Pilesize Dynamic One-pile Nim and Beatty's Theorem

In [4] we proved a generalization of Beatty’s Theorem which we stated came from the Nim value analysis of a game. In this paper we give the Nim value analysis of this game and show its relationship with Beatty’s Theorem. The game is a one-pile counter pickup game for which the maximum number of counters that can be removed on each successive move changes during the play of the game. The move size is bounded by a move function f whose arguments are pile sizes. After analyzing this game, we discuss a blocking version of this game as well as the mis`ere version