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 ${\displaystyle X}$ of the interval ${\displaystyle [0,2]}$ on the real line, then the players alternatively choose binary digits ${\displaystyle x_{0},x_{1},x_{2},...}$. Player I wins the game if and only if the binary number ${\displaystyle (x_{0}{}.x_{1}{}x_{2}{}x_{3}{}...)_{2}\in {}X}$, that is, ${\displaystyle \Sigma _{n=0}^{\infty }{\frac {x_{n}}{2^{n}}}\in {}X}$. See,^[1] page 237.

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

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