On Invariant Games

A 2-pile take-away game is played by two players who alternate in removing tokens from two piles according to given rules. The game is invariant if if the same moves can be played from any game position, provided only that there are enough tokens in the piles. Duchêne and Rigo in a paper in this issue conjectured that if the second player winning positions are given by a pair of complementary homogeneous Beatty sequences, then the game is invariant. We prove this conjecture.