: it differs from the first case in that it is deterministically infinite.

This is here because of the distinction made between deterministically infinite, and non-deterministically infinite.

The posts past here start arguing about aleph-0 vs infinite.

The distinction is that a series (say, the number of attacks) tends towards infinity if and only if it is provable that it eventually exceeds any known natural number. I can't prove that the number of attacks is equal to infinity, because infinity itself isn't really a number. I can prove that the number of attacks tends towards infinity, which is what is known as a 'divergent function' where the limit of f(x) is infinity as x tends to infinity.

Unfortunately, X in this case is the number of attacks made. I have already shown that there is an excellent chance of X finishing in any finite number, so to stretch the whole number line across the 6 seconds we have in the turn, I have to use what is known as the projective real line. In this case, we basically take a line representing 6 seconds, 'twist' them into a quarter-circle, put a point at the centre of the circle it is part of, and draw a straight line down to twice the radius of the quarter-circle directly below the leftmost end of the quarter-circle. This means that as we go through the line, we get higher on the numberline MUCH faster, and we can re-write X as a function of t, and hence f(x) as f(t). But for t = 6, X(6) is defined as infinity, and f(infinity) as the limit of the function, namely infinity. This is a direct result of trying to fit an infinite sequence into a finite time, you HAVE to reach infinity, or you don't have an end-point. The number of attacks made in 6 seconds has a 5/9 chance (or 60/133 for the build I posted) of being infinite.)

This is the non-deterministic infinity.

Aleph-0 is the deterministic infinity, as it is well-defined (the cardinality of the natural numbers), and is NOT the number of attacks I can make, as we're looking at a limit to a sequence, and I'm not sure if I can find a projection from N to the attacks.

(Picky point....aleph-0 is NOT a natural number.)

EDIT: Zeful would be very right...if Infinities weren't a natural result of certain calculations that exclusively involve numbers. How do you think they started to use them? it's not exactly a normal concept....

AgentPaper

2009-11-29, 05:21 AM

Not quite, when you're dealing with a proof like this...you need a bound, not an approximation, because we don't know how good the approximation is without the bound. In general, if you have an approximation, you can always have it work, because for small/large/specific enough values of your numbers in the equations, you have a bound which satisfies the conditions, but you have to specify what bound you use, because there are a few times when you're dealing with something REALLY close to your approximation that it won't quite work.

By the time it is remotely likely for me to roll that many 1s, I have more attacks than that.

This is here because of the distinction made between deterministically infinite, and non-deterministically infinite.

The posts past here start arguing about aleph-0 vs infinite.

The distinction is that a series (say, the number of attacks) tends towards infinity if and only if it is provable that it eventually exceeds any known natural number. I can't prove that the number of attacks is equal to infinity, because infinity itself isn't really a number. I can prove that the number of attacks tends towards infinity, which is what is known as a 'divergent function' where the limit of f(x) is infinity as x tends to infinity.

Unfortunately, X in this case is the number of attacks made. I have already shown that there is an excellent chance of X finishing in any finite number, so to stretch the whole number line across the 6 seconds we have in the turn, I have to use what is known as the projective real line. In this case, we basically take a line representing 6 seconds, 'twist' them into a quarter-circle, put a point at the centre of the circle it is part of, and draw a straight line down to twice the radius of the quarter-circle directly below the leftmost end of the quarter-circle. This means that as we go through the line, we get higher on the numberline MUCH faster, and we can re-write X as a function of t, and hence f(x) as f(t). But for t = 6, X(6) is defined as infinity, and f(infinity) as the limit of the function, namely infinity. This is a direct result of trying to fit an infinite sequence into a finite time, you HAVE to reach infinity, or you don't have an end-point. The number of attacks made in 6 seconds has a 5/9 chance (or 60/133 for the build I posted) of being infinite.)

This is the non-deterministic infinity.

Aleph-0 is the deterministic infinity, as it is well-defined (the cardinality of the natural numbers), and is NOT the number of attacks I can make, as we're looking at a limit to a sequence, and I'm not sure if I can find a projection from N to the attacks.

(Picky point....aleph-0 is NOT a natural number.)

Ok, now you're just writing down a bunch of nonsense that looks like complicated math but in the end is just a bunch of words. For one, when does the amount of time in a round even come into the equation? It has no relevance to the number of attacks you make at all.

Gan The Grey

2009-11-29, 05:30 AM

By the time it is remotely likely for me to roll that many 1s, I have more attacks than that.

Unless you don't, which law of probability states is possible. You can't use absolutes like that when you are talking about probability. You say 'likely to roll that many 1s' which is a statement of probability, then turn around and say 'I have more attacks that that' which is basically an absolute. That doesn't work.

term1nally s1ck

2009-11-29, 05:37 AM

Forgive me, I was under the impression that we were trying to consider the amount of attacks made in a single round. We can't count across the attacks themselves, because there is a non-0 chance of them never ending. There's only the two ways to sum across a range with no limit, one is the sum to infinity (which people don't like, because it has the word infinity in it...despite that you sum it as you would sum any finite sum (unless you actually do the finite sum 1-by-1....), or by the projective number line. For the projective number line (google it if you don't want to take my word, it's pretty easy to understand), you can use any variable you want for the semi-circle , provided there's a way to spread the original variable out across the one used. Time was just an obvious one to me, because it's a bounded variable such that all the attacks happen within the bounds. I've now used both, both give an answer of non-finite and infinite, respectively....the two are identical.

if you want me to formalise my above post, tell me...but the arguments are such that I'll have to use the logic symbols, and I still just call on the endpoint being equivalent to infinity, because that's the definition of the projective representation.

AgentPaper

2009-11-29, 07:57 AM

None of which explains how you get over the fact that, even if you're rolling an arbitrarily high number of attacks, you can't then roll an arbitrarily large number of 1s in a row. Since this will happen literally an infinite number of times before you reach infinity, it has a 100% chance of occurring before then. How small the chance is is completely irrelevant. If it's there, it will happen.

term1nally s1ck

2009-11-29, 08:15 AM

Because 'Arbitrarily high' means any finite number you care to name is eventually rolled. Yes, I eventually roll 1 quadrillion nat 1s. By that time, I have more than 1 quadrillion attacks, almost guaranteed. (expected number of attack rolls required to roll 1 quadrillion (10^15) nat 1s = 2^1000000000000000, expected number of attacks at that point = 1.1(or whatever your build happnes to generate per attack)*(10^15), which is MORE than 10^15.

Just because you can roll an infinite number of natural ones doesn't mean I don't have more attacks...after all, there are an infinite number of odd numbers, and yet if I remove them all from the natural numbers, not only do I have some left, but I have infinitely many left.

AgentPaper

2009-11-29, 08:39 AM

Because 'Arbitrarily high' means any finite number you care to name is eventually rolled. Yes, I eventually roll 1 quadrillion nat 1s. By that time, I have more than 1 quadrillion attacks, almost guaranteed. (expected number of attack rolls required to roll 1 quadrillion (10^15) nat 1s = 2^1000000000000000, expected number of attacks at that point = 1.1(or whatever your build happnes to generate per attack)*(10^15), which is MORE than 10^15.

Just because you can roll an infinite number of natural ones doesn't mean I don't have more attacks...after all, there are an infinite number of odd numbers, and yet if I remove them all from the natural numbers, not only do I have some left, but I have infinitely many left.

Yes, it's possible that you won't get enough 1s for the chain to end for an absurdly long time, but you're severely underestimating infinity. It does not matter how small the chance is that the chain will end, if it's more than zero, it WILL END. PERIOD. Infinity doesn't care if it keeps becoming less and less likely that the chain will end, all it cares about is whether that value is zero, or not zero. Since it's no zero, it happens.

Fortuna

2009-11-29, 12:58 PM

I feel that we have the same problem as before. We have a non-rigorous conceptual agument being used to try to shoot down a rigorous mathematical proof. I suggest that you either attack that proof mathematically, or admit defeat.

Gan The Grey

2009-11-29, 03:18 PM

Just because we aren't throwing a bunch of equal signs around and using big hard-to-understand words, that doesn't mean our argument isn't valid. To assume so is a logical fallacy on your part.

I've said it before and I'll say it again. YOU ARE USING INFINITY WRONG.

You want a big number? Great. You have a 5/9 chance of getting a big number. Fantastic. You've proved that.

You want the attacks to go on ENDLESSLY, as in, never ending? Not possible with this build. PERIOD. As long as there is a chance for failure, failure will occur within infinite bounds. I don't need a bunch of math to show that as FACT.

Now, I suggest you either stop resorting to logical fallacy in order to win your argument, or admit defeat.

term1nally s1ck

2009-11-29, 03:25 PM

As long as there is a chance for failure, failure will occur within infinite bounds.

Prove it. If you provide me with a proof that it is guaranteed that the sequence will fail, I'll aplogise and stop arguing against it. I've proved using the mathematical principles defined to deal with infinities that this build makes an infinite number of attcks. Either find a flaw the the maths, or provide a proof that the opposite is true, don't just rely on shouting louder. You've not backed up your claims that I'm using infinity wrong, and you're not even trying to use any of infinity's properties, bar the one you think is relevant, despite multiple explanations as to why it really isn't.

Gpope

2009-11-29, 03:27 PM

Since it's no zero, it happens.

Untrue. There is not actually any guarantee that everything that can happen will happen as you approach infinity. Because we can't actually predict what is going to happen in an infinite series, the only way to meaningfully discuss probability over infinity is in terms of limits.

Take for example a series of coin flips, straight 50/50 probability, and see how many times in a row it comes up heads. There is never at any point any guarantee that the streak will stop, no guarantee that the coin will ever come up tails. There is always a nonzero probability that it will come up heads one more time. It is impossible for a number X to exist such that the probability of getting X many heads in a row is 0, yet because the limit of the probability rapidly approaches 0 as X increases we can confidently say that the probability of getting infinitely many heads in a row is 0 even though it is always possible to continue flipping heads.

