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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. (If you haven’t looked at the puzzle yet, click here.)
We know all four shapes are the same before and after, but we assumed that the large right triangle also stayed the same size and shape. This assumption was our error! It turns out that the “before” triangle is not the same shape as the “after” triangle. Actually, neither of them is even a triangle at all! For both before and after, the supposed hypotenuse (the longest side) of the right triangle is not a straight line. To understand this, we need a geometry concept called similarity.
If two shapes are similar, the angle measures are the same and the side lengths are proportional. Here’s an example. Below we have a real right triangle with side lengths 20, 48, and 52. You can create a whole bunch of similar right triangles, like the two shown inside with dashed lines and then redrawn next to it.
Look at the ratios of each of the triangles’ “longer leg” to “shorter leg.” They are always the same. 48:20 for the biggest triangle, 24:10 for the medium, and 12:5 for the smallest. These ratios are each equivalent to 2.4:1. Don’t forget, this also means that they have equal angles! Now, look again at the triangles in our puzzle, laid out like we see our similar triangles just before. The side lengths are included for each.
The ratios of the longer to shorter are 13:5 (2.6) for the big triangle, 8:3 (2.667) for the medium triangle, and 5:2 (2.5) for the small triangle. The ratios of the sides of the triangles are not the same. This means the triangles are not similar, which also means that their angles are not the same. The big “triangle” that is made up by the smaller parts does not have a straight line for its hypotenuse. In the first position, the unequal angles cause the fake hypotenuse to bend in from the true triangle. In the second triangle, the unequal angles cause the fake hypotenuse to bend outside the true triangle’s hypotenuse. Here is an exaggerated drawing of the before and after triangles to show the bend. The dashed line shows where the true right triangle would be. Of course, because the differences are much slighter, we are fooled into believing that we see a true right triangle with a straight hypotenuse.
This bend answers how the extra square appeared in the middle of the second fake triangle. The bending in from the first triangle and the bending out from the second roughly adds up to one square unit. Imagine slicing up the square into little slivers that can line up into the space that the first image misses plus what the second image adds. This is how we get an extra area of one square unit.
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In the end, this puzzle relies on a very subtle change but, more importantly, an unchecked assumption that we made when viewing the video the first time. Working through mathematical puzzles like these sharpens our mathematical reasoning. It reminds us to tackle argumentation by critiquing any and all assumptions present in a given argument. We can then move logically through a question and answer process based on these assumptions. Use these approaches as you consider any mathematical argument.
You’ll soon notice that mathematical fallacies appear in many places.