The Longest Optimal Game of Nim
In the game of Nim you have several piles with tokens. Players take turns taking several tokens from one pile. The person who takes the last token wins.
The strategy of this game is well-known. You win if after your move the bitwise XOR of all the tokens in all the piles is 0. Such positions that you want to finish your move with are called P-positions.
I play this game with my students where the initial position has four piles with 1, 3, 5, and 7 tokens each. I invite my students to start the game, and I always win as this is a P-position. Very soon my students start complaining that I go second and want to switch with me. What should I do? My idea is to make the game last long (to have many turns before ending) to increase the chances of my students making a mistake. So what is the longest game of Nim given that it starts in a P-position?
Clearly you can’t play slower then taking one token at a time. The beauty of Nim is that such an optimal game starting from a P-position is always possible. I made this claim in several papers of mine, but I can’t find where this is proven. One of my papers (with Joshua Xiong) contains an indirect proof by building a bijection to the Ulam-Warburton automaton. But this claim is simple enough, so I want to present a direct proof here. Actually, I will prove a stronger statement.
Theorem. In an optimal game of Nim that starts at a P-position the first player can take one token at each turn so that the second player is forced to take one token too.
Proof. Consider a P-position in a game of Nim. Then find a pile with the lowest 2-adic value. That is the pile such that the power of two in its factorization is the smallest. Suppose this power is k. Notice that there should be at least two piles with this 2-adic value.
The first player should take a token from one of those piles. Then the bitwise heap-sum after the move is 2k+1−1. Then the Nim strategy requires the second player to take tokens from a pile such that its value decreases after bitwise XORing with 2k+1−1. All piles with the 2-adic value more than k will increase after xoring with 2k+1−1. That means the second player has to take tokens from another pile with 2-adic value k. Moreover, the second player is forced to take exactly one token to match the heap-sum. □
In the position (1,3,5,7) all numbers are odd, so I can take one token from any pile for my first move, then the correct move is to take one token from any other pile. My students do not know that; and I usually win even as the first player. Plus, there are four different ways I can start as the first player. This way my students do not get to try several different options with the same move I make. After I win several times as the first player, I convince my students that I win anyway and persuade them to go back to me being the second player. After that I relax and never lose. I am evil.
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