River Crossing Problems: Algebraic Approach

TL;DR

This paper applies algebraic and category theory methods to analyze classic river crossing puzzles, revealing their structural relationships and historical context.

Contribution

It introduces an algebraic framework using symmetry groups and category theory to study and connect the jealous husbands and missionaries and cannibals problems.

Findings

Symmetry group actions elucidate the structure of the jealous husbands problem.

Category theory describes the relationship between the two puzzles.

Historical insights link the problems to the development of group theory.

Abstract

We consider two river crossing problems, about jealous husbands and about missionaries and cannibals. The missionaries and cannibals problem arose a thousand years after the jealous husbands problem, although its solution had actually appeared several hundred years before its formulation. We apply an algebraic approach to study these problems, using a symmetry group action on the state set of the jealous husband problem; then category theory is used to describe the relationship between the two problems. Some historical issues are also touched, related to the fact that the missionaries and cannibals problem arose precisely when the group approach began to be widely spread and popularized. This is the approach that naturally connects both problems.