Nimbers!%20By:%20Alyssa%20Godesky

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Nimbers!By Alyssa Godesky
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• Nimbers come from a game called Nim, one of the
  oldest known strategy games.

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• 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 a is a
  itself.

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Nimber AdditionAB mex(A'B A'ltA U AB'
B'ltB)
Binary Decimal
0112 3 A
1002 4 B
1012 5 C
0102 2 ABC
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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.

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• 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.

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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.

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• 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.

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Sizes of Piles Sizes of Piles Sizes of Piles Moves
A B C
3 4 5 I take 2 from A (leaving a nim sum of 0, so I will win)
1 4 5 You take 3 from C
1 4 2 I take 1 from B
1 3 2 You take 1 from B
1 2 2 I take 1 from A
0 2 2 You take 1 from B
0 1 2 I take 1 from C
0 1 1 You take 1 from B
0 0 1 I take 1 from C, Win!
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• 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.

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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.