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Nimbers! By: Alyssa Godesky. Nimbers come from a game called Nim, one of the oldest known strategy games. .
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Nimbers come from a game called Nim, one of the oldest known strategy games.
The field of nimbers has characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal α is α itself.
How to create the nimbers… Start with ordinal 0. We can't add any new numbers using nimber addition - because there is only one. So we look for the mex of the set of defined ordinals: mex({0}). That's 1. So now we have two numbers, 0 and 1.
Using that, we get a set of new nimbers defined from the sums of existing nimbers. Each time we've exhausted the new numbers that can be generated using sums, we'll create a new nimber by taking the mex of the nimbers that we've defined so far.
How Nimbers relate to the game Nim: The winning strategy of Nim is to finish every move with a Nim-sum of 0, which is always possible if the Nim-sum is not zero before the move.
If there are only two heaps left, the strategy is to reduce the number of objects in the bigger heap to make the heaps equal. After that, no matter what move your opponent makes, you can make the same move on the other heap, guaranteeing that you take the last object.
In a normal Nim game, the first player has a winning strategy if and only if the nim-sum of the sizes of the heaps is nonzero. Otherwise, the second player has a winning strategy.
Fun Facts Nim was played for money on a TV show in Spain in the nineties. The show (called "Hola, Rafaella") had a magician who each week would challenge one viewer who called on the phone. With a different notation, they would play Nim, always starting with the position (2,3,4). The viewer could choose whether to go first or second. If the viewer beat the magician, he would win some money. This little game was part of the show for various months and only one viewer was able to beat the magician.
