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So I was reading "Graphs and their uses" by Oystein Ore and I came across a section about the game called Nim. Now the author takes as granted the binary digital sums as a way of solving the game. The thing is I'm unable to understand his intuition on this.

hedphelym
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    This is, to me, an interesting question, which is far from exhausted by the answer I have given. The answers to this question are likely to have enduring interest - why does Nim arithmetic work? It does, but how would anyone be motivated to think of that? It isn't a duplicate. Why close it? – Mark Bennet Jan 09 '15 at 00:10
  • Mathematical progress requires genius, luck and creativity sometimes. Someone made the connection first, so tautologically no one found it intuitive beforehand. – djechlin Jan 09 '15 at 15:31

2 Answers2

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The simple answer is "because it works".

While the binary digit parity is now a well-known approach to setting winning strategy for Nim, the game itself is far older than the knowledge of that strategy, so the use of this approach is not automatically intuitive.

It's a fairly natural extension of a parity-type approach that works in some simple games and puzzles.

Joffan
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Partly it is because there are two players. Two player impartial games with perfect information always relate to Nim, which has a special arithmetic related to fields of Characteristic 2.

Entertaining exposition of the wider theory can be explored via the Grundy numbers and the works of JH Conway and collaborators.

As Ross Millikan comments, Winning Ways by Conway, Berlekamp and Guy is a great source - the first volume of either edition.

Impartial games between two players are games like Nim where each player has the same moves from the same position (chess would be different, for example, because the players control different sets of pieces).

Each game position gets a number representing the value of the position. The number for a position is in some sense the simplest possible value taking into account the moves available to each player - the values of those options having already been computed. In impartial games it turns out to be the minimum excluded value, say $m$, and the position can then be treated like a pile of $m$ matches (or other objects) in the game of Nim. So the values get called "Nimbers".

The parity point comes about because there are two players. If player $1$ plays a winning move, then player $2$ has to play a losing move, and player $1$ will be in an equivalent simpler position where another winning move is possible. A winning move in such a game is always to a position of value $0$, from which there is no good option. From a position with value greater than $0$ it is always possible to move to a position with value $0$, because of the minimum excluded value rule ($0$ has been excluded, so there is a move to a position of value $0$). So in a well-played impartial game every other position will have value $0$.

So in a well-played game, pairs of consecutive moves cancel each other out in some way.

Mark Bennet
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