Introduction: 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

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

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

author
CaptainNemo (author)2017-04-01

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

author
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.

IMG_2396 (1).jpg
author
Lilleo87 (author)2016-12-24

Cool Puzzle AND easy to made

author
thedoc (author)2016-12-17

Yes but can you make a circle of the bits?

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emilevs (author)2016-12-17

now square the circel

author
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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author
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...

author
deluges (author)2016-12-15

Neat!

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jkcarter (author)2016-12-15

Super! Thanks for sharing.

author
hjjusa (author)2016-12-14

good job

author
seamster (author)2016-12-14

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

author
Anirudh Ralhan (author)2016-12-13

Awesome instructable!!! Really good :)

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