To cut an equilateral triangle into four pieces that can be rearranged to make a square. The Haberdasher's Puzzle, the greatest mathematical discovery of Henry Dudeney, was first published by him in the Weekly Dispatch in 1902 and then as problem no. 26 in The Canterbury Puzzles (1907).
The accompanying diagram shows the solution, which Dudeney describes as follows:
Bisect AB in D and BC in E; produce the line AE to F making EF equal
to EB; bisect AF in G and describe arc AHF; produce EB to H, and EH
is the length of the side of the required square; from E with distance EH,
describe the arc HJ, and make JK equal to BE; now from the points D and
K drop perpendiculars on EJ at L and M.
For a step by step construction, see Beyond Euclid
A remarkable feature of the solution is that the each of the pieces can be hinged at one vertex, forming a chain that can be folded into the square or the original triangle. Two of the hinges bisect sides of the triangle, while the third hinge and the corner of the large piece on the base cut the base in the approximate ratio 0.982: 2: 1.018. Dudeney showed just such a model of the solution, made of polished mahogany with brass hinges, at a meeting of the Royal Society on May 17, 1905.
