Wythoffʹs sequence and image-Heap Wythoffʹs conjectures Original Research Article

Define a Wythoffʹs sequence as a sequence of pairs of integers image such that there exists a finite set of integers T, image, image, and image. Structural properties and behaviors of Wythoffʹs sequence are investigated. The main result is that for such a sequence, there always exists an integer image such that when n is large enough, image, where image, the golden section. The value of image can also be easily determined by a relatively small number of pairs in the sequence. As a corollary, the two conjectures on the N-heap Wythoffʹs game by Fraenkel [Complexity, appeal and challenges of combinatorial Games, Theoret. Comput. Sci. 313 (2004) 393–415] on the N-heaped Wythoffʹs game are proved to be equivalent.