Additive Periodicity of the Sprague–Grundy Function of Certain Nim Games

Abstract

We deduce conditions for additive periodicity of the Sprague᎐Grundy function of Nim-like games, starting with Wythoff's classic game and ending up with a fairly large class of impartial games played on directed graphs. ᮊ 1999 Academic Press

More on the Sprague–Grundy function for Wythoff's game

“…This result was first proven by Norbert Pink in his doctoral thesis [8] (published in [3]), and Landman [4] later found a simpler proof. Both [3] and [4] derive an upper bound of 2 O.x 2 / for the preperiod and the period of row x.…”

“…See [6] for details. Blass and Fraenkel [2] obtained several results on the sequence T 1 of 1-values, as defined above (1)(2)(3)(4). They showed that the n-th 1-value is within a bounded distance to the n-th 0-value.…”

“…for 0 h g; (3)(4)(5)(6)(7)(8) and with the correct values of insert.r 1 /, insert.r 1 C 1/; : : : , then the algorithm will converge to the correct state within at most R g rows.…”

“…We tested Conjecture 3.10 experimentally as follows: For some constant r max we precalculated the state of row r and the value of insert.r / for all r between 0 and r max . We then ran Algorithm FSW starting from each row r , 0 r r max , with the dummy initial state (3)(4)(5)(6)(7)(8), comparing at each step whether the algorithm's internal state converged to the correct state of the current row. We carried out this experiment for different values of g. Our results are as follows: For g D 0 convergence always occurs after 0 rows; i.e., convergence is immediate.…”

On Fraenkel’s N-Heap Wythoff’s Conjectures

“…They can also be written as A n = nφ and B n = nφ 2 , where φ = (1 + √ 5)/2 (the golden section). Various generalizations and results on this game were done by Blass and Fraenkel [1], Blass, Fraenkel, Guelman [2], WW [3], Coxeter [4], Dress [5], Fraenkel and Borosh [8], Fraenkel and Ozery [9], Fraenkel and Zusman [10], Landman [12], Yaglom and Yaglom [14]. 1 Another generalization of Wythoff's game, involving more than two piles, was proposed by Fraenkel [7], which is listed in the survey article by Guy and Nowakowski [11] as one of the "unsolved problems in combinatorial games".…”