1. Introduction

This paper uses ideas from combinatorial game theory, Beatty sequences, and Sturmian words. We have in many cases given the pertinent information in this paper, but have chosen to omit some material on these subjects. The reader who wishes to have more background information on certain topics is directed to the following references.

For standard terminology of impartial removal games on heaps of tokens, see [WW]; for Beatty sequences, see [B]; for k-Wythoff Nim, see [W; F]; for Sturmian words, see [L]; and for continued fractions, see [K].

Our problem is an inverse to that of the main field of research, which for a given an impartial ruleset Γ, (for example, Γ = k-Wythoff Nim) is to determine the P-positions of Γ (within reasonable time-complexity). Here we rather start with a particular (candidate) set of P-positions and search for “simple” game rules. Let us explain the setting.

Throughout this paper, we will denote the position consisting of two heaps of x ≥ 0 and y ≥ 0 tokens as (x, y). When the values of x and y are known, we adopt the convention that x ≤ y, though in general we regard such a position as an unordered multiset, so we identify (y, x) with (x, y).

Similarly, we let the move (u, v) denote a removal of v >0 tokens from one of the heaps and u from the other, where 0≤u ≤v; thus from the position (x, y), the move (u, v) is ambiguous, being either (x, y)→(x −u, y −v), provided both x−u ≥0 and y−v ≥0, or (x, y)→(x−v, y−u), provided both x−v ≥0 and y −u ≥ 0.