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In a normal nim game, the player making the first move has a winning strategy if and only if the nim-sum of the sizes of the heaps is not zero. Otherwise, the second player has a winning strategy. Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+) and also satisfies an additional property, x ...
Fibonacci nim is played with a pile of coins. The number of coins in this pile, 21, is a Fibonacci number, so a game starting with this pile and played optimally will be won by the second player. Fibonacci nim is a mathematical subtraction game, a variant of the game of nim. Players alternate removing coins from a pile, on each move taking at ...
The nim-values are zero for -positions, and nonzero for -positions; according to a theorem of Tom Ferguson, the single-number positions with nim-value one are exactly the numbers obtained by adding the smallest value in the subtraction set to a -position. Ferguson's result leads to an optimal strategy in multi-pile misère subtraction games ...
Created Date: 8/30/2012 4:52:52 PM
Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in Nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which ...
The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game.
Dr. Nim was based on a mathematical game called NIM, which similarly consisted of twelve marbles. A simple strategy will always win as long as the opponent goes first. This is the strategy for single-pile NIM: If the opponent takes 3 marbles, the first player should take 1. If the opponent takes 2 marbles, the first player should take 2.
UPS's announcement that it will cut back on deliveries for its largest customer, Amazon (), sent its stock tumbling as much as 15% on Thursday.But the company says it made the change with the goal ...