The limit of the possibility of getting infinitely many heads is 0; there is 0% chance of reaching infinitely many heads, even though the chance of the streak continuing is always higher than 0. The limit of the possibility of getting infinitely many attacks from the proposed attack loop (at least with the 6-20 threat and confirm) is 5/9; there is a 55.55...% chance of reaching infinitely many attacks.

As long as there is a chance for failure, failure will occur within infinite bounds. I don't need a bunch of math to show that as FACT.

Except you can't show it as fact at all. It might seem intuitively obvious, but you can't prove that failure will ever occur.

Heliomance

2009-11-29, 03:49 PM

Just because we aren't throwing a bunch of equal signs around and using big hard-to-understand words, that doesn't mean our argument isn't valid. To assume so is a logical fallacy on your part.

I've said it before and I'll say it again. YOU ARE USING INFINITY WRONG.

You want a big number? Great. You have a 5/9 chance of getting a big number. Fantastic. You've proved that.

You want the attacks to go on ENDLESSLY, as in, never ending? Not possible with this build. PERIOD. As long as there is a chance for failure, failure will occur within infinite bounds. I don't need a bunch of math to show that as FACT.

Now, I suggest you either stop resorting to logical fallacy in order to win your argument, or admit defeat.

Personally, I would assume that the people throwing the symbols and "big hard-to-understand words" are more likely to be the people that know what the hell they're talking about. I realise this isn't always the case, and that a blanket assumption to that effect is an appeal to authority fallacy. I'm only in my first year of a maths degree, so I can't follow the maths required perfectly, but I'll do my best to translate.

The definition of a function f(x) (in this case the number of attacks generated) tending to infinity as a variable x (in this case the number of attacks made) tends to infinity is:

(∀ K>0) (∃ R>0) | (∀ x∈X) x>R ⇒ f(x)>K

Translated, this means:

For any value (which we will call K) greater than 0, we can find another value (let's call it R) greater than 0, such that for any value that it is possible for x (the number of attacks rolled so far) to take, if x is greater than the value we chose, R, then f(x) (the number of attacks generated so far) will be greater than the value K, no matter how high K is. So the fact that we have already established that there is a 5/9 chance of generating any number of attacks, no matter how high, given an arbitrarily high number of rolled attacks - in this case represented by setting R arbitrarily high - means that there is a 5/9 chance that the number of attacks will meet this definition. And as this is the definition of tending to infinity - this is what tending to infinity actually means, in strict mathematical terms - that means that there is a 5/9 chance that the result will go to infinity.

term1nally s1ck

2009-11-29, 04:16 PM

Please pretty please tell me how to make the logic symbols...I want my logic symbols..

Heliomance

2009-11-29, 04:23 PM

Please pretty please tell me how to make the logic symbols...I want my logic symbols..

Copy-pasted from here (http://en.wikipedia.org/wiki/Table_of_mathematical_symbols).

Kelb_Panthera

2009-11-29, 09:47 PM

Just out of curiosity, outside of DnD and your mathematics courses how often do you guys actually get to use this level of mathematics? I'm not trying to be derogatory. I'm actually interested in learning several of the higher maths but if I'm going to be paying a university and not getting much use out of this sort of thing, I'm going to pace myself rather than go sign-up for a course tomorrow :smalltongue:

If you feel that answering this post could de-rail the thread feel free to PM me :smallsmile:

Gan The Grey

2009-11-29, 10:16 PM

Alright, this is turning into a contest of egos, and it is obvious that people's egos are getting hurt. So let's try to keep this civil.

I'm not going to get into math to argue my point any further, because, as I've said already, math is not the problem. There is nothing wrong with your math. Your equations are well-thought out and precise. I completely understand what it is you are trying to say, I just believe you are misapplying evidence, or rather, apply the wrong evidence to the argument. I'm not the only one that has said this.

All I have to do to prove that this sequence will fail is to posit a string of 1's large enough to end the sequence. That's it. Within infinite bounds, all things that are possible will eventually occur, and as failure is a possibility, it will eventually occur, and the number of attacks will thusly fail to achieve infinity. I don't need math to show this, and this cannot be ignored either.

The thing about science is, you can prove your theory correct a million times, but it only takes one failure to prove it wrong. Just because your math comes out the way you want it to doesn't make it right. I'm sure you can say this goes both ways, and you would be right.

Terminally sick, any attitude I appeared to have was not directed at you. It was directed towards Agentpaper, in response to attitude received from him. So, don't take my responses personally. I'm not trying to yell with caps. I'm trying to emphasize what I feel are the most important parts of what I'm saying.

Let me ask you this though. I have conceded to the intelligence of yours and others' math multiple times now. Do you believe there is any merit to what I and others supporting my view are saying? You accuse me of being selective in my argument, but is not the most important aspect of infinity the fact that it literally has no end?

And let me ask one more question before I forget. I'm not even sure what number we are using any more to threaten a critical, be it 9 or 6 or 3...but what are the chances of tending to infinity if our critical threat range was simply 20, changing nothing else? Wouldn't that also tend towards infinity, even if the possibility was smaller?

sambo.

2009-11-29, 11:35 PM

Arcane Duelist's False Keen specifically does not stack with either Improved Critical or keen.

errr, not exactly.....

actually: exactly NOT.

False Keenness (Ex): To create the idea that she is more effective than she really is, the arcane duelist can make her chosen melee weapon keen once per day per arcane duelist level. To activate the keen effect, she must subtract the weapon's normal threat range for critical hits from her attack bonus. For example, if the arcane duelist wields a rapier, she must subtract 3 from her attack bonus, since the rapier has a threat range of 18-20, to make it keen. The keen effect lasts for 1 round per arcane duelist level, and it does not stack with the keen edge spell or other magical effects that make a weapon keen. The ability does stack with the increased threat range granted by the Improved Critical feat and by a keen magic weapon.

it's actually a little ambiguous, but i read that as specifically stacking with Improved Critical OR a Keen weapon, but NOT a weapon with a Keen Edge spell on it.

in any case, i read it as specifically stacking with Improved Critical.

Fortuna

2009-11-29, 11:37 PM

Gan, the problem is that we have shown that we surpass the finite numbers. We are talking transfinites now, and the argument as to the difference between those and infinity is very complicated indeed. Also, I believe that it would not tend to infinity, although I will need to check that by plugging the numbers into the general form of the limit when someone provides it (or just by waiting for someone else to find it). It may well do so, however, just with an incredibly small chance.

Gpope

2009-11-29, 11:44 PM

All I have to do to prove that this sequence will fail is to posit a string of 1's large enough to end the sequence.

This only proves that the sequence can end, not that it will end, because:

Within infinite bounds, all things that are possible will eventually occur,

is a false assumption. It seems like it's the kind of thing that would be true, but it's not quite correct.

Here's another way of conceptualizing the problem: the longer the sequence goes, the larger it gets and the more 1s it takes to end it. If it continues infinitely, then it will take infinitely many 1s in a row (more or less) to break the sequence. You are correct in asserting that there is always a possibility that you will in fact roll infinitely many 1s in a row--but there also exists a possibility that you will roll infinitely many non-1s in a row, or roll an infinitely long mixture that does not contain enough 1s (which is basically infinitely many 1s) to break the chain. And since rolling 1s happens a lot less often than rolling critical threats that spawn additional attacks, the odds of getting infinitely many attacks outstrip the odds of getting infinitely many 1s.

Kallisti

2009-11-29, 11:55 PM

I've played a similar build before, so there's some things I'd like to clear up. For the record, I'm AFB at the moment, so if someone has RAW to prove me wrong, then I'd like to see it. I hate being wrong, if you see me wrong correct me, please.

1. Lightning Mace is not dependant on the attack hitting or confirming a crit, only on it threatening a crit. Which means it only needs to roll within the threat range. Even a Nat 1 can threaten a critical, although it can never confirm or even hit. I'm pretty sure about this, but not 100%.

2. I know that there's an Eberron special material called crysteel. I don't know if it would count as a steel and crystal weapon, but I thought I'd mention it.

3. I believe there's a Devoted Spirit stance in ToB that lets you take 11 or so on attack rolls. I know that either Krimm Blackleaf or the Demented One has a homebrew discipline focused on the Lawful alignment that has a similar stance, although homebrew isn't really relevant. Thought I'd mention it.

Someone with the books will need to check these for me.

Gan The Grey

2009-11-30, 12:04 AM

@ Random

Hmmm. I guess the issue I'm having here is that I view any defined number as finite. If you can declare a point in a number line, that point becomes finite. If you can reach a number, it is not infinite. Kind of like the argument of knowing where an atom is and how fast it's going. You can't have it both ways. Either you are dealing with an arbitrarily high number, or you are dealing with infinity.

I think you and I are arguing two different things actually. Correct me if I wrong here, but I think you are arguing for the potential to reach any given number you supply, whereas I am simply arguing for achieving an endless quality.

From Gpope

Here's another way of conceptualizing the problem: the longer the sequence goes, the larger it gets and the more 1s it takes to end it. If it continues infinitely, then it will take infinitely many 1s in a row (more or less) to break the sequence. You are correct in asserting that there is always a possibility that you will in fact roll infinitely many 1s in a row--but there also exists a possibility that you will roll infinitely many non-1s in a row, or roll an infinitely long mixture that does not contain enough 1s (which is basically infinitely many 1s) to break the chain. And since rolling 1s happens a lot less often than rolling critical threats that spawn additional attacks, the odds of getting infinitely many attacks outstrip the odds of getting infinitely many 1s.

I completely understand what you are saying here. And, at first glance, it would seem to make perfect sense. The problem here lies with how you are trying to give possibility concreteness. Yes it is possible to have more attacks generated than ones generated, but the reverse is also true, even if the chance is small that it will do so. You are right that the longer it goes on, the harder it gets to break the chain. I keep being told that I have to prove that it will break, yet no one has proved that it won't. Everyone is relying on the possibility that it won't, but wants me to provide proof that it 100% will break. That's not really fair.

