  We prove three conjectures of Fraenkel and Ho regarding two classes of
variants of Wythoff's game. The two classes of variants of Wythoff's game
feature restrictions of the diagonal moves. Each conjecture states that the
Sprague-Grundy function is invariant up to a certain nim-value for a subset of
that class of variant of Wythoff's game. For one class of variants of Wythoff's
game, we prove that the invariance of the Sprague-Grundy function extends
beyond what was conjectured by Fraenkel and Ho.
