1. Introduction

At the workshop in combinatorial games in BIRS 2011, Aviezri Fraenkel posed the following intriguing problem: find nice (short/simple) rules for a 2-player combinatorial game for which the P-positions are obtained from a pair of complementary

Beatty sequences [Be]. We begin by solving this problem, by defining a class of heap games, dubbed bichromatic nim, or just chromatic nim, and then later in Section 4, we explain some background to the problem. In Section 3, we solve a similar game on arithmetic progressions. In Section 5, we discuss the general environment for chromatic nim on two heaps. At last, in Section 6, we study the famous evil numbers, also known as the indexes of the 0's in the Thue–Morse sequence.

2. Bichromatic nim finds a game for your complementary Beatty solution

Let S denote a subset of the positive integers. We let the i-th token in a stack be red if i ϵ S, and otherwise the i-th token is green. We play a take-away game on k ≥ 0 copies of such stacks of various finite sizes. Classical nim rules are always allowed; any number of tokens can be removed from precisely one of the stacks.