Backsliding Toads and Frogs

Summary

Abstract. Backsliding Toads and Frogs is a variant of Toads and Frogs in which virtually all positions are loopy. The game is an excellent case study of Conway's theory of sides. In this paper, we completely characterize the values of all natural starting positions. We also exhibit positions with the familiar values n and 2⁻ⁿ, as well as positions with temperatures n and 2⁻ⁿ, for all n.

Introduction

The game of Toads and Frogs was introduced in Winning Ways [Berlekamp et al. 2001]. It is played on a 1 × n strip, populated by some number of toads and frogs. Left plays by moving any toad one space to the right; Right by moving any frog one space to the left. If either player's move is blocked by the opponent, he may choose to leap over her, provided the next square is empty. Jumps do not result in capture. As usual, the winner is the player who makes the last move.

The variant Backsliding Toads and Frogs was also introduced in Winning Ways. Here both players have the additional option of retreating by one space, though reverse jumps are still prohibited. Unlike standard Toads and Frogs, the backsliding variant is loopy. As we will see, this additional rule has a monumental effect on the play of the game.

Figure 1 shows a typical position shortly after the start of the game. Each player has one advancing move and one backsliding move available, and Left has the additional option of leaping over Right's frog.