I think, here, we are going to have to agree to disagree. Correct me if I'm wrong, but you are saying that even given an endless amount of time, not all possible circumstances will be reached, whereas I'm saying that given an endless amount of time, all possible circumstances will occur. Your opinion holds much more water than mine even if you consider a time span equal to the life of the universe, but time is not an issue within our DnD example, as all attacks will somehow be resolved in a span of six seconds. Really, I think this comes down to a difference of opinion that can't really be solved at all. :smallsmile:

Gpope

2009-11-30, 12:40 AM

Everyone is relying on the possibility that it won't, but wants me to provide proof that it 100% will break.

Well, yes. You're saying that there is a 100% chance it will break as it continues towards infinity, so it is imminently fair to ask you to prove that there is a 100% chance it will break. Moreover, you're talking about reciprocal outcomes here. If there is not a 100% chance it will break as it continues infinitely, then there is a possibility that it will continue infinitely.

I think, here, we are going to have to agree to disagree. Correct me if I'm wrong, but you are saying that even given an endless amount of time, not all possible circumstances will be reached, whereas I'm saying that given an endless amount of time, all possible circumstances will occur.

You've summed up the difference succinctly. The idea that everything will eventually happen as you go through infinitely many trials seems obvious, but it's misleading. The problem is that "all possible circumstances" include contradictory outcomes. It's possible that you will roll infinitely many natural 1s in a row, and it's also possible that you will roll infinitely many natural 20s in a row. These two outcomes are mutually exclusive--if you're rolling infinite natural 1s, you never start rolling natural 20s, so you can't have both of them occur in the same infinite sequence. Of course, the odds of either one happening are infinitesimally small, so we can safely rule them out even though there is no logical reason that either one couldn't happen.

You could, on the other hand, say with confidence that everything with a finite probability will eventually happen given infinitely many trials. You can rest assured that you will eventually roll a natural 1 at some point in infinitely many rolls, 100% of the time. You can even say with confidence that you are guaranteed to eventually roll any arbitrarily large number of natural 1s in a row that you care to name. But you cannot assume that you will roll infinitely many natural 1s in a row, and it takes infinitely many natural 1s to guarantee that the attack chain breaks.

CockroachTeaParty

2009-11-30, 12:42 AM

I haven't bothered to read the whole thread, but I got the gist of it.

My question is, if this infinite critical hit attack thing worked, what would it look like?

Would it just be the character charging, his arms blurring into a spinning whirlwind of death? Would the blades be moving faster than the eye could see? Would the air combust?

Gan The Grey

2009-11-30, 12:53 AM

Well, yes. You're saying that there is a 100% chance it will break as it continues towards infinity, so it is imminently fair to ask you to prove that there is a 100% chance it will break. Moreover, you're talking about reciprocal outcomes here. If there is not a 100% chance it will break as it continues infinitely, then there is a possibility that it will continue infinitely.

That's a good point, and it really just illustrates the differences in our opinions on possibilities within infinite bounds.

You've summed up the difference succinctly. The idea that everything will eventually happen as you go through infinitely many trials seems obvious, but it's misleading. The problem is that "all possible circumstances" include contradictory outcomes. It's possible that you will roll infinitely many natural 1s in a row, and it's also possible that you will roll infinitely many natural 20s in a row. These two outcomes are mutually exclusive--if you're rolling infinite natural 1s, you never start rolling natural 20s, so you can't have both of them occur in the same infinite sequence. Of course, the odds of either one happening are infinitesimally small, so we can safely rule them out even though there is no logical reason that either one couldn't happen.

You could, on the other hand, say with confidence that everything with a finite probability will eventually happen given infinitely many trials. You can rest assured that you will eventually roll a natural 1 at some point in infinitely many rolls, 100% of the time. You can even say with confidence that you are guaranteed to eventually roll any arbitrarily large number of natural 1s in a row that you care to name. But you cannot assume that you will roll infinitely many natural 1s in a row, and it takes infinitely many natural 1s to guarantee that the attack chain breaks.

But were ARE talking about finite probabilities, because at no point in the line does it require an infinite amount of 1's to break the chain. At any point, the number of 1's needed to break the chain is a specific number. So we can't discount this.

term1nally s1ck

2009-11-30, 12:54 AM

I've played a similar build before, so there's some things I'd like to clear up. For the record, I'm AFB at the moment, so if someone has RAW to prove me wrong, then I'd like to see it. I hate being wrong, if you see me wrong correct me, please.

1. Lightning Mace is not dependant on the attack hitting or confirming a crit, only on it threatening a crit. Which means it only needs to roll within the threat range. Even a Nat 1 can threaten a critical, although it can never confirm or even hit. I'm pretty sure about this, but not 100%.

2. I know that there's an Eberron special material called crysteel. I don't know if it would count as a steel and crystal weapon, but I thought I'd mention it.

3. I believe there's a Devoted Spirit stance in ToB that lets you take 11 or so on attack rolls. I know that either Krimm Blackleaf or the Demented One has a homebrew discipline focused on the Lawful alignment that has a similar stance, although homebrew isn't really relevant. Thought I'd mention it.

Someone with the books will need to check these for me.

Aura of perfect order.

We have a winner. lets see, drop the weaponmaster, shove in swordsage 6, stick with a 9-20 crit threat, and take 11s. All you need to do now is buff your to-hit, which is less difficult. Just wow.

I take it this build would be accepted by all as an infinite loop, given that you hit on an 11 or less?

@Gan: Yes, there is some merit in having intuitive ideas about infinity, especially since you can get completely opposite ideas back, even with rigorous proofs.

http://en.wikipedia.org/wiki/Random_walk

Ironically, we're both right. There is simultaneously a <1 chance that the sequence terminates at a finite number, and a guarantee that the sequence terminates. I'm confused as hell how this is possible, but infinity is by its nature, counterintuitive.

Apologies to all for dsimissing so many people's ideas, apparently two completely opposite views can be correct in this case.

term1nally s1ck

2009-11-30, 12:57 AM

Just out of curiosity, outside of DnD and your mathematics courses how often do you guys actually get to use this level of mathematics? I'm not trying to be derogatory. I'm actually interested in learning several of the higher maths but if I'm going to be paying a university and not getting much use out of this sort of thing, I'm going to pace myself rather than go sign-up for a course tomorrow :smalltongue:

If you feel that answering this post could de-rail the thread feel free to PM me :smallsmile:

Suprisingly, to me, the more I learned about maths, the more I was able to apply it to everyday stuff. There are lots of little questions people ask about hings, usually not expecting an answer to be possible, and once you've done some advanced maths, you know how the probvlem could be solved.

Gan The Grey

2009-11-30, 01:00 AM

Aura of perfect order.

We have a winner. lets see, drop the weaponmaster, shove in swordsage 6, stick with a 9-20 crit threat, and take 11s. All you need to do now is buff your to-hit, which is less difficult. Just wow.

I take it this build would be accepted by all as an infinite loop, given that you hit on an 11 or less?

@Gan: Yes, there is some merit in having intuitive ideas about infinity, especially since you can get completely opposite ideas back, even with rigorous proofs.

http://en.wikipedia.org/wiki/Random_walk

Ironically, we're both right. There is simultaneously a <1 chance that the sequence terminates at a finite number, and a guarantee that the sequence terminates. I'm confused as hell how this is possible, but infinity is by its nature, counterintuitive.

Apologies to all for dsimissing so many people's ideas, apparently two completely opposite views can be correct in this case.

Nice research sir. +2 points. :smallwink: Now I'm going to take a look at that some more and wait for my brain to melt.

Oh...and +10 points to Kallisti for being a pimp. /Thread lol

Gpope

2009-11-30, 01:09 AM

But were ARE talking about finite probabilities, because at no point in the line does it require an infinite amount of 1's to break the chain. At any point, the number of 1's needed to break the chain is a specific number. So we can't discount this.

But by the same token, at any point in the chain there is a finite, nonzero (nowhere near zero, once you get going) probability of the chain continuing.

Suppose we look at someone who just takes Lightning Mace and absolutely no other threat range modifiers or critical hit triggers. If they roll a natural 20, they keep rolling; if they ever roll anything than a natural 20, they stop. At any given point there's a 5% chance that you keep rolling, so there is no finite limit to the number of attacks that you can get from Lightning Mace. Yet I think we both readily agree that you're never going to get an infinite number of attacks from Lightning Mace, because the probability of getting an arbitrarily high number of natural 20s is infinitesimally small.

Similarly, with an attack loop that only stops on a natural 1 and expands itself otherwise, once you get enough attacks going you require an arbitrarily high number of natural 1s (not technically an infinite number of 1s, you are correct there) to stop, a probability that is also infinitesimally small as you go higher.

Gan The Grey

2009-11-30, 01:20 AM

But by the same token, at any point in the chain there is a finite, nonzero (nowhere near zero, once you get going) probability of the chain continuing.

Suppose we look at someone who just takes Lightning Mace and absolutely no other threat range modifiers or critical hit triggers. If they roll a natural 20, they keep rolling; if they ever roll anything than a natural 20, they stop. At any given point there's a 5% chance that you keep rolling, so there is no finite limit to the number of attacks that you can get from Lightning Mace. Yet I think we both readily agree that you're never going to get an infinite number of attacks from Lightning Mace, because the probability of getting an arbitrarily high number of natural 20s is infinitesimally small.

Similarly, with an attack loop that only stops on a natural 1 and expands itself otherwise, once you get enough attacks going you require an arbitrarily high number of natural 1s (not technically an infinite number of 1s, you are correct there) to stop, a probability that is also infinitesimally small as you go higher.

Alright! Now we're getting somewhere!

This traditional use of Lightning Maces is really not much different than the build we are discussing. All you are doing is increasing the chances of additional attacks. Using this new example and your previous logic, you could say that you will infinitely NOT get an additional attack, simply because the chance is small. That's basically your argument for infinite attacks, right? You are going to get infinite attacks simply because chance favors you? The only REAL difference between the two is that chance favors you more and more the further in you go.

Let me ask this. Let's say the first time you try to set up your infinite attack routine, you roll a 1 each time you roll, until you run out of attacks. Doesn't that single circumstance right there disprove the idea that this build will afford you infinite attacks?

Oh, and as final proof for the validity of my argument, I cite 'Murphy's Law'.

'Anything that can happen, will happen, and at the worst possible time.'

