In 1903, Henry Dudeney published a puzzle: cut an equilateral triangle into pieces that rearrange into a perfect square. His solution used four pieces, connected by hinges so the triangle swings open into the square in a single motion. The solution was elegant. The question it left open was not.
For 120 years, nobody proved that four was the minimum. Dudeney's dissection was the best known result — a record, not a theorem. Records invite challengers. Someone might find a three-piece dissection, or a two-piece dissection, and the record would fall. The absence of a better solution proved nothing about the impossibility of one. Mathematics distinguishes sharply between “nobody has found fewer” and “fewer cannot exist.”
Uehara, Kamata, and Demaine closed the door. Their proof, presented in January 2025, shows that no dissection of an equilateral triangle into a square uses fewer than four polygonal pieces.
The proof method is what matters. They don't enumerate all possible three-piece configurations and check each one. Instead, they reduce the dissection problem to a graph structure — a matching diagram that captures how edges and vertices of the cut pieces must relate to both the triangle and the square simultaneously. The graph encodes the topology of the rearrangement: which edges of a piece form part of the triangle boundary, which form part of the square boundary, and which are interior cuts. Three pieces produce a graph with constraints from both shapes. The constraints are incompatible. The impossibility is structural, not computational. The proof doesn't say “we checked and nothing works.” It says “the topology doesn't allow it.”
This is a different kind of closure from finding a better solution. A better solution replaces the old record with a new one — the door stays open, just the threshold moves. A lower bound proof locks the door. The four-piece dissection is no longer the best anyone has found. It is the best that exists. The distinction changes the relationship between the result and future work: nobody will ever improve on Dudeney's 1903 solution, not because the problem is hard, but because the answer is final.
The 120-year gap is itself interesting. Dudeney's puzzle is simple to state, the answer is elegant, and the optimality proof uses techniques (graph matching, combinatorial topology) that have existed for decades. The tools were available long before the proof was found. The question sat open not because it was beyond reach but because it was not urgent. Records are comfortable. A four-piece dissection works, the hinged version is beautiful, and mathematicians moved on to other problems. The proof didn't require new mathematics — it required someone to decide the question was worth closing.