In mathematics, the binary game is a

topological game In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is ...

introduced by

Stanislaw Ulam Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapo ...

in 1935 in an addendum to problem 43 of the Scottish book as a variation of the

Banach–Mazur game In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spac ...

. 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

,2 /math> 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_1x_2x_3...)_2 \inX, that is, \Sigma^_\frac\inX . See, page 237.

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

Rényi–Ulam game Ulam's game, or the Rényi–Ulam game, is a mathematical game similar to the popular game of twenty questions. In Ulam's game, a player attempts to guess an unnamed object or number by asking yes–no questions of another, but ''one'' of the an ...

References

{{Reflist Topological games