Two variants of Wythoffʼs game preserving its -positions

We present two variants of Wythoffʼs game. The first game is a restriction of Wythoffʼs game in which removing tokens from the smaller pile is not allowed if the two entries are not equal. The second game is an extension of Wythoffʼs game obtained by adjoining a move allowing players to remove k tokens from the smaller pile and l tokens from the other pile provided l < k . We show that both games preserve the P -positions of Wythoffʼs game. This resolves a question raised by Duchêne, Fraenkel, Nowakowski and Rigo. We give formulas for those positions which have Sprague–Grundy value 1. We also prove several results on the Sprague–Grundy functions.