Fractal Sequences and Restricted Nim [article]

Lionel Levine
2004 arXiv   pre-print
The Grundy number of an impartial game G is the size of the unique Nim heap equal to G. We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these "fractal sequences" and give a bijection between these
more » ... es and certain upper-triangular arrays. As a special case we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap.
arXiv:math/0409408v1 fatcat:vcqnwkgafbcr3njdhdqjtitizu