Arrange the 4 pieces below so they fit perfectly inside the triangle.

https://mathigon.org/polypad/embed/UlyZAfqpyOaDfg?sidebar=hidden&toolbar=move,color&settings=hidden

Here's a hint if you need it:

https://mathigon.org/polypad/embed/ZKzyf8tOBtG7A?sidebar=hidden&toolbar=move,color&settings=hidden

Now rearrange the pieces into the same triangle using the new outline shown in the 2nd triangle:

https://mathigon.org/polypad/embed/13qicL0PGDZRpA?sidebar=hidden&toolbar=move,color&settings=hidden

What is happening here? The triangle didn't get bigger and the pieces stayed the same size. Yet, we have an empty square in the triangle!

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/1a5a42d7-3183-4a76-a1c3-f5534dc433bd/Missing_Square.gif

The exploration is called Curry's Paradox. A paradox is a seemingly contradictory situation. Here, we have the same 4 shapes that seem to have different area based on how we arrange them. That doesn't seem to fit with our understanding of area. Can you explain what is happening here? Mathematician Haskell Curry created this paradoxical situation.

Solution

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/9da98bbe-8317-4ea8-90f5-033c304097c9/Missing_Square.gif

Let's start making sense of these by finding the area of the triangle and the shapes.

https://mathigon.org/polypad/embed/AL9ZE4HEi0XHkQ?sidebar=hidden&toolbar=move&settings=zoom&copy=no&delete=no&rotate=no

https://mathigon.org/polypad/AL9ZE4HEi0XHkQ

We've found the issue! The triangle has an area of 32.5 while the four pieces have an area of 32. That must mean when arranged in this manner, the 4 pieces don't quite fill up the triangle:

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/e308517d-b7a8-476d-a192-09563b665428/Screen_Shot_2021-03-09_at_3.05.59_PM.png

And when arranged like this, the 4 pieces plus the extra square have an area of 33. So they must be extending beyond the triangle slightly:

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/26be6c69-0c4e-419b-a90c-4696c35ed009/Screen_Shot_2021-03-09_at_3.08.28_PM.png

The blue triangle and the red triangle do NOT make a straight line. To see this, drag the blue triangle on top of the red triangle and you'll see the longest sides (the hypotenuse) don't line up exactly. Zoom in as needed.