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that left can end the game immediately, or on the second move, but right can reach position G if allowed to move twice in a row. This is generally applied when the value of G is positive (representing an advantage to right); tiny G is better than nothing for right, but far less advantageous. Symmetrically, miny G (analogously denoted ⧿
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in many texts) is the game {0|{0|-G}}, using the bracket notation for combinatorial games in which the left side of the vertical bar lists the game positions that the left player may move to, and the right side of the bar lists the positions that the right player can move to. In this case, this means
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Tiny and miny aren't just abstract mathematical operators on combinatorial games: tiny and miny games do occur "naturally" in such games as toppling dominoes. Specifically, tiny
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yields {0|{{0|{{0|{0|-G}}|0}}|0}}. While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾
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that transform one game into another. When applied to a number (represented as a game according to the mathematics of
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133:= G." Conway's assertion is also easily verifiable with canonical forms and game trees.
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23:, a branch of mathematics studying two-player games of perfect information in
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is a natural number, can be generated by placing two black dominoes outside
117:+ ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾
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have certain curious relational characteristics. Specifically, though ⧾
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noted, calling it "amusing," that "↑ is the unique solution of ⧾
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is infinitesimal with respect to ↑ for all positive values of
148:
Lessons in Play: An
Introduction to Combinatorial Game Theory
146:; Nowakowski, Richard J.; Wolfe, David (2007).
50:For any game or number G, tiny G (denoted by ⧾
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179:Winning Ways for Your Mathematical Plays
123:= ↑. Similarly curious, mathematician
63:) is tiny G's negative, or {{G|0}|0}.
16:Operators in combinatorial game theory
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101:is equal to up. Expansion of ⧾⧾⧾
78: + 2 white dominoes.
1:
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197:Combinatorial game theory
21:combinatorial game theory
166:Berlekamp, Elwyn R.
150:. A K Peters, Ltd.
182:. A K Peters, Ltd.
144:Albert, Michael H.
125:John Horton Conway
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81:Tiny games and
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45:infinitesimal
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137:References
107:into its
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191:Category
176:(2003).
70:, where
47:values.
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95:, ⧾⧾⧾
152:ISBN
35:are
33:miny
31:and
29:tiny
19:In
193::
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168:;
83:up
27:,
160:.
130:G
120:G
114:G
104:G
98:G
93:x
88:G
76:n
72:n
68:n
60:G
53:G
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