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Published online by Cambridge University Press: 27 June 2025
We present a number of new results for the combinatorial game Toads and Frogs. We begin by presenting a set of simplification rules, which allow us to split positions into independent components or replace them with easily computable numerical values. Using these simplication rules, we prove that there are Toads and Progs positions with arbitrary numerical values and arbitrarily high temperatures, and that any position in which all the pieces are contiguous has an integer value that can be computed quickly. We also give a closed form for the value of any starting position with one frog, and derive some partial results for two-frog positions. Finally, using a computer implementation of the rules, we derive new values for a large number of starting positions.
1. Introduction
Toads and Frogs is a two-player game, played on a one-dimensional board. Left has a number of toads, and Right has a number of frogs, each on its own square of the board. Each player has two types of legal moves: he may either push one of his pieces forward into an adjacent empty square, or he may jump one of his pieces over an adjacent opposing piece, into an empty square. Jumps are never forced, and jumped-over pieces are not affected in any way. Toads move to the right, frogs to the left. The first player without a legal move loses the game.
Throughout the paper, we represent toads by T, frogs by F, and empty squares by the symbol □.
Here is a typical Toads and Frogs game. Left moves first and wins.
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