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I enjoyed your article about Nim. Did you know that the game still works if the rule for winning is reversed?
Here is a variation that I use when introducing the game of Nim to a novice. I lay out the piles (usually in the arrangement 2,3,5,7) and explain the rules of play. I then make the bold (and slightly risky statement) that I can always win, whether I move first or you move first and whether the winning player takes the last piece or the losing player takes the last piece. Once the player make those decisions and the game is underway, I get my Nim sum to zero as soon as I can and then play along until a minimum position is reached, such as 3,2,1 or 2,2. Then, I offer to let you reverse the rule about who wins, if you wish. With a little thinking, it's clear that I can win under either rule but my last move must have a Nim sum of one if rule is that the person who takes the last one loses.
It's a great way to introduce a little bit of maths!
If it seems that the person would be overwhelmed by a discussion of binary numbers, I have them visualize each of the piles of markers as being divided into powers of 2. Then you want to play such that after your move each such sub-pile must have a matching partner. For example, with the starting pattern of 2, 3, 5, 7 this becomes 2, 2+1, 4+1, 4+2+1. This is easier for some people than binary numbers.
==Gordon Stallings==

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