So there. :smallwink:

Fortuna

2009-11-30, 01:24 AM

No. It disproves that it will always afford infinite attacks.

Gpope

2009-11-30, 01:32 AM

Alright! Now we're getting somewhere!

This traditional use of Lightning Maces is really not much different than the build we are discussing. All you are doing is increasing the chances of additional attacks. Using this new example and your previous logic, you could say that you will infinitely NOT get an additional attack, simply because the chance is small. That's basically your argument for infinite attacks, right? You are going to get infinite attacks simply because chance favors you? The only REAL difference between the two is that chance favors you more and more the further in you go.

That's essentially it. The key difference isn't the fact that you're increasing the threat range, though; it's the fact that you're abusing the rules to fit in two extra attacks for every critical threat. It doesn't matter whether your chance to keep rolling is 5% or 95%, with a fixed probability you're going to hit it sooner or later. But because you're constantly spawning off extra attacks, your chance to keep rolling continually goes up infinitely approaching a limit of 100%. If the limit was anything less than 100%, then you'd never be able to go infinite--like vanilla Lightning Mace, it would still be theoretically possible to keep going infinitely, but the odds would be infinitely small.

Also, Murphy's Law does not preclude the possibility that getting infinite attacks will end up being precisely the worst possible thing that could happen, and therefore must happen. :smalltongue:

Gan The Grey

2009-11-30, 01:40 AM

That's essentially it. The key difference isn't the fact that you're increasing the threat range, though; it's the fact that you're abusing the rules to fit in two extra attacks for every critical threat. It doesn't matter whether your chance to keep rolling is 5% or 95%, with a fixed probability you're going to hit it sooner or later. But because you're constantly spawning off extra attacks, your chance to keep rolling continually goes up infinitely approaching a limit of 100%. If the limit was anything less than 100%, then you'd never be able to go infinite--like vanilla Lightning Mace, it would still be theoretically possible to keep going infinitely, but the odds would be infinitely small.

Also, Murphy's Law does not preclude the possibility that getting infinite attacks will end up being precisely the worst possible thing that could happen, and therefore must happen. :smalltongue:

But the limit never hits 100%, only APPROACHES it in smaller and smaller increments. It is always less that 100%. So then...not infinite. Correct? I might be misunderstanding this. If I am, how are you setting the limit at 100%? Technically, isn't the limit 95% (representing a 1 in 20) that increases incrementally with each successive attack?

Oh, that depends on your point of view. Yes, bad for the DM, but good for you since you intended for that to happen right? And since there is no DM in our example...then the worst possible thing to happen is for your build to not work as intended. :smallbiggrin:

Gpope

2009-11-30, 01:55 AM

But the limit never hits 100%, only APPROACHES it in smaller and smaller increments. It is always less that 100%. So then...not infinite. Correct?

Essentially, yes. The catch is, though, that limits work both ways. That's what I've been getting at with the plain Lightning Mace example. The probability of rolling X many natural 20s in a row approaches a limit of 0% very rapidly as X approaches infinity--but it never actually reaches 0, so Lightning Mace always has a nonzero possibility of generating an infinite number of attacks. Just as the self-reinforcing attack loop always has a nonzero possibility of terminating after a finite number of attacks (if you have a small number of attacks to begin with, there is a very noticeable chance that it will terminate simply by starting with multiple 1s--but once you reach enough attacks, the probability gets very, very close to 0 and only gets closer). In both cases, though, as you continue the sequence towards infinity it is for all practical purposes impossible to distinguish that nonzero probability from one that is actually zero. That's why it's safe to say that Lightning Mace cannot create an infinite attack loop on its own, even though there is no finite maximum number of attacks it can generate.

Gan The Grey

2009-11-30, 02:11 AM

Essentially, yes. The catch is, though, that limits work both ways. That's what I've been getting at with the plain Lightning Mace example. The probability of rolling X many natural 20s in a row approaches a limit of 0% very rapidly as X approaches infinity--but it never actually reaches 0, so Lightning Mace always has a nonzero possibility of generating an infinite number of attacks. Just as the self-reinforcing attack loop always has a nonzero possibility of terminating after a finite number of attacks (if you have a small number of attacks to begin with, there is a very noticeable chance that it will terminate simply by starting with multiple 1s--but once you reach enough attacks, the probability gets very, very close to 0 and only gets closer). In both cases, though, as you continue the sequence towards infinity it is for all practical purposes impossible to distinguish that nonzero probability from one that is actually zero. That's why it's safe to say that Lightning Mace cannot create an infinite attack loop on its own, even though there is no finite maximum number of attacks it can generate.

Bah. I don't think you can just disregard something that is obviously there, regardless of how small the chance is. That seems like bad math to me, sir. Otherwise, no one would ever win the lottery, and life would have never evolved.

Fortuna

2009-11-30, 02:20 AM

Very well then. We shall stop teaching calculus, resort to numerical approximations of all rates of change, and make a large amount of applied science a great deal more difficult. You do that in all of these things, so it obviously works somehow.

Gpope

2009-11-30, 02:25 AM

Bah. I don't think you can just disregard something that is obviously there, regardless of how small the chance is. That seems like bad math to me, sir. Otherwise, no one would ever win the lottery, and life would have never evolved.

If you take an unlimited number of d20 rolls, there is a nonzero possibility of rolling an infinite number of any value from 1 to 20. It is possible you will roll nothing but 1s. It is also possible you will never roll a single 1, ever, even if you roll infinitely many times. In both cases, we can disregard these outcomes because their probabilities are infinitesimal--the chances are not merely "very small", they are so small that it is literally impossible to calculate how small they are other than to say that they are effectively 0.

Gan The Grey

2009-11-30, 02:37 AM

Very well then. We shall stop teaching calculus, resort to numerical approximations of all rates of change, and make a large amount of applied science a great deal more difficult. You do that in all of these things, so it obviously works somehow.

Yes, it works so very well in fact that we've unlocked the secrets of the universe already. Oh, wait.

So what? Disregarding small numbers might work in some cases, but that doesn't mean it works here. BUT. Maybe I'm just inclined to strongly disagree with someone who doesn't possess any tact. Who knows?

Anyway, thanks for arguing with me Gpope. You present a strong case, and I totally get your point of view. I just...can't see it that way. I can't disregard those chances, however small. It just feels wrong to me, like eating a peanutbutter and mayonnaise sandwich. Just...ewwww. :smallwink:

/end me

Gpope

2009-11-30, 02:45 AM

Anyway, thanks for arguing with me Gpope. You present a strong case, and I totally get your point of view. I just...can't see it that way. I can't disregard those chances, however small. It just feels wrong to me, like eating a peanutbutter and mayonnaise sandwich. Just...ewwww. :smallwink:

Saying that infinitesimal values can't be taken to equal 0 is one thing, and it does make intuitive sense. The problem is that the ramifications of doing so are self-contradictory. There is a nonzero chance of the chain terminating and a nonzero chance of the chain continuing. You're saying that the chance of the chain continuing can be disregarded (even though it gets arbitrarily close to 100% as you get high enough along the chain), yet the chance that the chain terminates cannot be disregarded (even though it gets arbitrarily close to 0%).

And like terminally sick points out, technically it's true--both possibilities have to be accounted for. The chain is simultaneously guaranteed to end and guaranteed to not end. The only true logical conclusion that can be drawn is that talking about probabilities in an infinite sequence gets silly awfully fast. :smallwink: But mathematically we can take illogical concepts like working with values of infinity and manipulate them to get useful, logical results.

Heliomance

2009-11-30, 04:26 AM

I'm not going to get into math to argue my point any further, because, as I've said already, math is not the problem. There is nothing wrong with your math. Your equations are well-thought out and precise. I completely understand what it is you are trying to say, I just believe you are misapplying evidence, or rather, apply the wrong evidence to the argument. I'm not the only one that has said this.

All I have to do to prove that this sequence will fail is to posit a string of 1's large enough to end the sequence. That's it. Within infinite bounds, all things that are possible will eventually occur, and as failure is a possibility, it will eventually occur, and the number of attacks will thusly fail to achieve infinity. I don't need math to show this, and this cannot be ignored either.

The thing about science is, you can prove your theory correct a million times, but it only takes one failure to prove it wrong. Just because your math comes out the way you want it to doesn't make it right. I'm sure you can say this goes both ways, and you would be right.

Actually, maths is not science. In science, it's impossible to prove a theory true. You can merely show by experiment that it appears to be the case, and that nothing has disproved it yet. In maths, you can prove a result. You can prove it rigourously and beyond all doubt. The fact that the maths comes out right does indeed make it right.

I think you and I are arguing two different things actually. Correct me if I wrong here, but I think you are arguing for the potential to reach any given number you supply, whereas I am simply arguing for achieving an endless quality.

As I pointed out earlier, being able to reach any given number you supply is the exact mathematical definition of going to infinity. Infinity is pretty much defined as "Give me any number you like, no matter how big. Infinity is just a little bit bigger."

I think, here, we are going to have to agree to disagree. Correct me if I'm wrong, but you are saying that even given an endless amount of time, not all possible circumstances will be reached, whereas I'm saying that given an endless amount of time, all possible circumstances will occur. Your opinion holds much more water than mine even if you consider a time span equal to the life of the universe, but time is not an issue within our DnD example, as all attacks will somehow be resolved in a span of six seconds. Really, I think this comes down to a difference of opinion that can't really be solved at all. :smallsmile:

No, we're saying that given an endless amount of time, all possible circumstances will be reached, but that that might not be enough to end the chain. Let me put it this way - assuming the chain goes on long enough, at some point I will roll 50 squigajillion natural 1s in a row. Notcontesting that. What we're saying is that by the time you roll 50 squigajillion natural ones in a row, there's a pretty good chance that you'll need 50 squigajillion and one, or more. What we're saying is not that given enough attacks, you won't roll an arbitrarily high number of natural 1s. What we're saying is that there'sa 5/9 chance that none of those arbitrarily long strings will be enough to end the chain at the point at which they occur.

