After tens of thousands of games of pool, every time I rack the balls I seem to switch about half of them around. I know I'm wasting time. So, I want to know exactly how many balls I should
to swap. For those of you who aren't pool nerds like me, I've explained the 8-ball racking process inside.
A "proper" rack should look something like this
There are 15 balls. 7 "stripes", 7 "solids", and one black ball.
You dump them into the triangle at random, then swap them around until you have the "correct" pattern. There are two different recognised patterns (European and American), but we'll pretend there's just one.
You can either start that pattern with stripes at the front, or solids. You can also reverse the pattern in terms of symmetry. As such, I think there are four ways of racking the balls correctly.
However, once the balls are in, you can spin the triangle into three positions, giving you 12 total possibilities.
There is a complication: the black is the only ball that has to sit on one of the inside three spots.
So, if the balls happen to land in the worst possible way, what are the most number of ball swaps that you'll ever have to make? I'm talking about straight swaps — no moving three at a time.
A second, perhaps more-important question: what's the median starting position, in terms of how many ball swaps I'll have to make?