1. Introduction

The first author posed an intriguing problem at the GONC-workshop: “Describe nice/short rulesets for games with so-called complementary Beatty sequences as sets of P-positions.” The problem is the opposite of the main field of research in this area, which is to, given a game, search for its set of P-positions. Here we describe three such rule sets, which resolves the question for any pair of complementary homogenous Beatty sequences.

Let us recall the rules of d-Wythoff [10], d a fixed positive integer. The available positions are (x, y), x and y nonnegative integers. The legal moves are

• (I) Nim-type: (x, y) → (x − t, y), if x − t ≥ 0 and (x, y) → (x, y − t), if y −t ≥ 0; t > 0.

• (II) Extended diagonal type: (x, y)→(x −s, y−t) if |t −s|<d and x −s ≥ 0, y −t ≥ 0; s > 0, t > 0.

This game is a so-called impartial take-away game [2], vol. 1. We restrict attention to normal play; that is, the player first unable to move loses. For our games it means that the player called upon to move from (0, 0) loses.

Rules (I) and (II) imply that d-Wythoff is a so-called invariant [5; 16] (takeaway) game; that is, each available move is legal from any position as long as the resulting position has nonnegative coordinates. Every move in any invariant game is an invariant move. In this note we study another type of take-away game, where certain positions have some local restrictions on the set of otherwise invariant moves. Such games are called variant [5; 16]. We define these notions in Section 4.