| 词条 | Tiny and miny |
| 释义 |
In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G (denoted by ⧾G in many texts) is equal to {0||0|-G} for any game G, whereas miny G (analogously denoted ⧿G) is tiny G's negative, or {G|0||0}. 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 n, where n is a natural number, can be generated by placing two black dominoes outside n + 2 white dominoes. Tiny games and up have certain curious relational characteristics. Specifically, though ⧾G is infinitesimal with respect to ↑ for all positive values of x, ⧾⧾⧾G is equal to up. Expansion of ⧾⧾⧾G into its canonical form yields {0||||||0|||||0||0|-G|||0||||0}. While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾G + ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾G = ↑. Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "↑ is the unique solution of ⧾G = G." Conway's assertion is also easily verifiable with canonical forms and game trees. References
| first2=Richard J. | last2=Nowakowski | | first3=David | last3=Wolfe | author3-link= | title=Lessons in Play: An Introduction to Combinatorial Game Theory | publisher=A K Peters, Ltd. | year=2007 | isbn= 1-56881-277-9
| first2=John H. | last2=Conway | author2-link=John Horton Conway | first3=Richard K. | last3=Guy | author3-link=Richard K. Guy | title=Winning Ways for Your Mathematical Plays | publisher=A K Peters, Ltd. | year=2003 1 : Combinatorial game theory |
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