TOKYO -- A Japan-U.S. research team on March 10 offered a mathematical proof that "Dudeney's dissection," a challenge to cut an equilateral triangle and use the parts to create a square with the fewest pieces, cannot be solved using three or fewer pieces.

The challenge was first presented by English mathematician Henry Dudeney, who was also a puzzle maker, in a magazine's puzzle column in 1902. While many answers were submitted, just one solution was awarded a prize. It involved cutting the equilateral triangle into four pieces. However, whether this solution was optimal, meaning whether using fewer than four pieces was possible, remained an unresolved question until now.

The research team from the Japan Advanced Institute of Science and Technology (JAIST) and the Massachusetts Institute of Technology, led by JAIST assistant professor of mathematics Tonan Kamata, began work around 2018 to prove the impossibility of solving the puzzle with three pieces.

Since the number of possible division patterns appeared infinite, directly checking all configurations was out of the question. To overcome this, the team devised their own approach of classifying patterns based on how dividing lines connect and considering characteristics such as the lengths and positions of sides of the resulting shapes.

Through this method, they narrowed down what was initially thought to be an infinite number of patterns down to 37. Assuming that the puzzle could be solved using three pieces, they analyzed each of these configurations and discovered contradictions such as adjacent sides not aligning properly. They successfully proved the impossibility by using proof by contradiction, a logical proof method in which assumption that the hypothesis is false is shown to lead to a contradiction.

Various approaches to solving the puzzle have previously been proposed, but mathematically proving the impossibility of a three-piece solution had long been considered challenging.

Professor emeritus Jin Akiyama of the Tokyo University of Science, who was Kamata's academic adviser, described the research as "extremely ingenious." He suggested the technique could be applied to proving the impossibility of other puzzles as well. He also noted that the research holds potential practical applications, such as compacting solar panels into limited spaces for transportation and deployment in space, as well as improving research and development involving shape transformations.

Kamata noted, "When first encountering Dudeney's puzzle, I was deeply moved. However, there were very few theoretical examinations available that explained it clearly." Wanting to explain it in his own words, he decided to start researching it. He added, "Now that I've clearly articulated it, I'd like to apply what I've learned to resolving other problems."

The paper was published in the open research-sharing platform ArXiv at https://arxiv.org/abs/2412.0386.

(Japanese original by Kouki Matsumoto, Science & Environment News Department)