Further Hopping with Toads and Frogs

TL;DR

This paper analyzes the combinatorial game 'Toads and Frogs', providing new position values, discussing specific configurations, and proposing conjectures for future research.

Contribution

It introduces new values for starting positions and explores the values of specific configurations, advancing understanding of the game's combinatorial structure.

Findings

New position values identified

Analysis of specific configurations provided

Five new conjectures proposed

Abstract

We show the value of positions of the combinatorial game ``Toads and Frogs''. We present new values of starting positions. Moreover, we discuss the values of all positions with exactly one $\Box, \regT^{a}\Box\Box \regF^{a}, \regT^{a} \Box \Box \Box \regF \regF \regF,\regT^{a}\Box\Box \regF^{b}$, $\regT^{a}\Box\Box\Box \regF^{b}$. At the end, we post five new conjectures and discuss the possible future work.