1. ## Door Guessing GAME

Lot's of people have been discussing the math and optimum strategies behind searching the doors of Kraagor's Tomb, and whether MitD is helping or hurting Team Evil with his sabotage. I'm decided to help said people put their money where their mouth is and came up with a game for them to try! Playable with a normal deck of cards and two people!

The first player plays as Redcloak. He wants to find the Gate hidden within the deck of cards (The gate is the Three of Hearts! Why the Three of Hearts? It was decided arbitrarily.) Each day (or turn), he will pick one card from the deck to look at and discard.

The second player plays as the Monster in the Darkness. He wants to sabotage Redcloak's efforts so that it takes a long time for the Gate to be found. He does so by taking cards from the deck and sticking them into the discard pile.

Redcloak can use any strategy he wants for choosing which card to look at; it doesn't have to be the top one! He can go from the bottom and work his way up the deck, he can choose the card under the top one each turn, he can put them on a dart board and throw a dart to pick, he can do things Xykon's way and pick one at random. Whatever he wants! The only catch is that MitD gets to watch him do so, and the deck cannot change order (so you can't shuffle it).

At the end of Redcloak's turn, MitD gets to discard as many cards as he wants without looking at them. He can also use any strategy for choosing which card (or cards) to discard. Redcloak does NOT get to see what strategy MitD is using.

After that, Redcloak takes another turn. This back and forth happens until Redcloak finds the Gate or all the cards have been discarded. When all the cards are discarded, shuffle the discard deck into the starting deck and keep going.

Now for the important bit; how do you win?

As Redcloak plays, keep track of how many 'days' (turns) pass by. Redcloak wants this number to be low, and MitD wants this number to be high.

When Redcloak finds the Gate, play another round with the roles switched. The second player now plays as Redcloak, and the first player now plays as the MitD.

After two rounds have been played, see which player found their Gate faster as Redcloak. That player is the winner!

Important Bits:
You can use any strategy you can come up with for choosing cards. This strategy cannot include "look at the card and pick the Three of Hearts". That's cheating! Only look at the card you want to pick if you're Redcloak, and don't look at any of the cards if you're the MitD.
This game has a lot of luck involved. It may be best to play multiple series of rounds if you want to find out which player has the better strategy (Conversely, play just one game if you want to determine which player is the luckier bastard gentleman). I suggest playing best 3 of 5, then best 4 of 7, then 5 of 9, etc. until the entire afternoon has been taken up.

Game Variants:
Kraagor and Serini: Try playing with two decks shuffled together! This means there will be two gates for Redcloak to find.
Fast Mode: Remove all but twelve cards from the deck to play with. This will make the game go a lot faster.
Xykon!: Single Player Mode: The one player plays as MitD and picks a card at random to reveal for Redcloak. Try to get the longest number of days!

2. ## Re: Door Guessing GAME

Originally Posted by nerdoflogic
This game has a lot of luck involved. It may be best to play multiple series of rounds if you want to find out which player has the better strategy
Actually, it's entirely a game of luck :P Unless the deck is poorly shuffled, strategy doesn't change the outcome at all.

3. ## Re: Door Guessing GAME

Untrue, actually.
A Redcloak that always chooses the top card and a MitD that always chooses the bottom card will produce a different field curve of times than if they both just chose randomly.
Try it for yourself with a friend a couple times and see! The purpose of this is for experimentation.

4. ## Re: Door Guessing GAME

There is no strategy to picking face down cards from a shuffled pile. Whether you pick, top, bottom, or third from the top, it's still a random draw. There is no strategy for Redcloak. Strategy for MitD exists, but is entirely about how many cards to discard rather than how you pick cards randomly.

