Bologna Sandwich was worried about an upcoming test in Discrete Mathematics and was finding it hard to get to sleep. Bologna awoke early in the morning, aroused by devilish laughter, only to see an impish looking homunculus sitting at the bottom of the bed next to a seemingly infinite pile of chips. Hello, Bologna, it said, would you like to play a little game? This pile contains 43546758443209876 chips and the bottom chip represents your immortal soul. The rules are quite simple. The first player removes some chips, but not all of them. After that we take turns removing some chips.
The only rule now is that a player cannot remove more than the previous player removed in his last turn. The winner is the player who takes the last chip. If I win I get to keep your soul and if you win, you get an A in the test. Would you like to go first or second? This seemed a reasonable bet to Bologna.
Can you give Bologna a strategy for playing no matter how many chips there are? (What if there were just one more chip in the initial pile?)
What if the rule were that one is allowed to take up to twice the number of chips the previous player took?
If the pile was odd, the strategy would be to go first and take only one chip. It would take a while, but Bologna would eventually take the last chip and win.
So if it is Bologna's turn and there is an odd number of chips in the pile, he wins. So the strategy would be to force the imp to take a number of chips that leaves Bologna with an odd number. If the imp is smart, he could use this strategy to win also, so Bologna must never let the imp have an odd number in his pile.
Posted by Iain
on 2004-05-10 12:25:34