The arithmetic of reversed positional games Original Research Article

Berlekamp et al. (1982) and Conway (1976) showed that the real numbers can be regarded as the outcome of games. The purpose of this paper is to investigate the positional games introduced by Berge (1976), with the decision of winner reversed. We shall conclude that they are congruent to numbers modulo ∗ provided no draw is possible, where ∗ = {0|0} denotes the non-numerical game with the earliest birthday (cf. Berlekamp et al., 1982; Conway, 1976). We shall conclude also that a reversed positional game of second type is congruent to an integer modulo ∗.