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Here’s a visual proof for why 32.5=31.5 32.5=31.5

Here’s the animation

This puzzle came directly from Proofs without words on Mathoverflow.

Your Goal: Explain what’s wrong with the proof and where the “missing square” went! Good luck, this one drove me crazy for a bit!

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    $\begingroup$ I'm almost positive this has been asked before but can't find a duplicate! (If it hasn't, I'm impressed it's lasted out so long - this is kind of a classic...) $\endgroup$
    – Stiv
    Oct 29, 2020 at 19:49
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    $\begingroup$ @Stiv I looked too, I expected to find it but didn’t. $\endgroup$
    – Cotton Headed Ninnymuggins
    Oct 29, 2020 at 19:50
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    $\begingroup$ @Stiv. Very similar concept in this very popular question puzzling.stackexchange.com/questions/24848/how-can-64-65/… $\endgroup$
    – DrD
    Oct 29, 2020 at 20:15
  • $\begingroup$ @DrD does a similar concept mean a duplicate? $\endgroup$
    – Cotton Headed Ninnymuggins
    Oct 29, 2020 at 20:20
  • $\begingroup$ In this case the answer logic is very similar to the 64=65 question. So it may not be a literal duplicate but close. Just my opinion. $\endgroup$
    – DrD
    Oct 29, 2020 at 20:23

2 Answers 2

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The red and blue triangles don't have the same 'slope'.

So while in the first picture there is a 'dent' in the hypotenuse of the big triangle, in the second one there is a 'bump'.

That difference equates for the missing square.

Changing a bit the sizes of the pieces involved may help with visualising the illusion here. (sorry about the broken grid)

triangle illusion

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  • $\begingroup$ Aw man, you beat me by like 50 seconds $\endgroup$
    – Voldemort's Wrath
    Oct 29, 2020 at 19:48
  • $\begingroup$ Nice clip, it has different triangles but it exaggerates the differences in angles better $\endgroup$
    – Cotton Headed Ninnymuggins
    Oct 29, 2020 at 21:10
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It's not technically a triangle because the red triangle's hypotenuse has a different slope compared to the blue triangle's.

Rise over run for blue = 2/5

Rise over run for red = 3/8

These values would be the same if this were a triangle. Therefore, when the shapes are rearranged, the area changes.

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