5. ## Re: Door Guessing GAME

Originally Posted by fabiocbinbutter
There is no strategy to picking face down cards from a shuffled pile. Whether you pick, top, bottom, or third from the top, it's still a random draw. There is no strategy for Redcloak. Strategy for MitD exists, but is entirely about how many cards to discard rather than how you pick cards randomly.
With the "reshuffle and keep going" mechanic, the optimal strategy for the MitD is to discard the rest of the deck each time, forcing a reshuffle every turn. Redcloak's odds are marginally better for a second card from the same deck (1/51) instead of the first card from a reshuffled deck (1/52), which is true every time the deck is reshuffled; and the MitD is the only player that can act on that information (to deprive Redcloak of the marginal improvement of odds).

6. ## Re: Door Guessing GAME

Originally Posted by fabiocbinbutter
There is no strategy to picking face down cards from a shuffled pile. Whether you pick, top, bottom, or third from the top, it's still a random draw. There is no strategy for Redcloak. Strategy for MitD exists, but is entirely about how many cards to discard rather than how you pick cards randomly.
Originally Posted by Jasdoif
With the "reshuffle and keep going" mechanic, the optimal strategy for the MitD is to discard the rest of the deck each time, forcing a reshuffle every turn. Redcloak's odds are marginally better for a second card from the same deck (1/51) instead of the first card from a reshuffled deck (1/52), which is true every time the deck is reshuffled; and the MitD is the only player that can act on that information (to deprive Redcloak of the marginal improvement of odds).
For what it's worth, the guy with the math degree wholeheartedly agrees with both of you. (And cried when reading many of the reactions to the strip.)

And as this relates to the strip: obviously, this strategy wouldn't work, as the MitD would immediately be found out. So his best strategy there is to paint as many extra doors as he can get away with, reaching the "reshuffle phase" quickly. And Redcloak might be led to believe the "deck" is faulty and the gate is elsewhere, so... bonus.

7. ## Re: Door Guessing GAME

Originally Posted by Jasdoif
With the "reshuffle and keep going" mechanic, the optimal strategy for the MitD is to discard the rest of the deck each time, forcing a reshuffle every turn. Redcloak's odds are marginally better for a second card from the same deck (1/51) instead of the first card from a reshuffled deck (1/52), which is true every time the deck is reshuffled; and the MitD is the only player that can act on that information (to deprive Redcloak of the marginal improvement of odds).
At first, I thought this too, but then I realized that by picking a large number of cards on his first turn (for example 25), then MitD has a ~50% chance of discarding the gate, making it useful information. If he does discard the gate in this way, he can 100% safely guarantee another 25 turns using this information by not discarding anything and forcing Redcloack to pick duds for 25 turns.

We know that the expected value of the discard all strategy is (one plus) the mean of a negative binomial distribution (r=1,p=51/52), so 52 turns.

On the otherhand, if MitD's first action is to discard roughly half the deck (25 cards), he has a 25/51 chance to hit the gate and a (26/51) chance to miss the gate.

Mitd Hit: 25/51
MitD Miss: 26/51

Given that he misses the gate, Redcloak has one chance to get it at 1/26. He has a 25/26 chance to miss the gate (after which for simplicity of deriving the EV, we will say that MitD returns to the default 'discard all' strategy)

MitD miss & Redcloak hit: (26/51) * (1/26)
MitD miss & Redcloak miss: (26/51) * (25/26)

Going down the other branch, if MitD hit, redcloack has 100% miss chance:

MitD hit & Redcloack miss: (25/51) * 1

Now the EV's for each branch are:

MitD miss & Redcloak hit: (26/51) * (1/26) * 1 turn
MitD miss & Redcloak miss: (26/51) * (25/26) * (1+52) turns (because we just added one safe turn, then returned to the default EV52 strategy)
MitD hit & Redcloack miss: (25/51) * 1 * (26+52) turns (because we just added 26 safe turns, then returned to the default EV52 strategy)

Adding these all up we get an EV of
(26/51)*(1/26)*1 + (26/51)*(25/26)*(1+52) +(25/51) * 1 * (26+52) = 64.2
...in other words, a 28% improvement

And this is just modifying one turn with a not necessarily optimized initial discard number. If we were to apply this strategy each time the whole deck were shuffled rather than just the first play, the EV would be necessarily be higher.

