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?
In the first case, you advise Bologna to play second. Because there are an even number of chips on the pile, whatever H does, Bologna will be able to beat him (if H takes half or more from the pile, H loses. If H takes less than half from the pile, Bologna removes 1 chip if the remaining chips are odd numbered, thus winning, or if H takes an odd number of chips, Bologna takes an odd number of chips also).
In the second case (1 more chip on the pile), Bologna should go first, removing only one chip, thus securing a win.

Posted by Henry
on 20040730 16:21:45 