The haberdasher's problem is the name given to problem of dissecting an equilateral triangle into a square. Demaine et al. (2024) give a summary of the problem's history. It was posed by Dudeney in his April 6, 1902 column with no clear indication of if 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 if this dissection was due to Dudeney or McElroy (Frederickson 1997, Frederickson 2002, Demaine et al. 2024). The puzzle and solution appeared later in Dudeney (1907) with the name "The Haberdasher's Puzzle."
Label the dissection as shown above. Then the the vertices of the triangle 
correspond to a single point 
in the square. On the other hand, the points 
and 
in the triangule 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
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(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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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).
Demaine et al. (2024) proved that the equilateral triangle and square have no common dissection with three or fewer polygonal pieces, thus establishing that Dudeney's dissection is optimal.
