Mathematics for Teaching Algebra, Geogebra, Geometry The Pythagorean Theorem Puzzle

The Pythagorean Theorem Puzzle

Math puzzles are great activities for enjoying and learning mathematics. The following is an example of Tachiawase. Tachiawase is a popular puzzle in Japan which involve dissecting a geometrical figure into several parts and then recombining them to form another geometrical figure. The puzzle below is credited to Hikodate Nakane (1743). This was one of the puzzles distributed at the booth of Japan Society of Mathematical Education during the ICME 12 in Seoul this year.

Make a shape that is made from two different sizes of squares by dividing them into three parts  then recombine them into one square. [Reformulated version: Make two cuts in the shapes below to make shapes that can be recombined into a bigger square.]

two squares puzzle

Here’s how I figured out the puzzle: I know that it must have something to do with Pythagorean Theorem because it asks to make a bigger square from two smaller ones. But where should I make the cut? I was only able to figure it out after changing the condition of the puzzle to two squares with equal sizes. It reduced the difficulty significantly. This gave me the idea where I could make the cut for the side of the square I will form. The solution to this puzzle also gave me an idea on how to teach the Pythagorean theorem.

I made the following GeoGebra mathlet (a dynamic math applet) based on the solution of the puzzle. I think the two-square math puzzle is a little bit tough to start the lesson so my suggestion is to start the lesson with this mathlet and then give the puzzle later.  As always, the key to any lesson are the questions you ask. For the applet below, here’s my proposed sequence of questions:

  1. What are the areas of each of the square in the figure? Show at least two ways of finding the area.
  2. How are their areas related? Drag F to find out if your conjecture works for any size of the squares.
  3. Can you think of other ways of proving the relationships between the three squares without using the measures of the sides?
  4. If the two smaller squares BEDN and GFNH have sides p and q, how will you express the area of the biggest square LEJG in terms of the area of the smaller ones?
  5. Express the length of the sides LEJG in terms of the sides of BEDN and GFNH.

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After this lesson on Pythagorean relation you can give the puzzle. Once they have the correct pieces, ask the students to move the pieces using transformation in the least possible moves. They should be able to do this in three moves using rotation. Click here to download the applet. Note: If you don’t see the applet, enable java in your browser

Use the comment sections to share your ideas for teaching the Pythagorean relation. If you like this post, share it to your network. Thank you.


2 thoughts on “The Pythagorean Theorem Puzzle”

  1. That’s the original puzzle by Nakane – dividing them into 3 parts. Part of the solution is to draw the two squares and make the two cuts EJ and GJ making three pieces (shapes). The English translation from Japanese was not good. Maybe I should change the original instruction from “three parts” to “two cuts”.
    Thanks for the additional info re attribution of proof. I know it was one of many proofs of the Pythagorean theorem. I hope I did not make the reader think I discovered it:-) My claim to fame is having thought of reducing the problem into simpler one first, show the solution as a GeoGebra applet and yes having thought of a way of teaching it so students can formulate the theorem themselves:-)

  2. You have divided one square into 3 parts, the other into 2 parts, to the total of 5 parts. You need to be careful with formulating the problem. Also, it would have been nice to have any references. For example your dissection is often attributed to the 17th century Dutch mathematician Frans van Schooten. More references and more dissections could be found in G. N. Frederickson, Dissections: Plane & Fancy, Cambridge University Press, 1997

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