Please let me know if I made any mistakes in logic or calculations...

(Actually, re-reading the rules, it's not clear whether the discard pile is revealed or not. I guess I just assumed so from playing so much Magic the Gathering, lol)

8. ## Re: Door Guessing GAME

Originally Posted by fabiocbinbutter
MitD miss & Redcloak hit: (26/51) * (1/26) * 1 turn
This is the only branch that matters, the chance that Redcloak finds the Gate which ends the round. The 26s cancel out, for 1/51 chance that the MitD does not discard the Gate and Redcloak finds the Gate. And in fact, for any non-total discard it works out that way; the odds of the MitD not discarding the Gate in k cards is (51-k)/51, which leaves 1/(51-k) odds for Redcloak to choose the Gate out of the remaining cards; the (51-k) cancels out, leaving 1/51 odds.

So no matter what, if less than the remainder of the deck is discarded, Redcloak has a 1/51 chance of getting the Gate with the second card....Which is better odds for Redcloak than the 1/52 chance of getting the Gate on the first pick after a reshuffle.

Originally Posted by fabiocbinbutter
(Actually, re-reading the rules, it's not clear whether the discard pile is revealed or not. I guess I just assumed so from playing so much Magic the Gathering, lol)
I'm assuming "MitD gets to discard as many cards as he wants without looking at them" implies MitD still can't look at them after they're discarded.

9. ## Re: Door Guessing GAME

Originally Posted by Roland Itiative
Let's look at a small sample, shall we? Ten doors, numbered 0 to 9. Team Evil picks a door, opens it. They have 1/10 of chance of succeeding. If they fail, MitD picks 3 doors. They have a 3/9 of chance of marking the right door (they pick 3 out of a total of 9, since Team Evil already eliminated one door).

Next night, Team Evil picks one out of 6 doors. But their chance to succeed is not 1/6 this time. They only have a 1/6 chance assuming MitD failed last turn, a completely independent chance of 6/9. When you have two independent events, and you need them both to happen at once, you multiply the probabilities. (1/6)*(6/9) is 1/9, exactly the same probability Team Evil would have to find the right door on this night had the MitD not interfered before.

Why is that? It doesn't matter that the MitD is reducing the choice pool, but that they aren't increasing the amount of information available. Team Evil will not make a more informed choice the next day because of the MitD, so their odds of success should not, and are not, any greater.

Interestingly enough, for each night the MitD keeps doing that, they are getting increased chances because of Team Evil. Team Evil's actions give the MitD extra information about a door, and increases the MitD's odds (similar to the Monty Hall problem), but the reciprocate isn't true. Both groups are playing the game by different rules, and have different effects on each other.
I posted the above in the comic discussion thread, and it's still just as relevant here. The important thing for the probability of winning is not the amount of cards you can pick from, but the amount of information you have about the deck, and the MitD does not generate any information about the deck when he picks cards.

10. ## Re: Door Guessing GAME

In the 'game' as described in the OP, MitD's optimum strategy is to add all the remaining cards to the discard pile, thus ending the game with only a 1/52 change of Redcloak and Xykon winning.
In the comic situation, the more unexplored doors MitD paints crosses on, the more likely he/she/it will mark off the door hiding the Gate (what Gate?); however, the earlier this occurs the more likely it is that Redcloak and Xykon
(a) realise that additional doors have been marked that they haven't explored yet
(b) are able to recall which doors they have actually explored, and
(c) even if they can't remember and have to start from scratch with all the doors, they haven't wasted a huge amount of time starting again.

11. ## Re: Door Guessing GAME

Not speaking mathematically , MITD takes a risk
Barring interventions for the order , sooner or later all the doors will be marked
Some embarrassing questions might then be asked of the MITD

12. ## Re: Door Guessing GAME

Originally Posted by Jasdoif
With the "reshuffle and keep going" mechanic, the optimal strategy for the MitD is to discard the rest of the deck each time, forcing a reshuffle every turn. Redcloak's odds are marginally better for a second card from the same deck (1/51) instead of the first card from a reshuffled deck (1/52), which is true every time the deck is reshuffled; and the MitD is the only player that can act on that information (to deprive Redcloak of the marginal improvement of odds).
There is a very good possibility I am wrong here, but couldn't you make the game last longer than just 1/52 odds every turn?

