IVs range from 1-31 for each statistic and as far as I can tell are mostly random. However, if you're lucky enough to acquire several pokemon with naturally high IVs, you can breed them together and pass some of those IVs down. By default, three IVs are passed down, chosen randomly from both parents. If one of the parents holds a Destiny Knot, then five IVs are passed down instead. By chain-breeding forever, you can manipulate what IVs your eventual "perfect pokemon" ends up with. Ideally you'll want 31 in each stat, but this is borderline impossible to do.
Pokemon XY bring us a new treat when it comes to breeding for IVs, though. Destiny Knot did not allow you to pass down additional IVs before, so it is of course a godsend. That's not all, though; there's now a feature called the Friend Safari available in the post-game. For every 3DS friend you have registered, you have a Friend Safari of a particular pokemon type. It looks like this is mostly random. According to my friends my safari type is Fighting, whereas two of my friends have Poison. Pokemon caught from the Friend Safari not only have a higher likelihood of having "hidden" abilities (abilities that are not usually encountered in the wild) but they also typically have much higher IVs!
So, I spent most of the morning breeding dozens of Heracross and Scyther with the intent of ending up with pokemon with near perfect IVs. As a result, I ended up with a ton of extra pokemon who were perfectly usable but not up to my OCD standards. I decided to host a giveaway on one of the Pokemon subreddits. As a result, I gained about 25 new 3DS friends and Friend Safaris for each one--including a Bug safari, where I can catch Heracross. In a way this means all that breeding was a waste, but if I hadn't done it I wouldn't have made so many new friends. It works out either way.
There's at least a small possibility I am going completely mad.
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