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So, you're tired of navigating a Pokemon thread where everyone uses fancy words and stuff? Well no worries! Here's a guide on... BREEDING FOR DUMMIES!
One of the first things to remember in breeding is that there are natures that
help and ones that
hinder. For instance, a Special Attacker such as Mewtwo would be
hindered by an Adamant nature, but
helped in his stat growth by a Modest nature. To improve your chances for getting a particular Nature in a child Pokémon, use at least one parent with the Nature you want, and give it the
Everstone.
Brilliant Spreadsheet for Growth.
Some IVs are randomly inherited from the parents. You can lock in a maximum of
one IV by giving a parent the corresponding
Power training item (Power Bracer for Attack, Power Lens for Special Attack, and so forth). Use Flat Battles to determine the IVs of a hatched Pokémon, or if you only care about the highest IV and the sum of all of its IVs, you can take it to a guy who's in the lobby of either the Battle Tower or the Battle Subway, depending on your game. You're looking for them to judge it Relatively Superior overall and Can't Be Better in its best stat.
The third main thing with breeding is
Egg Moves. Egg moves are moves the parent has, and passes down to a child. For instance, in my breeding of Chimchar, the egg moves are Grass Knot and Swords Dance, which are passed down to the child who gets said moves from the start. In order to pass on most Egg Moves, the father must know the move (Genderless Pokémon never learn Egg Moves). These include all of the moves that are listed as the child Pokémon's Egg Moves in a move list chart, as well as any compatible TMs or HMs. You can also pass on moves that the child would learn by level-up that aren't in those two lists--for these, both parents must know the move.
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Don't feel the need to bother with this stuff unless you're playing against humans or at the Battle Frontier. You shouldn't need it for most of the game, and it can be kind of a pain.
Individual Values
Every Pokémon species has its own base stats, as you probably know. Aerodactyl are very fast but have weak defenses, Blissey have huge amounts of HP, Luvdisc are terrible at everything but speed, and so on. But, in addition to the normal stat allocation, every Pokemon gets a bonus of 0-31 points applied to each stat. So a level 100 Tyranitar could have an attack stat of 273, or it could have a attack of 304. That bonus is the IV.
A Pokémon has its IVs generated the moment you meet it, or accept its egg, and there's no way to change them. You're stuck with what you've got. However, you can influence them a little, if you're willing to breed. Bred Pokémon don't randomly generate all their IVs, they inherit up to 3 from their parents. Still doesn't help that much, especially if the parents have bad IVs, but what can you do? Typically you just end up hatching like a billion eggs until you get a good one.
Effort Values
Every time you win a fight, your Pokémon get experience, but that's not all they get. Every Pokémon also gives out effort points. For every 4 effort points in a particular stat, your Pokémon gains 1 point in it. So if that Tyranitar from earlier with 304 attack had 4 attack EVs, his actual attack would be 305. If he had 252 attack EVs, it'd be 367. Sounds handy, but there are limits. You can't have more than 255 points in a single stat, and you can't have more than 510 EVs in total.
You'll see a lot of Pokémon with 252 attack/special attack EVs, 252 speed, and 4 HP, but there are a lot of ways to configure them. Defensive Pokémon will obviously focus on different things, Pokémon which use both physical and special attacks will need to split EVs between them or possibly sacrifice speed, and so on.
It's a tremendous pain to work out what EVs your Pokémon has, so you'll likely want to start with a Pokémon that you freshly hatched and work from there, though you don't have to. There are 6 berries that reduce EVs in a single stat each, so you can use those to reset your Pokémon and start from scratch. After that, just keep a careful record of what you fight, and only take down Pokémon that you know give you the EVs you want.
EV training takes a while, but there are a few ways you can speed it up. The first is an astonishingly rare status condition called Pokérus. Any Pokémon with Pokérus gets double EVs from fights, and will spread the Pokérus to any Pokémon in your party for the first 72 hours after it catches it. After those 72 hours it stops being contagious, but still gets double EVs. Pokérus has something like 1/20000 chance to appear on wild Pokemon, but with the magic of the internet I'm sure you can obtain it.
The second way to speed up EV training is through Power Items. If a Pokémon is holding the right item, they get bonus EVs from every fight. For example, a Pokémon holding a Power Lens will get 4 special attack EVs. A Pokémon holding a Power Anklet will get 4 speed EVs. A Pokémon holding a Macho Brace will just get double the EVs of whatever they're fighting. These bonuses are doubled if you have Pokérus.
The third way to speed up EVs is through vitamins. Each vitamin given to a Pokémon gives them +10 EVs in a certain stat, up to a limit of 100. 1 protein gives 10 attack EVs, 10 give 100. These unfortunately aren't doubled by Pokérus or items, so you'll have to pay for the full amount. Luckily money is easy to come by in these games.
Just as an example, I'll show how I EV'd my Gengar.
First, I gave him 10 calcium, and 10 carbos, so 100 special attack and 100 speed already.
After that, I gave him Pokérus and a lucky egg, got him to meet a Floatzel (2 speed EVs, doubled to 4 because of Pokérus), and then sent out my Weezing to explode on it so Gengar could get all the experience, for a higher level and 100 special attack, 104 speed.
After that, I gave him a Power Lens, flew over to the Old Chateau, and killed 15 Gastly (1 special attack from Gastly, 4 from the lens, doubled to 10 from Pokérus) for 150 special attack EVs, bringing my total to 250 special attack, 104 speed.
After that I equipped a power anklet and killed one more Gastly (1 special attack from Gastly, 4 speed from the anklet, doubled to 2 and 8 from Pokérus), meaning Gengar had 252 attack, and 112 speed.
Leaving the anklet equipped, I flew to Route 201, and killed 14 Starly (1 speed from Starly, 4 from the anklet, meaning 10 speed per starly thanks to Pokérus), for a total of 252 attack, 252 speed.
All that was left after that was to kill a Bidoof with a Power Weight (1 HP from Bidoof, 4 from the Weight, 10 thanks to Pokerus, although I only gained 6 points because I hit the EV cap), leaving my Gengar with 252 special attack EVs, 252 Speed, and 4 HP, just how I wanted him.
Roughly an hour's work, and one Pokemon down.