Binary game explained

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 ^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

X of the interval [ 0 , 2 ] on the real line, then the players alternatively choose binary digits x _{0,} x _{1,} x _{2,} ... . Player I wins the game if and only if the binary number (x _{0{}.x} _{1{}x} _{2{}x} _{3{}...)} _{2} \in{}X , that is, \Sigma ^{inf ty} _{n =0} (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.

Notes and References

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