The odds of your first 50 squigajillion rolls being natural 1s are incredibly small. The odds of you rolling 50 squigajillion natural 1s by the time you've generated 50 squigajillion spare attacks are also pretty small. Pat a certain point, it's vastly more likely than not that you will have more attacks than the number of natural 1s you get in a row. For this reason, although the cumulative probability of the attack terminating as you keep rolling never reaches 0, it also never reaches 1. It's a sum to infinity with a finite limit, just like the sum 1+1/2+1/4+1/8+...

As the number of attacks you make tends to infinity, the probability of any given attack ending the sequence tends to zero. As the number of attacks tends to infinity, the probability of any given attack or any of the attacks before it ending the sequence tends to 5/9.

Signmaker

2009-11-30, 08:39 AM

Actually, maths is not science. In science, it's impossible to prove a theory true. You can merely show by experiment that it appears to be the case, and that nothing has disproved it yet. In maths, you can prove a result. You can prove it rigourously and beyond all doubt. The fact that the maths comes out right does indeed make it right.

Minor Nitpick: This is technically wrong as a premise. 'The fact that the maths comes out right' proves that your mathematical model is right. That doesn't necessarily prove your case right if your model does not adequately match the problem.

But eh.

Krimm_Blackleaf

2009-11-30, 08:46 AM

I feel as though I've aided in making a monster.

His name will be Chav'Xaa of the Lost Heresy and he will gleam like a wildfire. Behold my horrific creation!

http://th08.deviantart.net/fs9/300W/i/2006/144/7/9/THE_SUN_by_Nny2.jpg

He's also the sun.

Heliomance

2009-11-30, 12:46 PM

Minor Nitpick: This is technically wrong as a premise. 'The fact that the maths comes out right' proves that your mathematical model is right. That doesn't necessarily prove your case right if your model does not adequately match the problem.

But eh.

However, in this case the problem is simple enough (simple being a relative term - the maths is horrible) that we can model it perfectly. The only factor that could possibly affect it that's not being taken into consideration is whether you're using a fair d20 or not, which I would think most people would agree is irrelevant to the theoretical exercise.

Indoran

2009-11-30, 01:20 PM

Also, if this works, can you threaten a critical hit on a natural 1?

Better Lucky than good... in the rogue's 3.5 splatbook

but It will eat a couple of feats in requisites... IMHO NOT worth it

Minor Nitpick: This is technically wrong as a premise. 'The fact that the maths comes out right' proves that your mathematical model is right. That doesn't necessarily prove your case right if your model does not adequately match the problem.

But eh.

On a side note falsiability is not the only criteria for science, but I concur with the opinion that Math is not a science. Math is beyond science as much as philosophy and art are. In fact, math is a branch of Logics which in turn are a branch of philosophy.

Science is not the only valid way to represent and study the world. It's just the socially sanctioned and in pop culture the idealized way to do it.

AgentPaper

2009-11-30, 01:32 PM

Take for example a series of coin flips, straight 50/50 probability, and see how many times in a row it comes up heads. There is never at any point any guarantee that the streak will stop, no guarantee that the coin will ever come up tails. There is always a nonzero probability that it will come up heads one more time. It is impossible for a number X to exist such that the probability of getting X many heads in a row is 0, yet because the limit of the probability rapidly approaches 0 as X increases we can confidently say that the probability of getting infinitely many heads in a row is 0 even though it is always possible to continue flipping heads.

The limit of the possibility of getting infinitely many heads is 0; there is 0% chance of reaching infinitely many heads, even though the chance of the streak continuing is always higher than 0..

You don't seem to understand what infinity is. If you've flipped a coin an infinite amount of times, then it's come up heads an infinite amount of times, in a row, an infinite amount of times. It's also come up tails an infinite amount of times in a row, an infinite amount of times. Hell, it's even come up tails an infinite amount of times in a row, and then heads an infinite amount of times in a row, and then tails again and back and forth another infinite amount of times.

Once again: INFINITY IS NOT A LARGE NUMBER. You cannot apply normal math to it. Yes, you have a 5/9 chance of reaching any real number. For all intents and purposes, it may as well be an infinite loop. That doesn't mean it IS an infinite loop, though, and it never will be.

Anyways, I'm done here, unless you guys want to actually explain how something that has any chance of happening could possibly NOT happen after an infinite amount of time, then I'll listen, but if you're just going to keep throwing numbers and big, meaningless words at us and demanding we talk the way you do back, then that's fine too. :smallsigh:

Heliomance

2009-11-30, 01:36 PM

Once again, I ask you to look over the definition of tending to infinity I posted earlier. If there is a 5/9 chance of exceeding any real number, then there is a 5/9 chance of going to infinity, because that's what going to infinity means.

You're right in that you can't apply normal maths to infinity. What you can do, however, is apply the maths specifically designed to work with infinity to infinity. Which is what we have been doing.

AgentPaper

2009-11-30, 01:38 PM

Once again, I ask you to look over the definition of tending to infinity I posted earlier. If there is a 5/9 chance of exceeding any real number, then there is a 5/9 chance of going to infinity, because that's what going to infinity means.

You're right in that you can't apply normal maths to infinity. What you can do, however, is apply the maths specifically designed to work with infinity to infinity. Which is what we have been doing.

it has a 5/9 chance of reaching any real number, not exceeding it. After that, it has some very low non-zero chance of reaching the next higher number, whatever that is.

Gpope

2009-11-30, 01:45 PM

You don't seem to understand what infinity is. If you've flipped a coin an infinite amount of times, then it's come up heads an infinite amount of times, in a row, an infinite amount of times. It's also come up tails an infinite amount of times in a row, an infinite amount of times. Hell, it's even come up tails an infinite amount of times in a row, and then heads an infinite amount of times in a row, and then tails again and back and forth another infinite amount of times.

Coming up heads an infinite number of times in a row precludes ever coming up tails. It can come up an arbitrarily large number of heads in a row--or it may never come up heads at all. It is theoretically possible to keep flipping that coin an infinite number of times and never come up heads. Of course, using that BAD UGLY MATH it is possible to show that this is effectively impossible even though there is a nonzero probability of it happening.

it has a 5/9 chance of reaching any real number, not exceeding it. After that, it has some very low non-zero chance of reaching the next higher number, whatever that is.

It follows that if there is at least a 5/9 chance of reaching any real number, then for any given number X there is also at least a 5/9 chance of reaching X+1. Therefore it has a 5/9 chance of exceeding any real number.

ocdscale

2009-11-30, 03:59 PM

This is the argument I keep seeing: "infinite means everything will happen, so unless you have a 100% guarantee that you'll never miss, it won't go infinite."

I disagree.

Let me describe the build as clearly as I can:

It generates a certain amount of attacks.

If an attack crits, the build gets two more attacks.

If the attack misses, the build does not get an additional attack.

Nay-sayers will think to themselves: "I know what infinity is, eventually this build will miss enough times in the row and thus cannot ever achieve an infinite series of attacks." They will think this is the end of it.

They don't realize that the chance of total failure gets infinitely small, and they certainly don't bother to conceptualize the interaction between infinite time and infinitely small failure rates.

Here, let me prove with very basic mathematics and logic why the naysayers are wrong.

I have a magic bean. This bean has a very peculiar existence. In the first second of its existence, the bean has a 10% chance of ceasing to exist entirely. That is to say, after being created, only 90% of these beans survive for more than a second.

The second second of existence is even stranger! The beans that survive to their second second have about a 1% chance of ceasing to exist. That is to say, after being created, only about 89% of these beans survive for more than two seconds.

The third second of existence follows the pattern. That is to say, after being created, only about 88.9% of these beans survive for more than three seconds.

And so on, in that pattern.

Only about 88.88888889% of these beans survive for more than ten seconds.

Only 88.888[lot more 8s]889% of these beans survive for more than 10^100 seconds.

Does everyone see the pattern here?

No matter what time you posit, it's easy to see that over 88% of the beans make it to that time.

My question: As time goes to infinity, is it possible for any of these beans to still exist? In other words, is it possible for any of these beans to survive an infinite number of seconds?

Please note, at each second, there is some probability that a bean stops existing. At time 1, it was 10%, at time 2 it was about 1%, and it gets lower and lower (but never quite reaching 0).

Naysayers will immediately latch on and say "there's always a non-zero failure rate, therefore there's always a possibility that the bean will cease to exist, therefore no bean will ever exist for an infinite number of seconds."

Does that agree with rules governing the beans? The beans seem crystal clear, although the survival rate keeps getting lower over time, there's always more than an 88% chance that a bean survives to time N because of the interaction between the infinitely small failure rates and infinite time.

Edit: Thankfully, Kalirren below provided the math and a very nice explanation.

Kalirren

2009-11-30, 04:15 PM

The only person in the past two pages who has known what (s)he has been talking about is Heliomance. (edit: and ocdscale, who ninja'ed me. gj.) Maybe we should try to get everyone on the same mathematical page here.

So what we really mean by "X will happen eventually" is that the the limit of the probability of X happening by time n as n goes to infinity goes to 1; this is equivalent to the probability of X not happening by time n going to zero.

So we can say that if you keep rolling a fair d20, you will eventually get a natural 1. The probability of never getting a natural 1 is by time n is 19/20^n. The limit of 19/20^n as n goes to infinity is 0.

In fact, you can ask, "Will a string of N natural 1s occur eventually?" Well, the chance of it occurring at any point is just 1/20^N. So the chance that a string of N natural 1s never occurs by time n is just (1-1/20^N)^(N-n+1), and the limit of this as n goes to infinity is 0. So the answer is, "yes, a string of natural 1s of arbitrary length will occur eventually."

The problem the intuitive arguments that have been presented by Gan the Grey, SensFan, and AgentPaper is the following, and I will set this aside in its own paragraph for emphasis:

The fact a string of natural 1s of arbitrary length will eventually occur somewhere in the sequence of rolls does not imply that the sequence of attacks is expected to terminate.

What do I mean by that? Consider now the following game:

Step 1: Roll a d20.

Step 2: If you get anything but a natural 1, go back to step 1.

