Origami Haberdasher's Puzzle
- Edwin Corrie
- Mar 23
- 4 min read
Updated: Mar 30
The Haberdasher's Puzzle is a geometric dissection generally attributed to the English puzzle expert Henry Ernest Dudeney (1857-1930), in which the challenge is to cut an equilateral triangle into four pieces that can be reassembled to form a square.
It was first published in 1902 in The Weekly Dispatch magazine as a problem for readers to solve, before eventually appearing in Dudeney's first book The Canterbury Puzzles (1907) with the by now well known 4-piece solution.
Greg Frederickson has pointed out in his wonderful books on geometric dissections (in particular Hinged Dissections: Swinging and Twisting (2002)) that Dudeney alludes to a 5-piece solution but doesn't actually explicitly claim the 4-piece version as his own, and that his comments can be taken to suggest that it may have been devised - or at least independently discovered - by one of his correspondents by the name of Charles W. McElroy.
From The Canterbury Puzzles (1907)
A special feature of this particular dissection is that the pieces can be hinged together and pivoted round to form either of the two final shapes. Dudeney comments on this remarkable property in a footnote to his explanation of the solution:
I add an illustration showing the puzzle in a rather curious practical form, as it was made in polished mahogany with brass hinges for use by certain audiences. It will be seen that the four pieces form a sort of chain, and that when they are closed up in one direction they form a triangle, and when closed in the other direction they form a square.
A rather nice though not inexpensive wooden version is currently available from Tim Rowett's Grand Illusions website.
But it turns out that there are actually quite a lot of these "hingeable dissections" which transform one geometrical shape into another. Greg Frederickson found that many of the already known dissections were in fact also hingeable.
Italian visual artist Gianni Sarcone, author of several books on mathematical puzzles and optical illusions, has an intriguing variation of the Haberdasher's Puzzle, which he calls Geome-trick Puzzle. His books and Archimedes Lab website include a number of other interesting dissections as well as a lot of other material.
The Haberdasher's Puzzle in origami
In 1993 I came up with an origami version of the Haberdasher's Puzzle which I called Tri-Square. It came about after I discovered a method for folding a crease pattern for the square to equilateral triangle, described by Didier Boursin in the French origami magazine Le Pli No. 14 (March 1983). Putting the crease pattern inside a larger square and folding away the surrounding paper, I found a way to make the four pieces needed. Because they all start from squares of the same size the smallest piece is fairly thick (and in fact looking at it now I'd say the folding sequences for all four can probably be improved), but using thin foil paper it's possible to get quite good results. The pieces can be joined by the corners with thread to make the "hinges".
Tri-Square was due for inclusion in a BOS booklet called Origami Cornucopia, which was going to be a follow-up to my BOS Animal Origami series but was never completed. The diagrams were eventually published in Origami USA's PCOC Play Origami Collection (2014), but I've recently cleaned them up and added colour, so they're much nicer to look at and easier to follow now.
For a long time I thought this was the only existing origami version, but looking around just recently I discovered a couple of others. David Petty devised one called Tri-Puzzle, which I had completely forgotten about, and which appears in World's Best Origami (2010) by Nick Robinson (a great book that's still available and strongly recommended).
Jeremy Shafer has YouTube video tutorials , posted in 2022, explaining two excellent methods for recreating the Haberdasher's Puzzle. His Square to Triangle Origami Haberdasher Dissection is a very clean solution with pieces that have nice closed edges all round, while his Kirigami One-Piece Square to Triangle Haberdasher Puzzle is a brilliant one-piece version which involves cutting but has built-in hinges so that it can pivot round to form either the square or the triangle (you'll have to watch the tutorial to see how it works). I find myself wondering if it might be possible to use the same principle for other hinged dissections of a square, by simply making a different pattern of cuts.
One thing I didn't realise immediately was that in all these versions the triangle is isosceles, rather than equilateral as in the original Dudeney/McElroy dissection. Jeremy Shafer mentions this, but it wasn't until I got hold of Greg Frederickson's book on Ernest Irving Freese that I found it fully explained. It seems that the principle of this particular triangle-to-square transformation can be applied to any isosceles triangle within a certain range, an equilateral triangle being just a special case of an isosceles triangle that falls within the range.
Other geometric dissections in origami
Also in World's Best Origami by Nick Robinson is David Petty's method for making the Greek Cross to Square dissection from Dudeney's Amusements in Mathematics (1917).
Lindgren's Dodecagon to Square Dissection by Thomas E. Cooper can be found on the Origami USA website.
Origami Puzzles (2019) by Marc Kirschenbaum is a small booklet describing six puzzles based on existing geometric dissections.
In 1989, inspired by a reference in Martin Gardner's Mathematics, Magic and Mystery (1956), I came up with something which I naively called Dissection Puzzle, but which I now know is a classic dissection presented by the English mathematician Henry Perigal in 1872 as a proof of Pythagoras' theorem. Gardner points out that the point of intersection can be anywhere within the solid square, so there are various ways to make it in origami. My diagrams show how the pieces can be modified so that they form an octagon, and also that the pieces are hingeable. In 2002 I devised a neater version of the same thing with closed edges, calling it Square Dissection, and then in 2024 I made some improvements to the folding sequence. Nick Robinson also has a couple of versions in his Encyclopedia of Origami (2004).
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