The G-values of various games

A disjunctive combination of a finite set of two-person games Γ 1 , Γ 2 , …, Γ k may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that (i) a player loses if unable to move (in any game), (ii) the games are impartial , i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary), (iii) the games are of bounded play , i.e. for each game Γ i corresponding to any initial position P j there is an integer b ij such that the game must terminate after at most b ij moves, then Grundy (6) has shown that there is a function G ( P ) (called by him Ω( P )) of the positions P with the following properties: (1 a ) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Q i ) , where there is a permissible move from P to Q i , (1 b ) for a disjunctive combination of games, G for the combined position is the nim-sum of the G 's the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σ a r 2 r , and then in forming the sum, the a r 's are to be added mod 2 for each value of r , as in the theory of Nim((1), (7), (8), pp. 36–8).