Curry Triangle Paradox

Christopher DanielsonWe think math is fun!Leave a Comment

Five parts for a curry triangle lie slightly separated in order to make clear the decomposition. There is a 4-by- right triangle, a 9-by-7 right triangle, a 4-by-3 rectangle, a figure made of rows of 4, 3, and 2, and another of rows 1, 2, and 3.

Here’s the rap.

Me: Can we do a little math together?

You: Sure!

Me: Here’s a triangle, right?

A 13-by-5 triangle. Careful study reveals that it is cut into four pieces—a 5 by 2 triangle, an 8-by-three triangle, and two rectilinear hexagonal pieces that combine to make a 2-by-5 rectangles.

You: Yes.

Me: OK. You hold these pieces; I’m gonna swap these, and your job is to fill in the empty space to make the same triangle a different way.

An animation that shows swapping the position of the two triangles and removal of the two irregular hexagonal pieces, in preparation for a participant to place them in the new arrangement.

Eventually (anywhere from 10 seconds to 2 minutes later)

The same pieces as in the above image; rearranged so that the small triangle sits on the left, the larger triangle sits on top, and the two irregular hexagonal pieces nest but leave open one square inch.

Us: Oh no! What happened?!?

At this point, the conversation can take any number of directions. Many of these end in one or the other of us explicitly stating what’s going on, so I’ll do that for you now.

There’s a lie here, of course. The lie is all the way up at the top when I ask you to acknowledge that we’re starting with a triangle. We are not; it is actually a slightly concave quadrilateral.

Likewise, the ending shape is also not a triangle, but a slightly convex quadrilateral. We didn’t make the same shape made two different ways; we made two different shapes. These shapes unsurprisingly have two different areas.

Here’s a more extreme example of the same phenomenon.

Curry triangle generality

For simplicity, let’s call those four pieces up top, and the big triangle they appear to make a “Curry triangle”.

Now, we here at MathHappens are big fans of the Fibonacci numbers (numbers from the sequence, 1,1,2,3,5,8,13… and where you get each next number by adding the previous two numbers). Sure enough, the Fibonacci numbers appear in this paradox as the bases and heights of the triangular pieces: 2,3,5, and 8. In fact, any run of four numbers from that Fibonacci sequence will make a Curry triangle (try it!)

But it isn’t the Fibonacci-ness that makes this work. Instead, it is the fact that the middle numbers (3 and 5) have a product that differs from the outer numbers (2 and 8) by 1. That corresponds to the area I asked you to fill in at the outset—put these 15 squares into this 16 square units of space.

Now we know how to build Curry triangles backwards. Pick two composite numbers that differ by 1, say 27 and 28. These are the areas of your rectangles. Pick your favorite factor pair for each area, and you’re good to go.

All that remains is to dissect that smaller rectangle in some clever way so that it leaves a satisfying 1 square inch hole when you put the other pieces in the configuration.

For more information, the wikipedia entry on the Missing Square Puzzle is not bad. I am still hunting down a copy of the Martin Gardner book that mentions it.

This math brought to you by the many hours I spent with mathematicians, math graduate and undergraduate students, and various passers-by at the Math Communities table in the AIM booth at the Joint Math Meetings earlier this month.

Leave a Reply

Your email address will not be published. Required fields are marked *