2015/03/19 by Urban Larsson, Larsson, Urban
Computer Science · Mathematics · #Coding theory and cryptography #Computability, Logic, AI Algorithms #math.CO #msc:11B99 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1503.05751
2 pages
arxiv created 2015/03/25 · arxiv updated 2015/03/26
In a recent arXiv-manuscript Fox studies infinite subtraction games with a finite (ternary) and aperiodic Sprague-Grundy function. Here we provide an elementary example of a game with the given properties, namely the game given by the subtraction set \F2n+1-1\, where Fi is the ith Fibonacci number, and where n ranges over the positive integers. Our definition of nim-dimension reflects the precise number of power-of-two-components generated by the games; the group of nim-values is of order four so the dimension is two (in the classical definition this dimension would have been one). Thanks to Carlos Santos for an enlightening discussion on this matter.