combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a ''position'' ev ...

, a branch of mathematics studying two-player games of perfect information in

extensive form In game theory, an extensive-form game is a specification of a game allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the (possibly imperfec ...

, 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 ...

that transform one game into another. When applied to a number (represented as a game according to the mathematics of

surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...

s) they yield

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

values. For any game or number G, tiny G (denoted by ⧾_G in many texts) is the game , 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 . 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. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...

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.

Unicode

* * * (pronounced "tee-nie", /tiːniː/)Miscellaneous Mathematical Symbols-B
Page 5 * (pronounced "mee-nie", /miːniː/)

References

* * Combinatorial game theory {{Combin-stub