Instead of giving Redcloak a 1/52 shot of getting the right card every time, wouldn't the best strategy be to maximize your chances of discarding the card while still having him pick from the pile? This way he would have a 0% chance of getting the card (until the next shuffle) instead of a ~2% chance

(FYI, the OP should have made the game only use 50 cards so the math would be MUCH easier to figure out. screw it, i'll do that anyway)

You start out with a 1/50 chance of getting the right card. You miss, and the MitD discards 25 cards. You now have a 26/50 chance that you will NEVER get the right card and have to start over, and a 24/40 chance you will eventually get the right card this round. While he potentially makes the game shorter, he also potentially makes it MUCH longer. I have no idea if this evens out to a longer expected game or not. I'm good at statistics.

13. ## Re: Door Guessing GAME

That was what I suggested, but it's only viable if the discard pile is revealed information, otherwise, you're not actually changing any probabilities... can the OP please clarify if the players can see what's in the discard pile?

14. ## Re: Door Guessing GAME

I feel like it's important to note that MitD likely only began to paint doors after the first few doors were explored, because there were enough painted doors that he wouldn't be found out (if you did only 3 doors, to find 4 doors painted would be obvious, while if he waited until they had explored 30+, painting an extra door doesn't seem like too much of a stretch).

Also, if MitD is ever questioned about the extra paint, all he has to do is suggest that Kraagor's tomb is magically enchanted to spread the painted Xs to prevent any one adventuring group from finding the right door by process of elimination.

15. ## Re: Door Guessing GAME

Originally Posted by Valynie
Not speaking mathematically , MITD takes a risk
Barring interventions for the order , sooner or later all the doors will be marked
Some embarrassing questions might then be asked of the MITD
Actually, now that you mention it, do we know that the MitD is always on door-painting duty? If Redcloak, Oona, or Greyview (he can hold the brush in his teeth well enough to paint an X) sometimes paint the door, that adds yet another variable (I'm assuming Xykon is never on paint duty because he'd just find it boring and pointless and stupid).

16. ## Re: Door Guessing GAME

The monster being able to look in the discard pile (which he clearly can't in the comic, he's not dungeoneering by himself) makes the game mathematically much more interesting for the monster. It also makes the game itself a complete dud for Redcloack. "O, look, the first turn the discard pile grew by about ten cards, and then it stopped growing. Are you sure I have to fruitlessly pick another 40 cards before I have a chance again?"

The version without looking sounds like it could be fun for thoughtless moments or children. The optimal strategy as devised by Jasdoif is no fun to play with, so just hope they won't think of that. The rest of the game is purely chance based, like Monopoly (slight exaggeration) or Black Jack using optimal play without counting cards (no exaggeration).

17. ## Re: Door Guessing GAME

You know, just because there is a non-obvious optimal strategy doesn't invalidate the game. Look at tic-tac-toe : the game is solved and non-losable with 100% certainty as long as you don't do mistakes, and it still exists.

18. ## Re: Door Guessing GAME

Originally Posted by Lvl 2 Expert
The monster being able to look in the discard pile (which he clearly can't in the comic, he's not dungeoneering by himself) makes the game mathematically much more interesting for the monster. It also makes the game itself a complete dud for Redcloack. "O, look, the first turn the discard pile grew by about ten cards, and then it stopped growing. Are you sure I have to fruitlessly pick another 40 cards before I have a chance again?"
I agree that the monster being able to look in the discard pile would be much more interesting.

