In combinatorial game theory, a branch of mathematics studying two-player games of perfect information in extensive form, tiny and miny are operators that transform one game into another. When applied to a number (represented as a game according to the mathematics of surreal numbers) they yield infinitesimal values.
For any game or number G, tiny G (denoted by ⧾G 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 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 ⧿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.
Referencesedit
Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters, Ltd. ISBN1-56881-277-9.
tiny, miny, combinatorial, game, theory, branch, mathematics, studying, player, games, perfect, information, extensive, form, tiny, miny, operators, that, transform, game, into, another, when, applied, number, represented, game, according, mathematics, surreal. In combinatorial game theory a branch of mathematics studying two player games of perfect information in extensive form tiny and miny are operators that transform one game into another When applied to a number represented as a game according to the mathematics of surreal numbers they yield infinitesimal values For any game or number G tiny G denoted by G 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 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 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 editAlbert Michael H Nowakowski Richard J Wolfe David 2007 Lessons in Play An Introduction to Combinatorial Game Theory A K Peters Ltd ISBN 1 56881 277 9 Berlekamp Elwyn R Conway John H Guy Richard K 2003 Winning Ways for Your Mathematical Plays A K Peters Ltd Retrieved from https en wikipedia org w index php title Tiny and miny amp oldid 1224224781, wikipedia, wiki, book, books, library,
