Binary game

In mathematics, the binary game is a topological game introduced by Stanislaw Ulam in 1935 in an addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.

In the binary game, one is given a fixed subset X of the set {0,1}N of all sequences of 0s and 1s. The players take it in turn to choose a digit 0 or 1, and the first player wins if the sequence they form lies in the set X. Another way to represent this game is to pick a subset of the interval on the real line, then the players alternatively choose binary digits . Player I wins the game if and only if the binary number , that is, . See,[1] page 237.

The binary game is sometimes called Ulam's game, but "Ulam's game" usually refers to the Rényi–Ulam game.


  1. Telgársky, Rastislav (Spring 1987). "Topological Games: On the 50th Anniversary of the Banach-Mazur Game" (PDF). Rocky Mountain Journal of Mathematics. 17 (2): 227–276.
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