  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.
