The ‘Missing Square Puzzle’ Is Completely Mesmerizing. Can You Solve It?

·2 min read

The Missing Square Puzzle shows our rendition of a popular geometric conundrum. First, you see a right triangle made of four distinct shapes. Then, these are moved around to fit into the same triangle, but with an extra open square right in the middle.

What happened? Was there a sleight of hand, or maybe one of the shapes changed? Watch again, carefully tracking each shape’s movement. Is it possible that somehow, just by moving the shapes around, the area of the triangle grew by one square? Really? This can’t be.

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I’ve been a math educator for over 20 years. Math teachers delight in these puzzles because they give us the chance to teach reasoning and critical thinking skills. Most of my students see math class as following rules and practicing skills. Equally or more important to my job, though, is getting students to reason mathematically. Puzzles like these provide the opportunity for students to think thoughtfully with mathematics. Here’s how I’d go about getting my students to discover the fallacy in this puzzle.

First, pose the question:Why do we know the puzzle can’t be right?

The answer may be obvious, but it’s important. You have four different shapes and in the first position, you visually add their areas to get one value. In the second position, you visually add their areas and get the same value plus another square unit. We carefully watched the shapes and know that none of them have changed in size. We know area is preserved when we move shapes around. There is just no way to get that extra square unit.

Grounded by this certainty, we next ask: Was there anything in the video that we assumed, but shouldn’t have?

This is when I let my students think hard in small groups, so maybe take a minute before reading on. With time and perseverance, you can come up with the linchpin to this fallacy on your own.

Ready for the solution? Click here.

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