The Confusing Triangle Situation

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2023-07-20 / BJB

The Confusing Triangle Situation

Welcome to the enigmatic triangle situation. What’s going on here?

When the same shapes are repositioned, they appear to require less space and create an empty square. How is this possible?

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A friend shared this mind-boggling puzzle with me during college, and it baffled me until the solution finally clicked in. Here’s what’s really going on:

At first glance, it seems like a simple rearrangement of a single large triangle’s pieces:

Missing_Square_Animation

But, there’s a sneaky twist concerning the big hypotenuse in both triangles.

The crux lies in the slopes. The blue, red, and apparent big triangles all have varying slopes:

Blue triangle hypotenuse = slope of 2/5 = 16/40

Red triangle hypotenuse = slope of 3/8 = 15/40

Big triangle hypotenuse = slope of 5/13 = 15/39 = (15.38)/40

These slopes are close enough to visually appear identical, especially in diagrams with thick lines, but the big hypotenuse is not a straight line; it forms a wide V-shape.

In the upper triangle, the V angles inward, leaving no extra room (hypotenuse A). In the lower triangle, the V angles outward (hypotenuse B), creating space for an additional square. The area between the two hypotenuses shown in the diagram is equivalent to one square on the grid.

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The Stick Figure puzzle

The Three Jellybean Problem

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And one very big puzzle: