games • nim skulls • instructions • clue |

Think of the game in reverse. If you can leave one skull at the end of your turn, you win. If there were two skulls at the beginning of your turn, you’d take one and win. If
there were three skulls at the beginning of your turn, you’d take two and win. If
there were four skulls at the beginning of your turn, you’d take three and win. Let’s say there are five skulls at the end of your next-to-last turn. If the computer takes one, then there are four left and you can win. If the computer takes two, then there are three left and you can win. If the computer takes three, then there are two left and you can win. So, no matter what move the computer makes when it sees five skulls, you win. So, if you can leave five skulls after your turn, you win. By this same reasoning, if you can leave nine skulls after your turn, you win, because that allows you to get to five. You can figure out the other critical point. If you |