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Dudeney’s Dissection Is Optimal

Authors Erik D. Demaine , Tonan Kamata , Ryuhei Uehara

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LIPIcs.ITCS.2026.47.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
Keywords
  • Geometric Dissection
  • Dudeney Dissection
  • Dissection with Fewest Pieces

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Abstract

In 1907, Henry Ernest Dudeney posed a puzzle: "cut any equilateral triangle ... 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.

Cite As Get BibTex

Erik D. Demaine, Tonan Kamata, and Ryuhei Uehara. Dudeney’s Dissection Is Optimal. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 47:1-47:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.47

Author Details

Erik D. Demaine
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Tonan Kamata
  • Japan Advanced Institute of Science and Technology, Ishikawa, Japan
Ryuhei Uehara
  • Japan Advanced Institute of Science and Technology, Ishikawa, Japan

Funding

  • Kamata, Tonan: is mainly supported by JSPS KAKENHI Grants Number 22J10261 and 24K23857, and partially supported by Grants Number 20H05961 and 20H05964, and by JST ACT-X Grant Number JPMJAX25C8.
  • Uehara, Ryuhei: is mainly supported by JSPS KAKENHI Grants Number 24H00690 and partially supported by Grants Number 22H01423, 20H05961, and 20H05964.

Acknowledgements

The authors would like to thank the members of the Tokyo-based math club "sugakuday," a group that describes itself as a gathering of people who like math - or don’t. During a visit by Tonan Kamata, a discussion with the group - particularly with Motcho Ajisaka and Fueruma (@motcho_tw, @fueruma_suugaku on 𝕏) - led to the discovery of an oversight in the enumeration of square dissections in the previous version of this paper. We express our deep respect for their open, playful, and curiosity-driven community.

References

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