The Projectivity of $Y$-Games

Abstract

The first approach to the theory of regular Nim-type games was made by Grundy [2] and Sprague [5], where it was clarified that the Grundy number plays an important role. Guy and Smith [3] introduced the games parametrized by Grundy value and studied the periodicity of the series of Grundy numbers of games of fixed Grundy value with respect to the heap size. Examples of Grundy values whose series of Grundy numbers are non-periodic were found by Yoneda (see [4]). On the other hand Yamasaki [6] studied the mis ere Nim-type games, remarking the singular D-schemes, where a D-scheme D indicates a Nim-type game without decision of winners and D is said to be singular if the first player has a winning strategy in strictly one of the regular game and the mis ere game played over D. It is shown that the flat D-schemes form a unique maximal class of Z)-schemes satisfying the end game modification theory. The flatness, however, of a D-scheme D is proved when the whole restrictions of D of Grundy number 0 or 1 are known. A sufficient condition 'projectivity' of 'flatness' was introduced so that one can see the projectivity of a D-scheme D when he finds a set 2 of restrictions of D satisfying several conditions, where EF becomes clear to coincide with the set of singular restrictions of D. It was clarified that Nim, Restricted Nim, Block Nim, Tsyan-shizi (Chinese Nim, Wythoff's Nim) and Sato's Maya game (Welter's game) are projective and that Keyles is not flat. In this paper we shall see the projectivity of the generalized Yoneda's games which are equivalent to the games with Grundy value 0.*****