User:Jaberkow

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  in many texts) is equal to {0||0|-G} for any game G, whereas miny G (analogously denoted  ) 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  is infinitesimal with respect to ↑ for all positive values of x,  is equal to up. Expansion of  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  + ↓ will show that it is a second player win, and that, consequently,  ↑. Similarly curious, Conway noted, calling it “amusing,” that “↑ is the unique solution of .” Conway’s assertion is also easily verifiable with canonical forms and game trees.