Think you're a genius? Can you solve this? I took me a while, but there is an answer... I'll give you a clue, think back to high school trig and the ratios that apply to triangles.

OK, as things are digressing, here's the answer, the red and green triangles look like they're identical but at a different scale, but they're not.

If check the height/base (tan) you'll see they're slightly different.

The hypotenuse of the two composite triangles (meaning the green, red, orange and leafy green areas) are every so slightly different. In fact, they curve and cause the overall triangle to curve differently depending on where they are... and there in lies the extra space.

That's about all I figured out, the rest of the answer comes from the anonymous author of this brain teaser... __________________ Green triangle.... tan A = opp/adj tan A = 2/5 = 0.4

Red triangle.. tan A = opp/adj tan A = 3/8 = 0.375

The entire triangle shape: tan A = opp/adj tan A = 5/13 = 0.385

The tangent ratio shows that they are different. So how does this make the hole appear? It's an illusion.

In the first diagram, the shallower angle of the red triangle dips inwards and the steeper angle of the green triangle straightens the line back up to the corner of the full triangle. This leads the whole triangle to have almost a concave type of effect along the hypotenuse, losing about 0.5 of a square in area.

In the second diagram, the steeper angle of the green triangle swells the whole triangle out in a convex shape and the red triangle brings it back in. This gains an area of 0.5 over the full triangle.

The loss in the first and the gain in the second add up to the missing space. __________________

correct Oxy; I actually got a A grade on a test I didn't have to take after challenging my Algebra teacher in high school when she stated as FACT that there could only be 1 90 degree angle in ANY triangle :)

I got sent to the office for it also, but the free A was worth it.

(the office trip was for the space-time explanation, not the sphere one)

## 12 comments:

That's a good one. I got it after a bit of looking - but I won't spoil-the-fun / end-the-suffering by posting my answer ;-)

Yeah... right... :)

You must first check what assumptions you make.

There is actually no problem to solve if you make no false assumptions.

false assumptions will kill you quick though :)

Got one for you; can you make a triangle with 3 90 degree angles?

nazh, is that on a flat surface?

The discussion here is sounding distinctly Vulcan, Onyx's comment could have been a quote from any one of the Star Trek movies... :)

I'm impressed you guys find it straight forward, I thought it was scam at first, even now, as Pooh bear would say, "It hurts my thunker..."

onyx: can't give any more info to the problem (as that is part of the problem you have to figure out)

OK, as things are digressing, here's the answer, the red and green triangles look like they're identical but at a different scale, but they're not.

If check the height/base (tan) you'll see they're slightly different.

The hypotenuse of the two composite triangles (meaning the green, red, orange and leafy green areas) are every so slightly different. In fact, they curve and cause the overall triangle to curve differently depending on where they are... and there in lies the extra space.

That's about all I figured out, the rest of the answer comes from the anonymous author of this brain teaser...

__________________

Green triangle....

tan A = opp/adj

tan A = 2/5 = 0.4

Red triangle..

tan A = opp/adj

tan A = 3/8 = 0.375

The entire triangle shape:

tan A = opp/adj

tan A = 5/13 = 0.385

The tangent ratio shows that they are different.

So how does this make the hole appear? It's an illusion.

In the first diagram, the shallower angle of the red triangle dips

inwards and the steeper angle of the green triangle straightens the

line back up to the corner of the full triangle. This leads the whole

triangle to have almost a concave type of effect along the hypotenuse,

losing about 0.5 of a square in area.

In the second diagram, the steeper angle of the green triangle swells

the whole triangle out in a convex shape and the red triangle brings

it back in. This gains an area of 0.5 over the full triangle.

The loss in the first and the gain in the second add up to the missing

space.

__________________

So... how did you do... did you figure it out?

The puzzle only tricks you if you assume that the overall shape is a triangle.

It's not.

I started by looking at the ratio of the two right angle triangles. One is 3/8 the other is 2/5. They do not slope at the same angle.

Then the long answer is ... what you said. ;-) Good puzzle though, because its so easy to think the two overall shapes should be triangles.

And for Nazh, you can draw a triangle with three right angles on sphere. Or if you curve space time.... arrrggghh.

correct Oxy; I actually got a A grade on a test I didn't have to take after challenging my Algebra teacher in high school when she stated as FACT that there could only be 1 90 degree angle in ANY triangle :)

I got sent to the office for it also, but the free A was worth it.

(the office trip was for the space-time explanation, not the sphere one)

My brain hurts...

I must admit, I keep comparing both triangles and am still confused.

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