You figure it out

[Fist and Faith]

I don't understand why this is a problem. Why would we expect to be able to rearrange parts of a whole, yet still have exactly the same size/shape? It doesn't work in this case, and I don't expect it to work in most others. Yeah, if 16 squares are arranged in a square, we can rearrange them in a thousand ways, and always have a square if we want. But this isn't nearly the same situation.

I haven't read any responses above me yet, so maybe the mystery is explained by someone. I'll be back to do that. Heh.

[Vain]

OK....so I'm still not seeing a rational answer here - or did I just miss the point altogether?

[Montresor]

See my previous post:

I've seen this one before, and Lore's right. I forget the exact details, and maths bores me stiff, but there's a difference in size between the two triangles by about 3% or so (with the bottom one being slightly larger). Primarily, these things are always visual tricks.

This is a well-known one. I mean, I have at least seen it in a friend's puzzle book. I even took the liberty of chasing the answer up on the net (to ensure my memory was not running amok), but the answer is as above (the bottom triangle is minutely larger, creating the optical trick).

[The Laughing Man]

look at the length and the height of the first pic = 13x5. the red tr is 8x3, the green tr is 5x2. we must maintain the length and height in the second pic, and since the red tr is 3H, and the green one is 2H, we must move the yellow one down one. what was 2+1+2 H in the first pic, becomes 3+2 H in the second one. it is the length of the yellow one that creates the hole.

[Loredoctor]

look at the length and the height of the first pic = 13x5. the red tr is 8x3, the green tr is 5x2. we must maintain the length and height in the second pic, and since the red tr is 3H, and the green one is 2H, we must move the yellow one down one. what was 2+1+2 H in the first pic, becomes 3+2 H in the second one. it is the length the length of the yellow one that creates the hole.

It's an optical trick - the answer is online.

[Vain]

OK - in my simple terms:

Areas are as follows (sq cm):

Red triangle: 12
Green triangle : 5
Orange shape: 7
Green shape: 8


Reason for difference:

Red triangle * 2 = 24
Green Triangle * 2 = 10

Full shape size: 13*5 = 65

therefore: 65 - 24 - 10 - 7 - 8 = 16 empty spaces

Each full shape size has 16 empty spaces

If that made any sense at all

[The Laughing Man]

Hmm I printed the page, cut the pieces out (just to be sure) and damned they still match one on top the other, even with the extra space. It is a puzzler! I got a headache now.

this would seem to dispute that Lore, as does my ruler. It's not a trick, it's geometry. the individual shapes are maintained, but the collective shape is actually different, but thats where it tries to confuse you. The hole is the point, not the fact that the two collective triangles are in fact different.

[Loredoctor]

Hmm I printed the page, cut the pieces out (just to be sure) and damned they still match one on top the other, even with the extra space. It is a puzzler! I got a headache now.

this would seem to dispute that Lore, as does my ruler. It's not a trick, it's geometry. the individual shapes are maintained, but the collective shape is actually different, but thats where it tries to confuse you. The hole is the point, not the fact that the two collective triangles are in fact different.

Perhaps, but from the same site

The reason that people sometimes find this puzzling is because they start off making an assumption. Often they don't even realise they have made this assumption. The assumption is that we are comparing two triangles, and that they each have the same area. Certainly a quick inspection - helped by the grid - is that the 'triangles' are 13 squares across, and five squares high, and they kind of look the same.

In fact the top triangle is not a triangle at all, it is a quadrilateral. If you look closely along the line of the hypotenuse (the diagonal), you will see that it is not quite straight, it is slightly concave. Why? Well, the dark green triangle is 5 squares across, and 2 squares high. The red triangle is 3 squares high, and should be 7.5 squares across for the hypotenuse to have the same angle to the horizontal. In fact the red triangle is 8 squares across, and the angle of the hypotenuse is slightly shallower that that of the green triangle. Which is why the hypotenuse is not a straight line, when you look closely.

The bottom 'triangle' is not a triangle either, it is also a quadrilateral, but now the hypotenuse is slightly convex, since the position of the red and green triangles is now reversed.

Here is an image of the two triangles superimposed, which shows the difference in the two hypotenuse...

The area of the two 'triangles' is actually the same, since it is made up of the same pieces. But because the lower 'triangle' has a 'hypotenuse' that is actually convex, this extra area is the equivalent of one square, which is why the lower shape has a missing square.

