Variants of Wythoff's game translating its P-positions

We introduce a restriction of Wythoff's game, which we call F-Wythoff, in which the integer ratio of entries must not change if an equal number of tokens are removed from both piles. We show that P-positions of F-Wythoff are exactly those positions obtained from P-positions of Wythoff's game by adding 1 to each entry. We describe the distribution of Sprague-Grundy values and, in particular, generalize two properties on the distribution of those positions which have Sprague-Grundy value k, for a given k, for variants of Wythoff's game. We analyze the misere F-Wythoff and show that the normal and misere versions differ exactly on those positions which have Sprague-Grundy values 0, and 1 via a swap. We examine two further variants of F-Wythoff, one restriction and one extension, preserving its P-positions. We raise two general questions based on the translation phenomenon of the P-positions.

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