94:(in which each move may be played by either player) and asserts that every such game has an equivalent Sprague–Grundy value, a "nimber", which indicates the number of pieces in an equivalent position in the game of
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56:{0|0}, which is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.
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109:, but has value 0, since it is a second-player winning situation whatever the first player plays. It is not a
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98:. All second-player win games have a Sprague–Grundy value of zero, though they may not be the zero game.
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This article is about combinatorial game theory. For the novel entitled "The Zero Game", see
45:, the first player automatically loses, and it is a second-player win. The zero game has a
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is the game where neither player has any legal options. Therefore, under the
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Winning Ways for your mathematical plays, Volume 1: Games in general
49:
of zero. The combinatorial notation of the zero game is: { | }.
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105:with two identical piles (of any size) is not the
215:(corrected ed.), Academic Press, p. 44
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113:because first player has no winning option.
52:A zero game should be contrasted with the
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64:Simple examples of zero games include
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72:diagram with nothing drawn on it.
16:Game where both players can't move
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140:, Academic Press, p. 72
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234:Combinatorial game theory
35:combinatorial game theory
19:Not to be confused with
88:Sprague–Grundy theorem
82:Sprague–Grundy theorem
43:normal play convention
138:On numbers and games
101:For example, normal
76:Sprague-Grundy value
47:Sprague–Grundy value
201:Berlekamp, Elwyn R.
68:with no piles or a
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92:impartial games
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28:Brad Meltzer
90:applies to
239:0 (number)
228:Categories
117:References
111:fuzzy game
70:Hackenbush
178:, p. 124.
154:, p. 122.
107:zero game
54:star game
39:zero game
211:(1983),
190:, p. 73.
166:, p. 87.
136:(1976),
60:Examples
37:, the
86:The
103:Nim
96:nim
66:Nim
33:In
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125:^
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30:.
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