Introduction: Puzzle: Squaring a Triangle

Picture of Puzzle: Squaring a Triangle

You can cut a square into four pieces, and reassemble these pieces into a triangle. Or vice versa. This makes a great little puzzle. I’ll show the solution in step 2.

Step 1: Cut the Pieces

Picture of Cut the Pieces

From paper, cardboard, plywood, or acrylic cut the square into the four individual pieces using the attached template. I have included a pdf file for printing and a dxf file (in mm units) for laser cutting. The square is about 4" square :-)

  • Two minute version: Print the pdf file on paper, cut out the pieces, enjoy the puzzle.
  • Ten minute version: Laser-cut the pieces from plywood or acrylic, clean the edges, enjoy the puzzle.
  • Twenty minute version: print the pdf template, glue onto a piece of wood/plywood, cut out on a bandsaw or scroll saw, sand the edges, enjoy the puzzle.

Use a material that looks the same on both sides. Flip over some of the cut pieces to make the puzzle more difficult. The pieces are not symmetrical and only assemble with the right sides up!

Step 2: Solution

Picture of Solution

The puzzle is called Dudeney’s Dissection after Henry Ernest Dudeney. You can look up the history, background, and math on Wikipedia.

What is really cool about it: if you hinge the corners at A, B, and C the triangle can be transformed into a square, and vice versa by folding along the hinges. Another cool fact: if you cut this puzzle from wood, the grain direction will line up for both the triangle and the square solution.

You can build a tray with the triangle/square outline (shown in the last image) to assemble and store the pieces. Make the puzzle for a stocking stuffer or last minute nano-gift! Build a folding trivet or table using the idea with hinges.

Comments

HudsonR1 made it! (author)2017-09-16

Nice puzzle. I really liked it. Thanks.

rschoenm (author)HudsonR12017-09-19

Very nicely done. I like the coloring!

drdlh made it! (author)2017-03-07

Super neat puzzle. Thanks!

I wanted to make several for friends so took the time to build a jig. I scaled to 5 - 5/8" and am using 1/4" birch plywood. By mixing pieces from different iterations, grain doesn't give away the assembly.

HudsonR1 (author)drdlh2017-09-12

Does it work? I actually got your idea and I´ve tried to reproduce it. But After I cut the pieces using the jig, because the saw thickness, the pieces don´t match perfectly. Either square or triangle leaves borders offside. The individual cuts were easier to me so far. But I'll keep trying. :)

CaptainNemo (author)2017-04-01

finally the answer of how to get a square peg in a triangular hole!

Lilleo87 (author)2016-12-24

Cool Puzzle AND easy to made

thedoc (author)2016-12-17

Yes but can you make a circle of the bits?

emilevs (author)2016-12-17

now square the circel

Hobbytik (author)2016-12-15

Hello, this puzzel is known as Haberdasher puzzle. See http://www.craftsmanspace.com/free-projects/haberd...

Greetings from Holland.

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CreativeTinker (author)Hobbytik2016-12-15

That page has a wonderful construction/derivation of this puzzle. This puzzle is one of many due to Hilbert. He proved that any polygon can be cut into a finite number of pieces and reassembled into another polygon. see http://www.pleacher.com/mp/mlessons/geometry/hilbe...

deluges (author)2016-12-15

Neat!

jkcarter (author)2016-12-15

Super! Thanks for sharing.

hjjusa (author)2016-12-14

good job

seamster (author)2016-12-14

Brilliant little puzzle. I need to make a couple of these. Thanks for sharing this!

Anirudh Ralhan (author)2016-12-13

Awesome instructable!!! Really good :)

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