Dudeney's Dissection is Optimal

TL;DR

This paper proves that an equilateral triangle cannot be dissected into a square with three or fewer pieces, solving a century-old puzzle by analyzing graph structures representing the dissection.

Contribution

It provides the first proof that Dudeney's dissection requires at least four pieces, introducing a novel graph-theoretic approach to dissection problems.

Findings

No three-piece dissection exists between triangle and square.

Dissection complexity is at least four pieces.

Graph analysis confirms minimal dissection size.

Abstract

In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney demonstrated a beautiful four-piece solution, which today remains perhaps the most famous example of dissection. In this paper (over a century later), we finally solve Dudeney's puzzle, by proving that the equilateral triangle and square have no common dissection with three or fewer polygonal pieces. We reduce the problem to the analysis of discrete graph structures representing the correspondence between the edges and the vertices of the pieces forming each polygon.