The monster's initial optimal solution would then be to discard a large number (probably half) until the gate is discarded, then to discard the minimum amount.
Then, Redcloak can make a decision whether to search the draw pile or the discard pile next. This decision would be based on knowing how much the monster discarded that turn.
Once the monster realizes Redcloak is choosing where to search based on the number discarded, the monster has to discard large amounts or small amounts in an attempt to deceive Redcloak and force him to search in the wrong pile.

That way, it becomes a game of deception and prediction instead of a game of pure chance. I'd much rather play this than the pure-chance version where nobody can look at the discarded cards.

19. ## Re: Door Guessing GAME

Originally Posted by Cazero
You know, just because there is a non-obvious optimal strategy doesn't invalidate the game. Look at tic-tac-toe : the game is solved and non-losable with 100% certainty as long as you don't do mistakes, and it still exists.
Tic-tac-toe is always a tie unless someone screws up. This game amounts to "the person who won the coin flip to be the creature in the darkness wins a lot." Sorry, nerdoflogic, but I think the person who was stuck playing Redcloak would opt out as quickly as Redcloak would opt out of the in-comic version if he knew he was playing it.
Originally Posted by Grey Watcher
Actually, now that you mention it, do we know that the MitD is always on door-painting duty?
If we're talking about the comic rather than the card-game proposal, I took "Can I do the paint this time?" and Xykon's response as indicating that the creature in the darkness had rarely, if ever, done it before.

He could push to always be the one who does it from now on; like the other forms of overstepping that people in the comic discussion thread have been assuming he'll automatically do, it would be a mistake.

20. ## Re: Door Guessing GAME

Originally Posted by littlebum2002
There is a very good possibility I am wrong here, but couldn't you make the game last longer than just 1/52 odds every turn?

Instead of giving Redcloak a 1/52 shot of getting the right card every time, wouldn't the best strategy be to maximize your chances of discarding the card while still having him pick from the pile? This way he would have a 0% chance of getting the card (until the next shuffle) instead of a ~2% chance

(FYI, the OP should have made the game only use 50 cards so the math would be MUCH easier to figure out. screw it, i'll do that anyway)

You start out with a 1/50 chance of getting the right card. You miss, and the MitD discards 25 cards. You now have a 26/50 chance that you will NEVER get the right card and have to start over, and a 24/40 chance you will eventually get the right card this round. While he potentially makes the game shorter, he also potentially makes it MUCH longer. I have no idea if this evens out to a longer expected game or not. I'm good at statistics.
Here's the thing. The rules state the MitD doesn't get to look at the cards while discarding them, which I'm assuming means the MitD doesn't get to look at the cards after they're discarded. If that's the case, the chances that the MitD has not discarded the Gate is inversely proportional to the chances that Redcloak will find the Gate if it was not discarded; and the product of the two ends up at 1/51 regardless...unless MitD discards the entire deck. That's because, without being able to look at the MitD discards, there's only one piece of information available: If Redcloak drew a card and the game is still going, the card Redcloak already drew is not the Gate.

If you can look at the discard pile after the fact, or if the game was houseruled to allow looking at it once the cards are selected for discarding, then the possibility opens up for MitD knowingly discarding the Gate and dragging the rest of it out. (If you choose to do so, I recommend that if the Gate is discarded by MitD, add the count of the remaining cards in the deck to Redcloak's turn count and reshuffle, rather than make Redcloak go through them all....Or one third the count, if you want to simulate MitD not knowing he discarded the Gate and/or you want a possible incentive to dragging out the progress through the deck).

Originally Posted by Lvl 2 Expert
The monster being able to look in the discard pile (which he clearly can't in the comic, he's not dungeoneering by himself) makes the game mathematically much more interesting for the monster. It also makes the game itself a complete dud for Redcloack. "O, look, the first turn the discard pile grew by about ten cards, and then it stopped growing. Are you sure I have to fruitlessly pick another 40 cards before I have a chance again?"