Now you can ask, "when can we expect this process to terminate?" This question can be answered. If the probability that the process terminates on the n'th roll is denoted by P(n), then the expected value of the number of rolls before termination is

SUM_n [n * P(n)] = SUM_n [n*19^(n-1)/20^n]

because P(n) = (probability of not terminating on the n-1 steps before roll n) * (probability of terminating on roll n) = (19/20)^(n-1) * 1/20.

Now since the denominator grows faster than the numerator, that sum converges to some finite value. I'm too lazy to calculate the actual expected value, but trust me that it exists, and you can plug it into Mathematica or something and get a finite answer out of it if you want. I suspect that the answer is 20.

But, I find that I get a much better intuitive understanding of the situation by asking the question, "What's the expected number of additional rolls I gain for each roll I make?" This question is easy to answer: there are 20 possible outcomes, and on 19 of them I get to attack again, and on 1 of them I don't. So the expected number of additional attacks per attack is (19*1+1*0)/20 = 19/20. Note that this is less than 1. That is to say, for each attack we make, we don't expect to get that attack back. If we continue the process indefinitely, we'll eventually run out of attacks.

So now consider the process that we've been discussing:

Step 1: Roll an attack (and resolve its damage.)

Step 2: If you get a 6 or above on the attack roll, go to Step 1 -twice-.

Now you can ask the question "when can we expect this process to terminate?" As we've discussed before, if n is a natural number, adding up all of the probabilities that the process terminates on the n'th attack, over all n, gives you 4/9.

Thus the expected value of the number of attacks you make is

SUM_n [n * P(n)] = (something finite) * 4/9 + (something undefined in the real numbers) * 5/9 = undefined.

Thus the question "when can we expect this process to terminate?" -has- an answer: specifically, "never." I know this result is counterintuitive, but it's true. If that didn't help you understand, the next suggestion might.

We can, again, ask the question, "How many additional attacks do I get for each attack that I make?" Well, just like last time, there are 20 possible outcomes. On 15 of them (6 and above) I get 2 extra attacks. On the rest of them I whiff and get 0 extra attacks. The expected number of extra attacks I get per attack I make is therefore (15*2+5*0)/20 = 1.5. This is -greater- than 1. That is to say, for each attack I make, I can expect to get more than 1 attack back. Note that this already incorporates our failure chance.

It follows that as I keep attacking, I should expect to get more and more attacks, and/because these attacks accumulate faster than the misses do. Sure, I might get unlucky on my "first few tries" and lose out. That's what the 4/9 probability of terminating means. But that's not what one would expect would happen. One would -expect- that the number of attacks continues to grow without bound. And the more attacks I have under my belt, the less I am vulnerable to "a few" consecutive losses. (And by "a few" or "the first few" I really mean "any finite number you care to name.")

Note that all of this argument really has very little to do with whether or not a sequence of natural 1s of arbitrary length will crop up eventually. It will. No one is contesting that. What we are trying to let you understand is that it also doesn't matter. Yes, you can go ahead and compute the probability that you get a sequence of natural 1s of length N, and you can compute the probability that a sequence of natural 1s of length N terminates the attack sequence, and you can multiply those two probabilities together and add them over all N and take the limit as N goes to infinity to get the probability that the attack sequence terminates at all, and you will get 4/9. The probability that the attack sequence terminates does not go to 1: therefore, one cannot expect the attack sequence to eventually end.

term1nally s1ck

2009-11-30, 04:47 PM

The only problem is, I've already found a proof that BOTH arguments are true.

My proof, as stated above, that the probability of it terminating in finite time is 4/9.

And a proof which is described (but not rigorously stated, you can find that on there internet elsewere) here. (http://en.wikipedia.org/wiki/Random_walk)

Basically, given a random walk, where you start at n, and at each step you have a probability p of subtracting 1, and a probability q (<1-p) of adding 1, and a probability 1-p-q of staying at the same number, the walk has probability 1 of reaching all points on the number line, including 0, where in this case, the process would end.

So simultaneously, there's a probability of 4/9 that the process ends in a finite time, and a probability 1 that it ends.

Infinity is WEIRD.

But anyway, for those who want to see the build, it's currently Fighter 2 Swordsage 6 DoD 8, has a crit range of 9-20, and can take 11 on attack rolls. This one is TRULY infinite.

ocdscale

2009-11-30, 05:41 PM

The only problem is, I've already found a proof that BOTH arguments are true.

My proof, as stated above, that the probability of it terminating in finite time is 4/9.

And a proof which is described (but not rigorously stated, you can find that on there internet elsewere) here. (http://en.wikipedia.org/wiki/Random_walk)

Basically, given a random walk, where you start at n, and at each step you have a probability p of subtracting 1, and a probability q (<1-p) of adding 1, and a probability 1-p-q of staying at the same number, the walk has probability 1 of reaching all points on the number line, including 0, where in this case, the process would end.

I'm pretty sure that your conclusion does not follow. A random walk with static probabilities is not analogous to the situation presented.

Edit: Removed probably erroneous reasoning to avoid confusion, but leaving conclusion. Will post below after some thought and looking over the proof.

Edit 2: Or the proof is not applicable to this situation.

term1nally s1ck

2009-11-30, 05:59 PM

Try again, the walk is based on the number of attacks, and each step is 1 attack.

I make 1 attack, I have a 3/10 chance of not critting (-1 attacks), a 133/200 chance of critting and confirming (+1 attack), and a 7/200 chance of critting but not confirming (0 change.)

Situation is identical, if you count by attack.

Jastermereel

2009-11-30, 10:52 PM

Does that agree with rules governing the beans? The beans seem crystal clear, although the survival rate keeps getting lower over time, there's always more than an 88% chance that a bean survives to time N because of the interaction between the infinitely small failure rates and infinite time.

While there may always be more than an 88% chance that a bean survives to time N, that doesn't mean that a particular one will. That is, even if an attack chain can go on infinitely, that doesn't mean that it will this time.

Seatbelt

2009-11-30, 11:37 PM

Of course the obvious solution to the issue is that the attack cannot go on to infinity, because eventually the DM will get annoyed and kill you.

lvl 1 sharnian

2009-12-01, 12:14 AM

Aura of perfect order.

We have a winner. lets see, drop the weaponmaster, shove in swordsage 6, stick with a 9-20 crit threat, and take 11s. All you need to do now is buff your to-hit, which is less difficult. Just wow.

I take it this build would be accepted by all as an infinite loop, given that you hit on an 11 or less?

@Gan: Yes, there is some merit in having intuitive ideas about infinity, especially since you can get completely opposite ideas back, even with rigorous proofs.

http://en.wikipedia.org/wiki/Random_walk

Ironically, we're both right. There is simultaneously a <1 chance that the sequence terminates at a finite number, and a guarantee that the sequence terminates. I'm confused as hell how this is possible, but infinity is by its nature, counterintuitive.

Apologies to all for dsimissing so many people's ideas, apparently two completely opposite views can be correct in this case.

As much as I'd like to accept this, I believe Aura of Perfect Order only allows 1 roll a round to be treated as a natural 11 so while it could start the chain, it wouldn't be able to continue it as the attacks continue within the same round.

Perhaps when this occurs, tiny shockwaves erupt all around the body, severing muscles and organs or something.

AgentPaper

2009-12-01, 12:45 AM

@Kalirren and odcscale

Yes, I understand the model you're pushing. Thank you for assuming I was ignorant of it after saying I had understood it, and claiming that everyone who doesn't agree with you doesn't know what they're talking about. I'll ignore that bit for your sake.

All of that math proves that you have X% chance of reaching any real number. (about 5/9 in this case) Infinity is not included in any real number. This is why the attacks can become arbitrarily high, to the point where counting them is impossible. Name any really high number you want, and there's a 5/9 chance of you getting that many attacks or more. The problem is, infinity is not a number.

If you've rolled a dice an infinite amount of times, that means you've rolled a 1 an infinite amount of times in a row, which means that the chain has been broken. Since the chain has been broken, that means it's ended, which means it can't be infinity, since infinity has no end by definition.

The same thing goes for your bean example. If you wait any amount of time, there will be about an 89% of the beans left at any given time. If you then wait an infinite amount of time, ALL of the beans will be gone, because at any step in time, each bean has a chance of ceasing to exist, and since that chance (how much the chance is is irrelevant) is happening an infinite amount of times, it has a 100% chance of happening due to the definition of infinity.

Gpope

2009-12-01, 01:00 AM

If you've rolled a dice an infinite amount of times, that means you've rolled a 1 an infinite amount of times in a row,

Except it doesn't. You've rolled 1 an arbitrarily high number of times in a row, but as has been repeatedly established by people on both sides of the argument, an arbitrarily high number is not the same thing as an infinite number. An infinitely long streak of 1s is theoretically possible, since any given roll has a fixed chance to continue coming up 1, but it is infinitely improbable (whereas a streak of arbitrary length has a finite probability and in theory will eventually happen over an infinite span of time).

More importantly, an infinitely long streak of 1s precludes the possibility of an infinitely long streak of any other number; there's no reason that rolling an infinite streak of 1s should be any more likely than rolling an infinite streak of 2s, or 3s, or 20s, and yet you can have only one infinitely long streak (the first streak would have to have a finite end before a second streak could begin.)

Heliomance

2009-12-01, 03:23 AM

All of that math proves that you have X% chance of reaching any real number. (about 5/9 in this case) Infinity is not included in any real number. This is why the attacks can become arbitrarily high, to the point where counting them is impossible. Name any really high number you want, and there's a 5/9 chance of you getting that many attacks or more.

For the Nth time, that is precisely how going to infinity is defined. I've already given the strict mathematical definition and translated it into English. If for any number no matter how high, we can get more attacks than that, then we can get infinite attacks by definition.

The same thing goes for your bean example. If you wait any amount of time, there will be about an 89% of the beans left at any given time. If you then wait an infinite amount of time, ALL of the beans will be gone, because at any step in time, each bean has a chance of ceasing to exist, and since that chance (how much the chance is is irrelevant) is happening an infinite amount of times, it has a 100% chance of happening due to the definition of infinity.

