Technology Category
Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningSearch and Optimization: Combinatorial Optimization
Dudeney's Dissection is Optimal
🤖 AI Summary
This paper resolves the classical geometric dissection problem posed by Dudeney (1907): determining the minimum number of polygonal pieces required for a mutual dissection between an equilateral triangle and a square. Employing a discrete graph model grounded in edge-vertex correspondence, the authors integrate computational geometry, combinatorial topology, symbolic computation, and exhaustive verification to rigorously prove that no dissection exists with three or fewer polygonal pieces. This yields the first formal optimality proof of Dudeney’s celebrated four-piece solution, thereby establishing the theoretical lower bound and settling this long-standing open problem—unresolved for over a century. The core innovation lies in a novel graph-theoretic modeling framework that formally encodes dissection constraints, enabling a complete enumeration and exclusion of all candidate configurations for dissections with fewer than four pieces.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Prove Dudeney's four-piece dissection is optimal
Show no three-piece dissection exists for triangle-to-square
Analyze graph structures representing polygon edges and vertices
Innovation
Methods, ideas, or system contributions that make the work stand out.
Proving Dudeney's four-piece solution is optimal
Analyzing discrete graph structures for dissection
No three-piece solution exists for triangle-square
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