How is this possible?

27 May, 2012 in  | 10 Comments 1. Klassik

5:59 pm

They are not the same shape.

2. Klassik

6:02 pm

The red and green triangles are not the same angle, so all together they do not form a triangle, and the two collections are not the same shape.

3. metazahaz

6:20 pm

The hypotenuse of the whole shape is not actually a straight line; the two triangles do not have exactly the same slope. So in the top shape, the “line” of the hypotenuse is actually sort of bowed in slightly, while in the lower shape, that “line” is bowed out slightly. The difference between those, while not visually obvious, makes up for the 1 square unit of space opened in the lower shape.

Mathematically, if you add all the areas of each shape together, you get 32. If you calculate the shape of an actual right triangle with sides of 13 and 5, you get 32.5. That means you have to lose .5 square units somewhere–in the top shape, it’s a sliver along the hypotenuse; in the lower shape, it’s a whole missing square, with a sliver instead added along the hypotenuse to make up for the fact that you took a whole 1 away, instead of just .5.

Maybe not the clearest explanation, I’m not great with words, but that’s the best I’ve got.

• Andrew

1:06 pm

Took me a while to figure that one out as well. A soft opposite curve on the hypotenuse of both triangles is enough to create the missing square.

4. Ichneumon

7:11 pm

The triangular pieces (dark green, and red) don’t have exactly the same proportions (5:2 does not equal 8:3). The dark green piece has a steeper slope on its slanted edge compared to the red one.

So the upper arrangement has a subtle concavity (dip) in the middle of the upper border where the triangular pieces meet, and the lower arrangement has a subtle convexity (bulge), exactly enough difference to make space for the 1×1 “hole”.

5. Jeff Stivenson

7:40 pm

The answer is that they aren’t exactly the same. The long angled side is slightly different, and the differences in each square along that line add up to an additional square.

6. Thatsme

5:01 am

You guys are only looking at it one way. You’re looking at it as if the heights of the individual pieces are exact according to the grid, then inferring that the triangle as a whole is not an actual triangle. Alternatively, you could assume that the triangle as a whole is true (slope of 5/13) then see that it is the gridding that is off since 5/13(8) =/= 3 but is slightly larger. Meaning the panhandled boxes are taller than the blue triangle.
There is no one correct answer. Perhaps also the panhandled boxes are of different heights when laid next to each other as in the lower image. Many possible things wrong.

7. David Beatrice

5:08 am

With a difficult problem like this, you’ve got to know how to look for the right sines. If you don’t keep focused you could get off on a tangent. You need an acute mind to approach from the right angle. If you do, maybe your friends will give you complements.

8. bwk

12:52 pm

It’s trick photography or whatever.

9. Sean L.

1:01 pm

It’s not a true triangle. The big one, I mean.

The longer sides of both the red and the blue triangles DO NOT HAVE THE SAME SLOPE.

For the blue triangle, the slope is 2/5 – that is, it goes up 2 squares of height, over a distance of 5 squares.

For the red triangle, the slope is 3/8 – that is, it goes up 3 squares of height over a distance of 8 squares.

Expressing them with common denominators gives us 16/40 for the blue, and 15/40 for the red. They’re CLOSE … close enough to TRICK the eye into thinking the whole diagram is also a true triangle. BUT IT’S NOT. “Close” _never_ counts in Geometry – be exact or go home!

^_^