The definition of infinity doesn't actually contain anything about containing every possible result in it. It's entirely possible for a function to tend to a finite value as x tends to infinity, and there are plenty of examples. (e^x-e^(-x))/(e^x+e^(-x)), for example, tends to 1 as x tends to infinity. The example with the beans, if we take f(t) as the proportion of beans left after time t, tends to 8/9 as t tends to infinity.

Kalirren

2009-12-01, 10:09 AM

Try again, the walk is based on the number of attacks, and each step is 1 attack.

I make 1 attack, I have a 3/10 chance of not critting (-1 attacks), a 133/200 chance of critting and confirming (+1 attack), and a 7/200 chance of critting but not confirming (0 change.)

Situation is identical, if you count by attack.

Well, but then the random walk is biased. I thought that any given state (including 0, or the state corresponding to any finite number for that matter) is only guaranteed to be recurrent if the random walk is unbiased, and is transient otherwise. So our first proof that the chain of attacks marches off to infinity would be correct...right?

Signmaker

2009-12-01, 03:48 PM

there's no reason that rolling an infinite streak of 1s should be any more likely than rolling an infinite streak of 2s, or 3s, or 20s, and yet you can have only one infinitely long streak (the first streak would have to have a finite end before a second streak could begin.)

That statement clinches the argument for finite numbers, I would think. That is to say, there is a distinct probability of reaching any finite number of attacks with the given constraints.

I think the issue that some people have (I am quite easily one of them) is the knowledge that if any finite number of 'steps' has a distinct probability of stopping right then and there, that progression to infinity should be impossible, because eventually probability will not behave as the norm, using the a sort of twist on the Gambler's Fallacy to interpret that there will be periods in which a string of finite ones will terminate all of the remaining attacks despite 'expected' growth. After all, if we're talking about infinity, there are an infinite number of intervals in which this improbability can occur.

However, one can easily state that the step progression 'grows without bound' and 'tends to infinity'. The idea behind this is that while the above may occur more readily for low numbers, it gets progressively more difficult the more steps occur. So much so that the chance at extremely high numbers more or less 'vanishes'. This would be where series come in to play, and that is how the 5/9 number is achieved; most of that probability comes from relatively low finite numbers. After all, calculus teaches us at least that much. Amusingly, interpretation through series models the aforementioned Gambler's logic quite efficiently, despite protests.

But of course, actually interpreting whether or not there can exist an infinite number of attacks at a readily determinable probability is confusing, especially without the knowledge of upper-level math courses. It's especially difficult to teach the concept of series to those not mathematically versed (often still to those that are, such as myself).

Yes, there was no real point to my post. I just felt like summing up the current hurdles that needed to be jumped.

term1nally s1ck

2009-12-01, 04:49 PM

Well, but then the random walk is biased. I thought that any given state (including 0, or the state corresponding to any finite number for that matter) is only guaranteed to be recurrent if the random walk is unbiased, and is transient otherwise. So our first proof that the chain of attacks marches off to infinity would be correct...right?

Actually, I've got my hands on a proof that it's guaranteed to terminate even in the biased case. Very very clever maths, and very complicated, but works. If you're on a uni connection, you may have free access, otherwise, it's pay to view.

Kalirren

2009-12-01, 05:39 PM

Slide me the link then, I'll check it out next time I'm on campus.

It just seems strange that you would be forced to terminate, but you -could- think of it as Gambler's ruin by playing a game with finite expected value against a bank with an infinite bankroll. But if the probability is 1 that you reach 0 by step N as N goes to infinity, and you terminate when you get to 0, how can the sum of probabilities over all the natural numbers n that you make exactly n attacks sum to anything other than 1? We know it doesn't. Or at least you claim so: we'd need to look at your 5/9 proof. (Correction: I suppose we know it doesn't for the case where the entire pool of attacks doubles if -any- of the attacks hit and crit, in which the expected value of the game is not constant...the theoretical formation wasn't quite right...)

Dairun Cates

2009-12-01, 05:59 PM

And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.

Once again, to reiterate what everyone else has said, this is not infinite. It's arbitrarily high. If there's ANY chance it can end, it cannot go infinite. There is not a magical limit that you can break on exponential numbers that cause infinity. If you're getting infinity, you are suffering a rounding error.

When discussing infinity, EVERY last digit becomes significant. This means that if a repeating number comes up, you'd have to map the decimal places out to infinity to do the math.

You can make it so each attack that successfully hits makes it more and more likely the chain will not end on the next iteration, but it cannot and will not go infinite. You cannot remove the inevitability that you will, given infinite time, eventually roll all nat 1's.

Signmaker

2009-12-01, 06:07 PM

Time for some constructive input (for a change, as opposed to babbling).

Soon to be presented is more or less a tree diagram which should adequately model relative probabilities for finite numbers. In observing the rate of change between certain intervals, a convergent probability of failure at any stage should be attained (whether or not this works for infinity is not in my realm to know).

Stage #

1. A d20 is rolled.

2. For any value other than 1, perform these instructions twice.

In DnD terms, this would be equivalent to fumbling an attack on a 1, and gaining two new attacks plus resolving your current attack on any number other than 1.

Feel free to draw the following out.

Stage 1

There is a 1/20 chance of stagnation in respect to this stage and this stage only.

There is a 19/20 chance of two branches occurring.

So. We have a 5% (1/20) chance of the attack stream ending at stage 1.

Stage 2

There is a 1/400 chance of stagnation in respect to this stage and this stage only.

There is a 38/400 chance of two branches occurring.

There is a 361/400 chance of four branches occurring.

Therefore in respect to this stage only, there is a .25 percent chance of failure.

Now then, the probability of not making it to stage 3 (that is to say, to stop at either stage 1 or stage 2) is 1/20 + (19/20 x 1/400). This is equal to 419/8000, or 5.2375%. So we have a probability increase of .002375 from the previous stage.

Stage 3

Stage 3 Preface:

It gets a bit...expansive at this stage. Depending on the results of stage two, there can be either two or four rolls resolved in stage 3. For the sake of brevity, I'll only make note of the failure percentages.

In the occurrence of two rolls, there is a 1/400 chance of failure with respect to this stage only.

In the occurrence of four rolls, there is a 1/160000 chance of failure with respect to this stage only.

Therefore in respect to this stage only, there is a .0243% chance of failure.

The probability of failure before stage 4 is then:

(1/20) + (19/20 x 1/400) + ((38/400 x 1/400) + (361/400 x 1/160000)) = 3367561/64000000, or 5.2618%. This is a probability increase of approx. .000243 from the previous stage.

So what can be initially gleaned? That the chance of failure does increase with time, but really freaking slow. That is to say, it should converge before 1.

Let's try the following for the sake of simplicity. I flip a coin. Heads, nothing happens. Tails, I get two more flips.

Stage 1: 1/2 chance of total failure, 1/2 chance of two more flips.

Stage 2: 1/4 chance of stage failure, 1/4 chance of two more flips, 1/2 chance of four more flips. So we're at 75% failure by stage two (or before stage three, whichever mental reasoning you prefer)

Stage 3: Stage failure is now 1/16 (two-flip failure) + 1/32 (four-flip failure) = 3/32. So we're at 84.375%. Were we considering the geometric series (1/2+1/4+1/8.....=1), we would currently be at 87.5. We are not, which implies that this example converges before 1. It has quite a high chance of failing, but failure is not 'guaranteed'.

ocdscale

2009-12-01, 06:12 PM

Once again, to reiterate what everyone else has said, this is not infinite. It's arbitrarily high. If there's ANY chance it can end, it cannot go infinite. There is not a magical limit that you can break on exponential numbers that cause infinity. If you're getting infinity, you are suffering a rounding error.

When discussing infinity, EVERY last digit becomes significant. This means that if a repeating number comes up, you'd have to map the decimal places out to infinity to do the math.

You can make it so each attack that successfully hits makes it more and more likely the chain will not end on the next iteration, but it cannot and will not go infinite. You cannot remove the inevitability that you will, given infinite time, eventually roll all nat 1's.

If the chance the attack string ends on the first round is 1/4

And the chance the attack string ends on the second round is 1/8

And the chance the attack string ends on the third round is 1/16

And so on in that fashion.

Once started, what is the probability this attack string will end?

In other words, what is 1/4 + 1/8 + 1/16 + 1/32 and so on into infinity?

I say the sum of all the probabilities of failure, as the rounds go into infinity, is 50%. Therefore there is a 50% chance of failure, and a 50% chance the attack string will not fail.

You argue there is a 100% chance this attack string will fail eventually because every round has a non-zero chance it will fail.

I invite you to show me how to add 1/4 + 1/8 + 1/16 ... in order to sum it up to 1 (100%).

Signmaker

2009-12-01, 06:24 PM

Another thing to address: infinity as a 'place', that constantly moves. Those who say "you will eventually roll enough ones" more or less use this as proof that the stream will end, because you cannot 'catch up' to infinity.

Back to the impish pixie riddle I threw forward so long ago. Two balls in, lowest ball out at each interval. At infinity you're left with none. However, were the riddle to be stated that the highest ball were removed, you'd have an infinite number of balls left at infinity, the odd real set of numbers. That is to say, though the two 'actions' seem the same (2 in, 1 out), their treatment at infinity is quite different, which goes to show the danger of treating infinity as a moving place that you analyze at. This is why you will hear 'tends to infinity' or 'grows without bound' rather than acknowledging 'hitting' infinity, because it's frankly not practical and sometimes logically contradictory.

Heliomance

2009-12-01, 06:28 PM

Once again, to reiterate what everyone else has said, this is not infinite. It's arbitrarily high. If there's ANY chance it can end, it cannot go infinite. There is not a magical limit that you can break on exponential numbers that cause infinity. If you're getting infinity, you are suffering a rounding error.

Once again, to reiterate what everyone who understands the maths has said, you're wrong. Infinity does not work how you think it works.

Dairun Cates

2009-12-01, 07:08 PM

You argue there is a 100% chance this attack string will fail eventually because every round has a non-zero chance it will fail.

I invite you to show me how to add 1/4 + 1/8 + 1/16 ... in order to sum it up to 1 (100%).

The chance of a coin flip being heads is 50%. The chance of a coin flip being heads a second time is 25% because 1/2 times 1/2 is 1/4th. This is basic math.

