What’s the big idea, 32.5 does not equal 31.5

Question

Here’s a visual proof for why 32.5=31.5

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!

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...) — Stiv, Oct 29 '20 at 19:49
@Stiv. Very similar concept in this very popular question https://puzzling.stackexchange.com/questions/24848/how-can-64-65/24851#24851 — DrD, Oct 29 '20 at 20:15
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. — DrD, Oct 29 '20 at 20:23
@DrD I agree the answer logic is very similar, it seems so hard to find a duplicate, I don’t know you all find duplicates so easily. — Cotton Headed Ninnymuggins, Oct 29 '20 at 20:25
You are right. Hard indeed. Unless a couple of key words pop out. But I have noticed that while the OP may not find a duplicate the readers almost certainly will! — DrD, Oct 29 '20 at 20:27
If you look carefully you could see the lines of two triangle at different sections of the squares — AntsPiano, Oct 30 '20 at 01:24

Cristian Marian · Accepted Answer · 2020-10-29T20:06:54.300

3

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)

edited Oct 29 '20 at 20:06

answered Oct 29 '20 at 19:45

Cristian Marian

Aw man, you beat me by like 50 seconds – Voldemort's Wrath Oct 29 '20 at 19:48
Nice clip, it has different triangles but it exaggerates the differences in angles better – Cotton Headed Ninnymuggins Oct 29 '20 at 21:10

score 2 · Answer 2 · answered Oct 29 '20 at 19:46

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.

2 Answers2