A new heap game

Given $k\ge 3$ heaps of tokens. The moves of the 2-player game introduced here are to either take a positive number of tokens from at most $k-1$ heaps, or to remove the {\sl same} positive number of tokens from all the $k$ heaps. We analyse this extension of Wythoff's game and provide a polynomial-time strategy for it.Comment: To appear in Computer Games 199

Cubic Pisot Unit Combinatorial Games

Generalized Tribonacci morphisms are de ned on a three letters alphabet and generate the so-called generalized Tribonacci words. We present a family of combinatorial removal games on three piles of tokens whose set of P-positions is coded exactly by these generalized Tribonacci words. To obtain this result, we study combinatorial properties of these words like gaps between consecutive identical letters or recursive de nitions of these words. Beta-numeration systems are then used to show that these games are tractable, i.e., deciding whether a position is a P-position can be checked in polynomial time