• Source: Binary game

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



X


{\displaystyle X}

of the interval



[
0
,
2
]


{\displaystyle [0,2]}

on the real line, then the players alternatively choose binary digits




x

0


,

x

1


,

x

2


,
.
.
.


{\displaystyle 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


∈



X


{\displaystyle (x_{0}{}.x_{1}{}x_{2}{}x_{3}{}...)_{2}\in {}X}

, that is,




Σ

n
=
0


∞





x

n



2

n




∈



X


{\displaystyle \Sigma _{n=0}^{\infty }{\frac {x_{n}}{2^{n}}}\in {}X}

. See, 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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