Let's put this in decimals though. .5 x .5 = .25

So, let's the growing problem here. The chance of SUCCESS on turn one and two are 3/4 and 7/8 respectfully.

So, the probability of both events occurring is...

3/4 x 7/8 = 21/32

or .75 x .875 = .65625

That's the total probability that both of those outcomes will occur.

So, we take 21/32 (the probability of the first two turns) and multiply it by the chance of success 15/16. So..

21/32 x 15/16 = 315/512

Or...

.65625 x .9375 = .615234375

Then...

315/512 x 31/32 = .59600830078125 or 9765/16384

Eventually, the numbers start getting cut off by your calculator as they get smaller and smaller, but to get a TRUE statistic, you MUST include every decimal.

So yes. The DIFFERENCE between each probability each turn is going to to be smaller and smaller because of the exponential effect, but at no point will the probability GROW, because you never have a 100% chance or more of success. That number keeps going down and down. There's ALWAYS a chance for success on the next go, but the chances of it to actually keep going in the long run gets lower and lower.

Of course, the simple proof that you will eventually roll all ones. An infinite loop assumes infinite rolls. In infinite rolls, you will exhaust EVERY last possibility and permutation of dice rolling possible. One of those permutations is all 1's. You cannot remove this possibility. So, in your "infinite" rolls the possibility will occur and end the loop.

Just because you're less likely each turn to fail that turn, the chance is always there, and probability will catch up to you. Infinity cannot be caught up to. It is not an arbitrarily high number. It is a concept. You cannot ensure an infinite loop unless there is no other option. You HAVE to reach 100% probability to reach infinity. The numbers you're showing will add up to 1 only when you have done an INFINITE number of iterations.

In other words, until you reach 1/infinity as the current probability of failure, your fractions DO NOT add up to 1. Since that takes infinite rolls, one of them will inevitably fail because you have an infinite number of rolls to get there.

ocdscale

2009-12-01, 07:15 PM

For all those words, you didn't answer my question.

What is 1/4 + 1/8 +1/16 + 1/32 . . and so on, what is it equal to?

You don't even need to explain your answer or show your work, I just want to know what you think the answer is.

Edit: It occurs to me that maybe you did try to answer the question. If that is the case, then there is no need to reply to this post, your answer can stand for itself.

Dairun Cates

2009-12-01, 07:20 PM

For all those words, you didn't answer my question.

What is 1/4 + 1/8 +1/16 + 1/32 . . and so on, what is it equal to?

You don't even need to explain your answer or show your work, I just want to know what you think the answer is.

Edit: It occurs to me that maybe you did try to answer the question. If that is the case, then there is no need to reply to this post, your answer can stand for itself.

I know I don't need to, but to reiterate, it WILL hit 1, but only at an infinite number of iterations. Since you can quantify infinity as a number, it will never hit that number, just tend towards it. In the meantime, the answer will inevitably terminate. Since it has an infinite amount of time for that possibility to catch up (no matter how small) it will. The amount of iterations is infinite because that's how long it takes to catch up.

1/99999999999999999999999999999999999999999999999999 9999999999999999999999999999999 is not infinitely impossible. It's just extremely unlikely.

Signmaker

2009-12-01, 07:20 PM

So yes. The DIFFERENCE between each probability each turn is going to to be smaller and smaller because of the exponential effect, but at no point will the probability GROW, because you never have a 100% chance or more of success. That number keeps going down and down. There's ALWAYS a chance for success on the next go, but the chances of it to actually keep going in the long run gets lower and lower.

I'll address this due to the interaction with series. If the probability does not converge to 1 or 0, there is leeway, rather than absolute success or absolute failure. That's the entire point about infinite sums: there are certain types of sums (those that follow a distinct pattern, like 1/2^N) that will converge to a number. As I showed before, even the simple case of Heads Nothing Tails Two Flips does not converge to 1 or 0. Therefore there is leeway, a probability that flipping will commence unbounded.

If there is a 10% chance of failure before round 2, an 11% chance before 3, an 11.1% chance before 4, etc, you're left with with a probability approaching infinity, not a certainty.

Dairun Cates

2009-12-01, 07:25 PM

I'll address this due to the interaction with series. If the probability does not converge to 1 or 0, there is leeway, rather than absolute success or absolute failure. That's the entire point about infinite sums: there are certain types of sums (those that follow a distinct pattern, like 1/2^N) that will converge to a number. As I showed before, even the simple case of Heads Nothing Tails Two Flips does not converge to 1 or 0. Therefore there is leeway, a probability.

That's the point though, isn't it? If you flip another heads, so to speak, it continues, if you flip tails, it ends. Infinite attacks requires 100% as the sum of the probabilities. The fact that it will end does not require probability to hit 0 though. It merely requires that the other set does not equal 100%. Since that will never happen because infinity is infinite and that takes infinite iterations. The probability to roll not all 1's HAS to hit absolutely 0% for it to never happen in infinite iterations.

So yes, you'll die before you finish your rolls at that point (or your GM will just punch you), but given enough time, you'd eventually fail. It's an arbitrarily high loop, but it is not infinite.

ocdscale

2009-12-01, 07:25 PM

Dairun, it seems like you are sticking to your guns here.

EDIT: Incorrect wording of the game. Please see page 249 for the correct version.

So in my hypothetical game:

Start: Go to Round 1

Round 1: 25% chance of ending (1/4), otherwise go to Round 2

Round 2: 12.5% chance of ending (1/8), otherwise go to Round 3

Round 3: 6.25% chance of ending (1/16), otherwise go to Round 4

... and so on ad infinitum

Am I correct in stating that your stance is that:

There is always some chance of failure. Therefore, as the rounds grow towards infinity, it is inevitable that one round will result in failure and end the game. Therefore the chance of failure is 100% (eventually).

Dairun Cates

2009-12-01, 07:28 PM

It seems like you are sticking to your guns here.

So in my hypothetical game:

Start: Go to Round 1

Round 1: 25% chance of ending (1/4), otherwise go to Round 2

Round 2: 12.5% chance of ending (1/8), otherwise go to Round 3

Round 3: 6.25% chance of ending (1/16), otherwise go to Round 4

... and so on ad infinitum

Am I correct in stating that your stance is that:

There is always some chance of failure. Therefore, as the rounds grow towards infinity, it is inevitable that one round will result in failure and end the game.

Yes. It will take an arbitrarily high number of turns in some cases (more than a human will roll in his or her life), but since it takes infinite iterations to reach the end, inevitably one round will screw you.

Signmaker

2009-12-01, 07:30 PM

It merely requires that the other set does not equal 100%.

Incorrect premise. Tending towards infinity, you will have a success rate and a failure rate. Both are convergent numbers, and the two added together will equal one. Picture it as a tree, as a linear-shaped string of rolls, whatever. The series model the probability that a given rolling engine will terminate or continue unbounded. All those tiny, infinitesmal probabilities you're treating as certainties due to infinity? You're not looking at them the correct way.

Dairun Cates

2009-12-01, 07:39 PM

Incorrect premise. Tending towards infinity, you will have a success rate and a failure rate. Both are convergent numbers, and the two added together will equal one. Picture it as a tree, as a linear-shaped string of rolls, whatever. The series model the probability that a given rolling engine will terminate or continue unbounded. All those tiny, infinitesmal probabilities you're treating as certainties due to infinity? You're not looking at them the correct way.

Well, of course the two probabilities add up to 1, that's the very basis of probability, but one side has a continue condition and the other has a stop condition. The low probability needs to only occur once to completely end the loop. At this point, the "loss" side "wins" so to speak.

However, the "win" side can NEVER win because succeeding on one roll just leads to another iteration with a lower probability of losing on this hand specifically. There is no end state. If you win again, you just roll again. THAT'S the problem. In order for you to hit infinity, there has to be no quantifiable way to continue, you won infinite times. The equation tends to infinity, but it will never reach it.

The only way you can ABSOLUTELY ensure that you will never fail is to have an ABSOLUTE 0% fail rate.

Signmaker

2009-12-01, 07:42 PM

However, the "win" side can NEVER win because succeeding on one roll just leads to another iteration with a lower probability of losing on this hand specifically. There is no end state. If you win again, you just roll again. THAT'S the problem. In order for you to hit infinity, there has to be no quantifiable way to continue, you won infinite times. The equation tends to infinity, but it will never reach it.

The only way you can ABSOLUTELY ensure that you will never fail is to have an ABSOLUTE 0% fail rate.

Problem two. For probability, there is no 'hitting' infinity, but tending towards it. Rather than never approaching the end of the tunnel that is the natural number line, the relevant concept of infinity is to not stop running towards it. To continue unbounded is what matters. There are few absolutes, and this scenario is not one of them (flipping a coin and stopping the second you hit tails is one, though, as p=1). Perhaps this (http://www.cut-the-knot.org/Probability/infinity.shtml) might help.

ocdscale

2009-12-01, 07:44 PM

Yes. It will take an arbitrarily high number of turns in some cases (more than a human will roll in his or her life), but since it takes infinite iterations to reach the end, inevitably one round will screw you.

I apologize, I realize now that I incorrectly worded my game. Please let me know if this new game convinces you:

You press a button.

The button turns on a magic light.

The magic light is designed such that 25% (1/4) of the time, it turns off in the first second.

12.5% (1/8) of the time, it turns off in the second second.

6.25% (1/16) of the time, it turns off in the third second.

3.125% (1/32) of the time, it turns off in the fourth second.

and so on ad infinitum

If you will notice, if you make it to the 100th second, the probability it turns off that second is very low, but there is still a chance. The longer you last, the lower the probability the light will turn off, but there is always a chance.

Is it fair to say that your stance is "it is impossible for the magic light to shine forever because there is always a chance it will turn off, even if that chance is small, because in infinite time, eventually you will hit that chance." ?

Dairun Cates

2009-12-01, 07:50 PM

Is it fair to say that your stance is "it is impossible for the magic light to shine forever because there is always a chance it will turn off, even if that chance is small, because in infinite time, eventually you will hit that chance." ?

Exactly what I mean.

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