The version without looking sounds like it could be fun for thoughtless moments or children. The optimal strategy as devised by Jasdoif is no fun to play with, so just hope they won't think of that. The rest of the game is purely chance based, like Monopoly (slight exaggeration) or Black Jack using optimal play without counting cards (no exaggeration).
If I were to modify this to serve as a game, I'd incorporate both the possibility that Redcloak found (relatively) easy monsters and opened more than one door in a day, and a risk that Redcloak notices MitD marking extra doors to put a stop to the whole thing, and set it up as a solo game for MitD.

Spoiler: Because I Don't Know How to Shut Up
Say....On Redcloak's turn, draw a card. If it's a number card (not A/J/Q/K, as in not a strong dungeon), draw a second card. If that's not a number card either, draw a third card. If any of the drawn cards (no matter how many are drawn) is the Gate, the round ends. Otherwise, discard the drawn cards and add one to number of turns Redcloak's taken.

On MitD's turn...choose one to five cards, without revealing them until they've all been chosen. Once revealed, if the number on the lowest non-Gate number card (Aces are considered 1 here) is less than the number of cards drawn, MitD has been caught and the round is about to end.

• If the MitD reveals the Gate and is not caught, congratulations! Add half the count of remaining cards in the deck as a penalty to Redcloak's turns taken, add another 10 as a penalty while Redcloak scouts around elsewhere after the search is finished, reshuffle the deck and go back to Redcloak's turn with the new deck.
• If the MitD reveals the Gate and is caught, the round ends (Redcloak's going to check the ones he saw MitD messing with first).
• If the MitD is caught without revealing the Gate, add one half the count of remaining cards in the deck as a penalty to Redcloak's turns taken (he still has to find the Gate), and the round ends.
• Otherwise (not caught, no Gate), go back to Redcloak's turn.

And of course, as MitD is the player the goal is to maximize how many turns it's taken Redcloak to find the Gate during a round. It might be a good idea to set an upper limit along the lines of "MitD wins if the deck has to be reshuffled when Redcloak has over two hundred turns", if only for the psychological benefit of there definitely being an end in sight.

Intentional design points:
• There's no dead play: All cards are revealed before they're discarded, and the deck is reshuffled if the hand lasts past the Gate being revealed, so the unplayed deck will contain the Gate at all times it's being drawn from.
• With all cards revealed before being discarded, keeping track of which cards have been discarded can inform discard decisions.
• MitD discarding one card is always safe, so the player has a not-directly-losing option available at all times.
• Redcloak draws multiple cards more often than one, so MitD constantly playing safe isn't viable long-term.

If I were doing this seriously, I'd want to do more number crunching and trials on it*, but I think this is sufficient for mediocre expectations.

*
Spoiler: But the Monte Carlo Method is FUN
So I ran ten million trials on a shuffled deck (with one of the 3s replaced with an arbitrarily high number since the Gate card doesn't count here), to see what the maximum safe number of discards off a fresh deck would be.

Discard Chain Length Max Safe Chains Safe Chains Approx. Safe Chance
1 1494410 10000000 100.00%
2 2513349 8505590 85.06%
3 2252006 5992241 59.92%
4 2062564 3740235 37.40%
5 1134391 1677671 16.78%
6 426931 543280 5.43%
7 102068 116349 1.16%
8 13460 14281 0.14%
9 804 821 0.01%
10 17 17 0.00%

So discarding 3 is safe more often then not, and drawing above 5 would be quite risky (hence the limit of 5 cards in the rules). This, of course, is assuming a fresh deck so it doesn't account for the odds of Redcloak having discarded multiple cards at the beginning, nor how the odds of drawing a particular card increase as other cards are discarded.

21. ## Re: Door Guessing GAME

Originally Posted by Alcino
For what it's worth, the guy with the math degree wholeheartedly agrees with both of you. (And cried when reading many of the reactions to the strip.)

