When are translations of P-positions of Wythoff's game P-positions?

2014/03/11 by Fraenkel, Aviezri S., Ho, Nhan Bao
#91A46 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.2512

Abstract

We study the problem whether there exist variants of \sc Wythoff's game whose ¶-positions, except for a finite number, are obtained from those of \sc Wythoff's game by adding a constant k to each ¶-position. We solve this question by introducing a class \\W_k\_{k ≥ 0} of variants of \sc Wythoff's game in which, for any fixed k ≥ 0, the ¶-positions of \W_k form the set \(i,i) | 0 ≤ i < k\∪ \(\lfloor ϕn \rfloor + k, \lfloor ϕ² n \rfloor + k) | n≥ 0\, where ϕ is the golden ratio. We then analyze a class \\T_k\_{k ≥ 0} of variants of \sc Wythoff's game whose members share the same ¶-positions set \(0,0)\∪ \(\lfloor ϕn \rfloor + 1, \lfloor ϕ² n \rfloor + 1) | n ≥ 0 \. We establish several results for the Sprague-Grundy function of these two families. On the way we exhibit a family of games with different rule sets that share the same set of ¶-positions.

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