Here’s the rap.
Me: Can we do a little math together?
Me: Here’s a triangle, right?
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.
Eventually (anywhere from 10 seconds to 2 minutes later)
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.
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.