And as this relates to the strip: obviously, this strategy wouldn't work, as the MitD would immediately be found out. So his best strategy there is to paint as many extra doors as he can get away with, reaching the "reshuffle phase" quickly. And Redcloak might be led to believe the "deck" is faulty and the gate is elsewhere, so... bonus.
How is this affected if the MITD can no longer discard cards once the deck has been shuffled? It was under these conditions I was assuming the best strategy is for MITD to discard half the deck but I'm starting to feel less sure now.

22. ## Re: Door Guessing GAME

Originally Posted by Cazero
You know, just because there is a non-obvious optimal strategy doesn't invalidate the game. Look at tic-tac-toe : the game is solved and non-losable with 100% certainty as long as you don't do mistakes, and it still exists.
Which sounds to me like you've just reinforced what Lvl 2 Expert wrote: "The version without looking sounds like it could be fun for thoughtless moments or children."

When is the last time you saw two adults play tic-tac-toe with each other? Or even two older kids? It's very much a game for small children. The point at which kids figure out how the game works--and that it can only go one way if both participants share that understanding-- is also the point at which they lose all interest in playing it. The game hasn't lasted so long because it's fun for adults to play; it's lasted so long because it's fun for young children right up until the point where they figure the game out.

23. ## Re: Door Guessing GAME

Originally Posted by Elkins
Which sounds to me like you've just reinforced what Lvl 2 Expert wrote: "The version without looking sounds like it could be fun for thoughtless moments or children."

When is the last time you saw two adults play tic-tac-toe with each other? Or even two older kids? It's very much a game for small children. The point at which kids figure out how the game works--and that it can only go one way if both participants share that understanding-- is also the point at which they lose all interest in playing it. The game hasn't lasted so long because it's fun for adults to play; it's lasted so long because it's fun for young children right up until the point where they figure the game out.
There are adult versions of Tic-Tac-Toe.
Keeping up the concentration on strategy is more the point than figuring out strategy...or so I have been told.

24. ## Re: Door Guessing GAME

Hi a math student here. Instead of studying for finals i did an equasion here:
If RC and Xykon dont find the gate on the first walktrough then they take another walk trough all the gates at random again.
(if they don't this equaison is meaningles )
Becaouse finding the gate is certain lets look how many doors team evil must go trough to get to the gate. Before OOTS gets to them.

let D be the nuber of doors
let M be the nuber of doors MitD crosses whitout RC and Xykon knowing
then average doors that team evil must check to find the door in
-first walktroug: (D-M)/2 that is if MitD doesnt cross the gate, then RC and Xykon must on average check half the reamaining doors
-second walktroug (D-M)/2 + (D-M) that is if MitD doest cross the gate on the second walktrough (same as above but plus the whole first walktroug)
and so on
Note this is if MitD crosses the doors before Team evil starts checking (they still have pure luck if they stumble on the gate) and if the monsters do indead respawn as said by Oona

lets look at the chance of any of this options to happen
found on first walktroug: (D-M)/D chance of MitD not hiting the gate
found on second: M/D * (D-M)/D chance of MitD hiding the gate on first try and failing on the second.

So average nuber of doors checked is:
(D-M)/D *((D-M)/2) + M/D * (D-M)/D * ((D-M)/2 + (D-M)) + (D-M)/D *(D-M)/D *((D-M)/2 + 2(D-M)) + ....
so its a infinite sum:
(M/D)^n (D-M)/D * (n+1)/2 * (D-M) where n starts at 0 and goes to infinity.
if we derive this by M and solve for when the sum equals zero to get the extreme we get:
D = V * SquareRoot[n/(n+2)] meaning that
the more doors MitD crosses off the more doors RC and Xykon will have to check.

If MitD crossed all doors every night then Team Evil will have to pick a random door every day, leaving them only a small chance of luck to find the door on the first try.
If MitD did't cross any doors Team Evil would found the gate after on average of D/2 checked doors.

So in conclusion the more doors MitD crosses the better.

TLDR: the more doors MitD crosses the better his chance of hiding the gate. As was noted above. Of coure we need to consider other factors like suspicion and RC using a more foolproof method.

