Further Hopping with Toads and Frogs

Abstract

We prove some new results about the combinatorial game "Toads and Frogs". We give a finite number of recurrence relations for computing the values of all positions with exactly one $\Box$. We show that ${\rm T}^{a}\Box\Box{\rm F}^{a}$ is an infinitesimal for $a\geq 4$. At the end, we make five new conjectures and describe possible future work.