On Restricted Wythoff’s Nim

We shall study the following restricted Wythoff’s Nim. Let Si (1 ≤ i ≤ 3) be the set of positive integers. Each player can remove the number of tokens s1 ∈ S1 from the first pile and s2 ∈ S2 from the second pile and remove the same number of tokens s3 ∈ S3 from both piles. We shall identify (m, n) to a position of this nim, where m is the number of tokens in the first pile and n is the number of tokens in the second pile. In the case |S2| < ∞, we will show the Sprague-Grundy sequence(or simply G-sequences) gS(m, n) is periodic for fixed m.