Mathematical solution sent in by a visitor to the Grand Illusions web site

Kim Westh, who describes himself as a Danish Viking, sent in the following explanation, which will be of interest to the mathematicians among you

The two triangles do not have the same proportions. So the first image may appear as a triangle, but is in fact a quadrilateral.

When you do a little calculation (let's say the grid-unit is centimeters), the quadrilateral (the real thing) obtain the area of :

5 cm * 2 cm * 0,5 + 8 cm * 3 cm * 0,5 + 7 cm2 + 8 cm2 = 32 cm2

and below the octahedron obtain exactly the same area; the hole is just a way of explaining the illusion.

the dark-green triangle partition (rectangular), has an angle of : arctan (0,40) = 21,801°
the red triangle partition (also rectangular), has an angle of : arctan (0,375) = 20,556°

We hereby conclude that the intersection between the dark-green and the red triangle, is not a straight line, but two lines, presented with an outside angle of 181,245° - and this appears as one line.

[The Laughing Man]

print it out Lore, cut the pieces out individually, and rearrange them. A child playing with blocks knows this. Those mathmeticians are focusing on the irrelevant straight line bit. It's not supposed to be a straight line. They never said it was a triangle to begin with. They just said "look at the hole".

[I'm Murrin]

Yes, and that's the point--as it says in the explanation Lore quoted. The viewer assumes the line is striaght, when it is not, and it is that fact that means the two shapes have different areas. It's an optical illusion--as the title of the page says--in that they look like two identical triangles but are not.

[The Laughing Man]

then why is it pointing at the space? why does it say "from where does this hole come from? the point is supposed to be same blocks, same area, how did the hole get there?

[I'm Murrin]

That's the question. The point is to show how the illusion of identicality between the two make the hole seem impossible.

[hierachy]

I SOLVED IT WITH THE FIRST REPLY, NO MORE SOLVING NECESSARY!

[The Laughing Man]

oh, so now the question of the illusion is irrelevant because the mathmeticians are obsessed with angles, is that it?

[hierachy]

oh, so now the question of the illusion is irrelevant because the mathmeticians are obsessed with angles, is that it?

No. The angles explain the illusion. That's it.

[The Laughing Man]

I was speaking to Murrin.

We hereby conclude that the intersection between the dark-green and the red triangle, is not a straight line, but two lines, presented with an outside angle of 181,245° - and this appears as one line.

this has absolutely nothing to do with "where did the hole come from."

dude: where did the hole come from?

math guy: it's not a straight line.

dude: what?

[I'm Murrin]

I just think that what you are suggesting:
Q: "Where did the hole come from?"
A: They swapped the pieces around."
Is a less than adequate answer to the problem as intended.

It should in fact be:

"Where did the hole come from?"
"What do you mean?"
"How did an extra space appear in a shape of the same size?"
"It's not actually the same size--the two triangles slope at slightly different angles, so that when they are swapped they produce a shape with a slightly increased internal area, which means that the four pieces no longer fill the whole shape."
"Oh."

[The Laughing Man]

the hole is produced by moving the yellow block, not by increased internal area Murrin. look at it! it's supposed to make you feel dumb, and a trick for children to amaze them, that is all.

[Montresor]

It should in fact be:

"Where did the hole come from?"
"What do you mean?"
"How did an extra space appear in a shape of the same size?"
"It's not actually the same size--the two triangles slope at slightly different angles, so that when they are swapped they produce a shape with a slightly increased internal area, which means that the four pieces no longer fill the whole shape."
"Oh."

Precisely. The two triangles are not the same. It's a simple trick.

[I'm Murrin]

The point I'm trying to get across, Esmer, is that the image is intended to mislead the observer. I'm not disputing that the shape of the blocks is what resulted in a single square in that position appearing. I'm saying that the way this is presented is deliberately designed to make it seem like an impossible situation--which it would be, if the two large shapes were both triangles. If the top shape was a triangle, rearranging the blocks could not possibly produce a shape that was identical to the original shape.
Yes, that is the reason there is a hole. But there would be no point to the excercise whatsoever if it hadn't been designed to give the illusion that two shapes are identical when they are in fact not. To take the simple explanation and ignore the subtler nuances of the exercise--to ignore the context of the explanation--is to miss its point entirely.