Combinatorial games and the golden ratio on digraphs

Abstract: We introduce a new combinatorial game, named Triangle Game. In this game, a directed $3$-cycle graph is given, and tokens are placed on each vertex. The player chooses an edge and takes at least one token from the initial vertex. At the same time, the player is allowed to return some tokens to the terminal vertex of the edge, as far as the total number of the tokens decreases. We describe the set of \Pps~under both normal play and mis`ere play. The golden ratio $\phi=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description.