Mr. Jonathan Pearce, Please don't think me impolite for answering your question about
the 12 billiard ball problem. You can determine the heavier ball by: weighing #1 place the balls on a platform balance where there are 6 on each pan. If one pan is heavier than the other, it means that the heavy billiard ball is one of the 6 balls. weighing #2 get the 6 balls and place them on the balance once again,but this time there must be 3 balls on each pan. If one pan is heavier, the billiard ball must be there, so weighing #3 put 2 of the 3 remaining balls on each pan. If one is heavier than the other then that must be the ball you're looking for. If the 2 balls have equal weights then the remaining one must be the one you're looking for. I know this may be kind of weird because you may already have the answer to your question, but I enjoyed answering it. Thank you.