For over a century, a simple yet tricky math problem had continued to baffle experts.
Mathematicians struggled to find the fewest number of pieces needed to cut an equilateral triangle and rearrange it into a perfect square.
The problem, known as “Dudeney’s dissection” or the “haberdasher’s problem,” was first posed in 1902 when a self-taught English mathematician and puzzle columnist Henry Dudeney challenged his readers to cut an equilateral triangle into the fewest pieces possible and rearrange them into a square.
Two weeks later, he shared a solution from Charles William McElroy, a Manchester clerk who often sent him puzzle answers. McElroy had found a way to do it in four pieces.
As another two weeks passed, the puzzle columnist confirmed that no one had discovered a better solution. The record stood, but it remained uncertain whether a solution with fewer pieces was possible.
In graph theory, a graph is a network of lines, called edges, and points where they meet, called vertices.
Century-old math puzzle
After 122 years, a team of mathematicians has finally solved the puzzle. They proved that no smaller solution exists.
Tonan Kamata, a mathematician at the Japan Advanced Institute of Science and Technology (JAIST), along with Ryuhei Uehara and Erik Demaine, a colleague from Massachusetts Institute of Technology, had been developing a new approach to tackle origami-folding problems using graph theory.
By comparing the edges and vertices of different graphs, mathematicians can uncover deeper connections between structures.
Kamata believed this method could help solve Dudeney’s dissection.
“I believe many who appreciate mathematics would agree that the simpler an unsolved problem appears, the more profoundly captivating it becomes to those who love mathematics,” Kamata said.
While solving the puzzle, the experts found out that a two-piece solution was not possible due to the problem’s constraints.
To begin with, the triangle and square must have the same area since they are made from the same pieces.
The longest possible cut in a square is its diagonal.
However, simple calculations show that the diagonal is too short to match the edge of an equal-area triangle. This ruled out the two-piece solution.
To prove that a three-piece solution was trickier, the mathematicians found that there were an infinite number of ways to cut up the triangle in this solution.
“Each of those pieces could have arbitrarily many edges to it, and the coordinates of those cuts start at arbitrary points,” Demaine said.
“You have these continuous parameters where there are lots and lots of infinities of possible choices that make it so annoyingly hard. You can’t just brute-force it with a computer.”
Riddle solved, 122-year debate ends
To solve the puzzle, the trio grouped possible dissections of an equilateral triangle based on how the cuts intersect its edges. First, they narrowed down the infinite ways to cut the triangle into five distinct classifications.
They then applied the same method to a square and identified 38 unique classifications.
After this, the mathematicians attempted to match a triangular graph to a square by tracing all possible paths in each shape and comparing the edge lengths and angles.
If a path in the square had aligned with one in the triangle, it would have confirmed the existence of a three-piece solution.
This strategy almost changed a continuous problem into a discrete one.
“Within each classification, there are still infinitely many places all these vertices could go,” Demaine says.
The researchers developed complex lemmas—intermediate steps in a theorem to solve the math problem. Using proof by contradiction, they showed that no matching paths existed.
If the authors simplify their proof, the matching-diagrams technique could help solve many other origami-like open questions.
“These problems remind us how much there is yet to discover,” Kamata says. “Anyone can become a pioneer in this frontier.”
The result was posted on arXiv.org in a December 2024 preprint entitled “Dudeney’s Dissection Is Optimal”.