PS.: Sory for my bad english, feel free to point any mistakes in logic or language. And i hope i didn't bore you with my math

25. ## Re: Door Guessing GAME

Originally Posted by pidgy
If MitD crossed all doors every night then Team Evil will have to pick a random door every day, leaving them only a small chance of luck to find the door on the first try.
If MitD did't cross any doors Team Evil would found the gate after on average of D/2 checked doors.

So in conclusion the more doors MitD crosses the better.

PS.: Sory for my bad english, feel free to point any mistakes in logic or language. And i hope i didn't bore you with my math
Mathmatically yes, but once all the doors have been crossed and the gate hasn't been found, Team Evil will know something is up. At that point they will either go through logically (perhaps starting at the top row and going across or something), or assume the gate is elsewhere.

To make the card game more similar to the problem at hand, have the game only go through the deck once and the MiTD is trying to get as many turns as they can out of the game. It would still be a luck based game, but more like in-comic situation.

26. ## Re: Door Guessing GAME

As an almost physicist (6 months to go), I agree with the mathematician, and the math student. Both Roland Itiative and Jasdoif got it right.

I repeat NOISE CAN'T ADD INFORMATION.

28. ## Re: Door Guessing GAME

Originally Posted by Jasdoif
Smart stuff
So in other words, the benefit the MitD gets by not discarding all the cards and letting Redcloak have a chance at having to finish drawing the deck and shuffle again is negated by the possibility that the card is still in the deck and therefore the game will be MUCH shorter.

Gotcha (I think)

29. ## Re: Door Guessing GAME

The main problem I'm seeing from a gaming point of view is the lack of game. The Redcloak player has no options available to him. Really all he's doing is watching the MitD player do things. This isn't a game.

If there was a mechanic for 'calling out' the MitD and forcing a recount, it'd give the Redcloak player more options. Here's my suggestion;

Redcloak player looks at up to two cards. If it isn't the gate, he closes his eyes and lets MitD put it in the marked pile. While his eyes are closed, MitD can add extra cards to the marked pile (He can't search the pile for the gate card; he must do this blind). Redcloak can 'call out' the marked pile and name a number; if the pile has more cards than that number, MitD loses a life and the game starts from the beginning. If the number of cards in the pile is equal or less than the named number, Xykon gets angry at Redcloak for wasting his time, Redcloak loses a life and the game starts from the beginning. Redcloak can also 'call out' a card he's revealed; if it's not in the marked pile (Because MitD put a revealed card back into the deck and marked another on), MitD loses a life. If it is in the marked pile, Xykon gets mad, Redcloak loses a life.

Thoughts?

30. ## Re: Door Guessing GAME

Originally Posted by littlebum2002
So in other words, the benefit the MitD gets by not discarding all the cards and letting Redcloak have a chance at having to finish drawing the deck and shuffle again is negated by the possibility that the card is still in the deck and therefore the game will be MUCH shorter.

Gotcha (I think)
That's pretty much it. There's a direct relationship between the chances that MitD discards the Gate, and the chances that Redcloak draws the Gate from the non-discarded cards if MitD didn't discard the Gate. The numbers work out that the exact number of cards MitD discards makes no difference, as long as there's a card for Redcloak to draw afterwards.

To demonstrate....If MitD didn't discard any of, say, 51 cards; Redcloak's chance of getting the Gate next is 1 out of 51. If, on the other hand, MitD discarded 50 of 51 cards, there's only one card left for Redcloak to draw, so the Redcloak's chance of getting the Gate next is the same as the chance for MitD to not discard the Gate: (51-50) out of 51...or 1 out of 51, again. And it works out the same for any number in-between.

Because the only cards we can know for sure aren't the Gate are the ones Redcloak draws and discards for not being the Gate, Redcloak's chances of finding the Gate improve each time he gets to draw from the same deck (because there are fewer cards to choose from, and the number of cards MitD discards doesn't affect the odds as we just showed). So for MitD, discarding the entire deck to force drawing from a new deck each time keeps Redcloak with the lowest possible chances.