The optimal solution to the problem of dividing an equilateral triangle into the minimal number of pieces and recombining them into a square, known as "Dudeney’s Puzzle," has been proven after 120 years.
The Japan Advanced Institute of Science and Technology (JAIST) announced on the 11th that "professors Ryuhei Uehara and Tohann Kamata proved that the minimum number of pieces required to convert an equilateral triangle into a square is four." Professor Erik Demaine from the Massachusetts Institute of Technology (MIT) also participated in the study. The results of this research were uploaded to the preprint site arXiv on Dec. 5 of last year.
The history of Dudeney’s Puzzle dates back to 1907. British mathematician and puzzle author Henry Ernest Dudeney posed the question: "Is it possible to divide an equilateral triangle into the least number of pieces and convert it into a square?" Dudeney announced a method to convert it using just four pieces four weeks after presenting the puzzle, but whether it could be cut into three pieces or fewer remained an unsolved problem for 120 years.
To solve this challenging problem, the researchers took a step-by-step approach. They first analyzed whether it was possible to assemble a square by dividing an equilateral triangle into two pieces, concluding that it was impossible due to geometric constraints, so they excluded that option. They then examined the possibility of converting it into three pieces by narrowing down the possible cutting combinations using the basic properties of cuts.
In this process, the researchers applied the matching diagram technique. A matching diagram converts the relationships between the edges and vertices of the cut pieces into a graph structure for analysis, allowing a systematic review of whether a cutting method exists that can form an equilateral triangle and a square. The researchers used this technique to prove that no method could form a perfect square by dividing the equilateral triangle into three pieces or fewer.
This research is expected to contribute not only to the study of geometric puzzles but also to the development of optimization algorithms in manufacturing, textile design, and engineering cutting processes. Furthermore, the analytical technique developed by the researchers could be applied to various geometric transformation problems, making it potentially useful for future research.
Baek Jin-eon, a postdoctoral researcher in the Mathematics Department at Yonsei University, noted, "Problems involving arbitrary shapes are often not well-studied because their possibilities cannot be described by a finite number of variables. This research is significant as it provides a rare mathematical proof that explores all these shapes, and it could offer new inspiration or techniques for addressing shape-related problems due to its potential to handle infinite-dimensional shape spaces."
Dr. Baek gained attention in the mathematics community for solving the well-known 60-year problem called the "sofa moving problem," which appears in American high school math textbooks. The problem asks what the largest area plane figure that can pass through a right-angled corridor of width 1 is, under the condition that the sofa cannot be stood upright, disassembled, or tilted. It was proposed by Canadian mathematician Leo Moser in 1966.
References
arXiv (2024), DOI: https://doi.org/10.48550/arXiv.2412.03865
arXiv (2024), DOI: https://doi.org/10.48550/arXiv.2411.19826