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Wythoff's game is a two-player mathematical game of strategy, played with two piles of counters. Players take turns removing counters from one or both piles; in the latter case, the numbers of counters removed from each pile must be equal. The game ends when one person removes the last counter or counters, thus winning. Martin Gardner claims that the game was played in China under the name 捡石子 ''jiǎn shízǐ'' ("picking stones").〔(Wythoff's game at Cut-the-knot ), quoting Martin Gardner's book ''Penrose Tiles to Trapdoor Ciphers''〕 The Dutch mathematician W. A. Wythoff published a mathematical analysis of the game in 1907. ==Optimal strategy== Any position in the game can be described by a pair of integers (''n'', ''m'') with ''n'' ≤ ''m'', describing the size of both piles in the position. The strategy of the game revolves around ''cold positions'' and ''hot positions'': in a cold position, the player whose turn it is to move will lose with best play, while in a hot position, the player whose turn it is to move will win with best play. The optimal strategy from a hot position is to move to any reachable cold position. The classification of positions into hot and cold can be carried out recursively with the following three rules: #(0,0) is a cold position. #Any position from which a cold position can be reached in a single move is a hot position. #If every move leads to a hot position, then a position is cold. For instance, all positions of the form (0, ''m'') and (''m'', ''m'') with ''m'' > 0 are hot, by rule 2. However, the position (1,2) is cold, because the only positions that can be reached from it, (0,1), (0,2), and (1,1), are all hot. The cold positions (''n'', ''m'') with the smallest values of ''n'' and ''m'' are (0, 0), (1, 2), (3, 5), (4, 7),(6,10) and (8, 13). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wythoff's game」の詳細全文を読む スポンサード リンク
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