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Subtraction games are generally impartial games, meaning that the set of moves available in a given position does not depend on the player whose turn it is to move.For such a game, the states can be divided up into -positions (positions in which the previous player, who just moved, is winning) and -positions (positions in which the next player to move is winning), and an optimal game playing ...
Subtract-a-square can also be played as a misère game, in which the player to make the last subtraction loses. The recursive algorithm to determine 'hot' and 'cold' numbers for the misère game is the same as that for the normal game, except that for the misère game the number 1 is 'cold' whilst 2 is 'hot'.
Wythoff's game is a two-player mathematical subtraction game, played with two piles of counters. Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile must be equal. The game ends when one player removes the last counter or counters, thus winning.
The game "21" is played as a misère game with any number of players who take turns saying a number. The first player says "1" and each player in turn increases the number by 1, 2, or 3, but may not exceed 21; the player forced to say "21" loses. This can be modeled as a subtraction game with a heap of 21 − n objects. The winning strategy for ...
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 ...
Subtraction game SMIL: Image title: Interactive SVG using SMIL of the subtraction game by CMG Lee. Players take turn removing 1, 2 or 3 objects from an initial pool of 21 objects. The player taking the last object wins. Width: 100%: Height: 100%