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?

Call a position L if you are guaranteed to lose if presented with it and W if you are guaranteed a win. Winning positions are generally those that present the opponent with a losing position. Its a little more complicated here because if you take a third or more you will lose also.

I have the following worked out:

2L
3L
W4
5L
6W
7W
8L
9W
10W
11W
12L
13W
14W
15W
16W
17W
18L

I thought the number of W's in a row was doubling every time until the 17W (strategy=take5). It may be a more complicated exponential or even Fibonacci. I don't have time to finish right now. Thats what I's gots for now.
-Jer

Posted by Jer on 2004-05-10 14:47:50