Английская Википедия:Binary game

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In mathematics, the binary game is a topological game introduced by Stanisław 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 <math>X</math> of the interval <math>[0,2]</math> on the real line, then the players alternatively choose binary digits <math>x_0, x_1, x_2, ...</math>. Player I wins the game if and only if the binary number <math>(x_0{}.x_1{}x_2{}x_3{}...)_2 \in{}X</math>, that is, <math>\Sigma^{\infty}_{n=0}\frac{x_n}{2^n}\in{}X</math>. See, page 237.

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