Haberdasher's Problem

The haberdasher's problem asks for a dissection of an equilateral triangle into as few pieces as possible that can be rearranged to form a square of the same area. Demaine et al. (2026) give a summary of the problem's history. It was posed by Dudeney (1902) in his April 6, 1902 column with no clear indication of whether a solution was known. In the next issue of his column (April 20, 1902), Dudeney gave a five-piece solution while noting that C. W. McElroy of Manchester had found a four-piece solution.

In the following column (May 4, 1902), Dudeney presented the four-piece solution illustrated above, though without a clear indication of whether this dissection was due to Dudeney or McElroy (Frederickson 1997, Frederickson 2002, Demaine et al. 2026).

The name comes from a fictional haberdasher who challenges the other Canterbury pilgrims to cut an equilateral triangle of cloth into four pieces that form a square (Dudeney 1907). The story appeared as "The Haberdasher's Puzzle" in The London Magazine in 1903 (Frederickson 2019) and was later collected in Dudeney (1907).

Label the dissection as shown above. Then the vertices of the triangle correspond to a single point in the square. On the other hand, the points and in the triangle each bifurcate into two separate points corresponding to different vertices of the square. In the diagram, and , so and bisect and , respectively. Furthermore, as a result of the fact that points and become vertices of the square, the angles , , , and are all right angles.

The diagram above due to Stan Wagon (pers. comm., Feb. 18, 2025) shows how to superimpose triangles on squares to obtain the dissection.

The four pieces of the dissection consist of three distinct quadrilaterals and a triangle. Letting the triangle have unit edge length, there are six distinct edge lengths, given from smallest to largest by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

Amazingly, not only does this dissection allow the equilateral triangle to be dissected into a square with only three cuts, but the resulting four pieces can be hinged so that they collapse into either the equilateral triangle or the square (Gardner 1961, p. 34; Stewart 1987, p. 169; Wells 1991, pp. 61-62).

Kaurov (2026) gives an animation combining the planar hinged motion with a three-dimensional return. During the spatial motion, a small translation supplements the rotations to keep all four connections closed. One connection opens during the planar motion and closes again at the endpoint.

Demaine et al. (2026) proved that the equilateral triangle and square have no common dissection with three or fewer pieces that are polygons when rearrangement is restricted to translations and rotations without turning pieces over. Thus, Dudeney's dissection is optimal under these restrictions. Whether three pieces can suffice when turning pieces over is allowed, or when pieces with curved boundaries are allowed, remains open (Demaine et al. 2026).