In mathematics, tiny and miny are

operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...

that yield

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...

values when applied to numbers in

combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...

. Given a positive number G, tiny G (denoted by ⧾_G in many texts) is equal to for any game G, whereas miny G (analogously denoted ⧿_G) is tiny G's negative, or . 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 In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...

yields . 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 John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

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

* * {{cite book , first=Elwyn R. , last=Berlekamp , authorlink=Elwyn Berlekamp , 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 Combinatorial game theory