We can consider nim as a countably infinite-dimensional vector space over , the field with two elements. To do this, we write out a nim heap in binary, and then we make the digits into an element of by putting the least significant bit in the first entry, the second-least significant bit in the second entry, and so forth. In this case, the operation of nim sum is simply vector space addition on .

But we can do even better: we can turn this vector space into a field of characteristic 2. We define multiplication as follows (I will use for nim product): for any , , and a nim product of distinct Fermat 2-powers (numbers of the form ) is their ordinary product. Then we can use distributivity to compute any nim product. For example, . (Note that, unlike nim addition, nim multiplication does not preserve parity.)

In fact, we can say even more. Suppose we limit our game of nim game to piles of size at most . Then nim addition and multiplication are closed in this subclass of games. Therefore we have canonically(?) embedded in our nim field. (I am not so sure about the canonical part, because I don’t know if we could define multiplication in some other natural way that would still give us a field.) This tells us that is isomorphic to the field of finite nimbers.

So here’s a question for myself or my (possibly nonexistent) audience: what does nim multiplication tell us about actually playing games? Anything? Nothing at all? I would like to believe that it has some implication as far as playing games goes, but at the moment, I don’t see anything.

If we extend nim to ordinal piles, we can do something similar and end up with an algebraically closed Field (Conwayese for something that would be a field except that its elements form a proper class). Ordinal nim piles make sense as far as game play goes because the ordinals are (very conveniently!) well-ordered, so games cannot last forever.

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Saturday, May 19, 2007 at 2:12 pm

RuxorConsider the following game. You have a certain (finite) number of piles, each pile containing a number (a) of black tokens and a number (b) of red tokens, but neither a nor b can be zero. Each player in turn selects a pile and does the following: the player removes a certain number of black tokens, leaving a’

Saturday, May 19, 2007 at 2:13 pm

Ruxor[Sorry, previous comment got mangled because of an unescaped less-than sign.]

Consider the following game. You have a certain (finite) number of piles, each pile containing a number (a) of black tokens and a number (b) of red tokens, but neither a nor b can be zero. Each player in turn selects a pile and does the following: the player removes a certain number of black tokens, leaving a'<a, and a certain number of red tokens, leaving b'<b (one must remove tokens of both sort), so the pile now has (a’,b’) black and red tokens, but one also _adds_ two new piles to the game, one having (a’,b) and one having (a,b’). So where once was one (a,b) pile now there are three, (a’,b’), (a’,b) and (a,b’). Having a’=0 or b’=0 is legal, but one does not add piles having zero tokens of one sort, so in effect a’=0 means just removing black tokens and b’=0 means just removing red tokens. Having both a’=0 and b’=0 is also legal and that means the pile is entirely removed (any pile can be removed in a single move). The winner is the one who removes the last pile. Of course, a pile with a single red token is equivalent to a nim pile (one can just remove black tokens from the pile or remove the pile completely). Well it turns out that a pile with a black tokens and b red is equivalent to a nim pile with a*b tokens, where ‘*’ is nim multiplication. Admittedly this game is very strange…

Another game is this: you have a two-dimensional array of coins, each being on heads or on tails. Call (a,b) the coin on row a and column b (both being nonzero natural numbers). On each move you can: turn a coin (a,b) from heads to tails; or turn two coins provided both are on the same line ((a,b) and (a,b’) with b'<b) or the same column ((a,b) and (a’,b) with a'<a) and the one with the greatest coordinate (a,b) goes from heads to tails; or turn four coins, (a,b), (a’,b), (a,b’), (a’,b’), with a'<a and b'<b, provided coin (a,b) (the one with the greatest coordinates) goes from heads to tails. Again, the configurations where player two has a winning strategy are those for which the nim sum of all the nim products a*b ranging over all heads is zero.