Hiroki Inazu | Searcharxiv

arXiv2025

$\mathcal{L}\mathcal{R}$-Ending partisan rulesets

In this paper, we consider $\mathcal{L}\mathcal{R}$-ending partisan rulesets as a branch of combinatorial game theory. In these rulesets, the sets of options of both players are the same. However, there are two kinds of terminal positions. If the game ends in one kind of terminal position, then a player wins, and if the game ends in the other kind of terminal position, the other player wins. We introduce notations for positions in $\mathcal{L}\mathcal{R}$-ending partisan rulesets including disjunctive sum of terminal positions depending on the parity and basic definitions and show their algebraic structures. We also introduce some examples of $\mathcal{L}\mathcal{R}$-ending partisan rulesets and show how our results can be used for analyzing the rulesets.

math.CO↗

arXiv2025

Combinatorial games and the golden ratio on digraphs

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 mis\`ere play. The golden ratio $\phi=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description. We also show that Triangle Game is tame.

math.CO↗