[Flasken]

Ok, someone helt me out here. I'm dying to find out WHAT MAKES THIS POSSIBLE:

 

[slindeman]

The extra space comes from the diagonal bits, i.e. the incompletely filled squares. The slope of the diagonal line is higher to begin with in the bottom picture, which would have the effect of raising the total area beneath the "curve" so to speak, if it were not offset (which it is) by that hole. Makes perfect sense if you look at it for awhile and think about it.

Edit:

Another way to think about it. Forget about puzzle pieces and just think of a diagonal line sloping up from an origin. In this case, each picture has just 2 line segments that make the diagonal. If you were to calculate the area beneath the two you would find the bottom picture would have more area. This is because its higher sloped segment comes before its lower sloped segment. * Thus to make the area equal some space has to be removed from the bottom picture. It just so happens they thought of a clever puzzle piece arrangement to make this scenario happen.

* To see this even clearer, imagine a horizontal line and a vertical line. Arrangement 1 has the horizontal first, then the vertical, and as such has zero area under the curve. Arrangement 2 has the vertical first and the horizontal second and has a large area under the curve.

 

[chych]

lol, this problem has been going around the school. It is damn near impossible to figure it out on pencil and paper, but easier on this.

But yeah... those aren't even triangles, the hypotenuses are not lines, in the top they are convex and on the bottom they are concave.

 

[raymondlin]

Oh, I see it, it isn't a striaght line from end to end.

 

[morphsci]

Nevermind

 

[chych]

This diagram should illustrate it better. All parts ARE in fact identical, as the problem says. However, the big and small triangles are dissimilar.

The whole thing is highlited by the two identical blue highlights, which are both true triangles.

 

[Neruda]

oh yeah, I remember seeing that back in high school. the slopes of the first two triangles (red/green) are different.

 

[Odin]

Fine, I'll be the first stupid person posting here and proudly say "I still don't get it" I get the concave/convex part then I thought "This is it!" But then chych comes in and say the two shapes are identical, and now I'm confused, how are the two small triangles dissimilar???

 

[chych]

No, I mean each "part" of the triangle is identical, like how it says in the question. The two smaller triangles (the red and green that is) are dissimilar because the two non 90deg angles are not the same, even though it looks like they are the same. Maybe I should have said that a little differently...

 

[ai0tron]

Geez mess up the slope of a triangle by a tenth and you end up thinking you can redefine the logic of space.

 

[Dusty Chalk]

Alright, gotta pipe in.

All four individual pieces are exactly the same in both diagrams. However, the overal triangle in both diagrams is not the same -- it's not even really a triangle.

The slope of the green triangle is 2/5. The slope of the red triangle is 3/8. So the slope of the "overall" triangle (which isn't really a triangle) changes midstream.

Here, add up the individual surface areas: red triangle 3x8/2==12; green triangle 2x5/2==5; ugly 70's orange piece == 7; ugly 70's green piece == 8. Total surface area of both objects is 12+5+7+8==32. If that triangle were to exist, looks like it would be something like 5x13/2==32.5. So the one with the whole is slightly bowed out compared to the triangle that that thing looks like, and the one without the whole is slightly bowed in, in both cases, the "bowing" making up for ~0.5 square blocks. Make sense now Odin?

 

[Flasken]

Ok, take a look at this:

(Stare at the image and move your head forward while concentrating on the center dot.)

 

[Flasken]

Watch the image and black dots will magically appear.

 

[Flasken]

 

[AdamP88]

The first two are very cool - I love the concentric circles one. Too freaking cool. However, I didn't see myself trying to fight to say the color instead of the word on the last one. Guess my left brain wins out there.

I love optical illusions like those first two, though.