A triangle, made up of different building blocks, is either perfectly filled or, when rearranged, has a section missing. How is this possible? Welcome to Curry's Triangle Paradox.

Curry's Paradox was invented in New York City in 1953 by Paul Curry. Curry was a magician, and so you might expect some sleight of hand. But none seems apparent. It's a very simple-seeming object. A triangle is made up of a bunch of different polygons. All of the polygons fit together perfectly, with no gaps between them. No problem.

Until you rearrange the thing, switching the triangles and rearranging the blocks, it fits together again - with one missing space in the middle. This makes no sense. The entire thing is generally done on graph paper, so there can be no sneakily hiding a square in one of the blocks in the middle. And the entire shape keeps the same size at the beginning and at the end. There's just no space where an extra square can go, but you count up the dimensions and they're all the same in each triangle. What's happening?

The only thing we can't measure off the grid immediately with the naked eye is the hypotenuse of both triangles, so naturally that's where the funny business is going on. The two slopes aren't exactly identical. For every one block over on the blue square, you rise 0.4 blocks. For every block over on the red square, you rise 0.375 blocks, a shallower slope. Neither Curry's Triangle has a straight hypotenuse. The upper triangle has a slope that is slightly bowed inwards, and the lower triangle has a slope that is slightly bowed outwards. The difference between them? A single square. And that's what shows up in the triangle below. Check out this quick animation that makes it clear.

The moral? Don't take your eyes off magicians, especially when they dabble in math. Well, that and always calculate the slope of your hypotenuse. But I'm sure I'm preaching to the choir on that last one.

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Top Image: Mark Boyce

Graph Image: Wikipedia

Via Wolfram Math World and Cut the Knot.