Great Magic Triangle Math Puzzle

[jedishrfu]

This is quite an interesting puzzle. You know it's wrong but you don't know why by inspection:

The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry.

It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it. [source]

http://twistedsifter.com/2017/07/profs-use-this-puzzle-to-teach-lesson-about-problem-solving/

Can you figure out an easy way to inspect it?

I spotted one way.

 

[Orodruin]

##2/5\neq 3/8##

The original shape is not a triangle.

Edit: You can also rather easily see that the original vertex where the smaller triangles meet is inside the second shape and not on its border. This only works if like here you have access to the grid.

 

[jedishrfu]

Yeah that's what I noticed after staring at it for awhile that putting the smaller triangle inside the larger one and using the grid you can see they aren't similar.

It reminds me of the common failure of many trisection angle constructions where you can't prove that three intersecting arcs/lines cross at the same point and consequently the proof fails.

 

[fresh_42]

I'm a formula guy. Some basic computations show that both triangles are quadrangles. Or: the long side isn't differentiable on neither figure.

 

[StoneTemplePython]

Playing around with "equivalent" angles and tangents got me the same ##\frac{2}{5} \neq \frac{3}{8}##

If you take a belt and suspenders approach with proofs -- i.e. have a proof two independent ways of proving / verifying something-- then picture proofs frequently make a nice set of suspenders.

 

[jerromyjon]

I recently saw similar "paradox" regarding an infinite chocolate bar... I'm not good with math proof but I could tell the bar was shorter after you remove a piece.

 

[stoomart]

The original shape is not a triangle.

As a former fence and deck builder, this was apparent by eyeballing down the long sides of each "triangle", where A appears concaved and B appears convexed.

Edit: You don't need the grid for this inspection.

 

[Janus]

As a former fence and deck builder, this was apparent by eyeballing down the long sides of each "triangle", where A appears concaved and B appears convexed.

Edit: You don't need the grid for this inspection.

I saw the same thing. Here are the triangles overlaid:

 

[Cathr]

I noticed that the hypotenuse "slope" of the second triangle goes a little bit higher, that means it's not a perfect triangle.