Combinatorial games and the golden ratio on digraphs

Abstract

We introduce a new combinatorial game called Triangle Game. In this game, a directed $3$-cycle graph is given, and stones are placed on each vertex. The player chooses a directed edge and takes at least one stone from the initial vertex. At the same time, the player is allowed to return some stones to the terminal vertex of the edge, as long as the total number of stones decreases. We describe the set of \Pps~under both normal play and misère play. The golden ratio $ϕ=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description. We also show that Triangle Game is tame.