emails recieved from people explaining the "hole" illusion.

I was directed to your site by a friend, and saw your hole puzzle (the one with the two triangles and two other shapes, which comes very close to forming a large triangle. I had never seen this puzzle prior to today, but I love a good thought exercise, and thought I'd try my stab--much different from the other explanations on the site, and I feel it is much more plausible.

First of all, in response to those who believed that 1.2 degrees difference (not even noticeable to the eye) is enough to make enough space for a hole of that magnitude--nice try. Unfortunately, that answer is flawed. The figure is, technically, a quadrilateral, but not enough of one to make the extra space. Therefore, I will call the figure, in this explanation, "the large triangle."

The true answer is actually much simpler, and does not rely on the angles of the triangles, and has little to do with the triangle as a whole (no pun intended), but rather the arrangement of its parts.

First, let's establish what we know about the two triangles--without doing any calculations. The height of the green triangle is 2, and the width of its base is 5. The height of the red triangle is 3, and the width of its base is 8. Just to verify the statement on the site, you can easily count the squares to ensure this.

Okay, now let's take a look at the first large triangle. The green triangle is on top, and its base forms the top of the rectangle (the two shapes inside are irrelevant for right now). The red triangle is to the left, and its right edge forms the left side of the rectangle. This yields a rectangle of length 5 and height 3, and with an area of (5 * 3) = 15.

The second triangle switches the red triangle and the green triangle. This means the bottom of the red triangle forms the top of the rectangle (still ignoring the two shapes inside), and the right side of the green triangle forms the left side of the rectangle. This yields a rectangle of length 8 and height 2, and with an area of (8 * 2) = 16.

Okay, well now this says that there is one extra space, but why is there a hole? This is simply because the two shapes, while forming a rectangle of area 15 in the first triangle, still have a combined area of 15 in the second. Why? The way the shapes were separated allows them to be combined so as to look like a 8 x 2 rectangle, but, since you can't create extra area that didn't exist before, the shapes naturally have a "hole."

Mike

Alright eBaum?

Like the site - cute family :)

Just thought I'd let you know about the solution to this illusion if you're interested, 'cos the page says you don't get it yet.

The reason it works is because in neither arrangement is the resulting shape a triangle. The shape is nearly a triangle, but the 'hypotenuse' is not a straight line, so it is actually a quadrilateral.

Proof:

The angle at the bottom left of each coloured triangle, theta can be worked out by taking the inverse tan of the (height of the triangle / base).

The small triangle has a base length of five and a height of two => theta = inv tan (0.4) = 21.8deg
The larger triangle has a base of eight and a height of three =>
theta = inv tan (0.375) = 20.6deg

So, when the little triangle is top right of the large 'triangle', its 'hypotenuse' is concave. When the big triangle is on top, the 'hypotenuse' is actually convex. The difference in the shape of the top surface causes a difference in the area of the 'triangle', allowing exactly one square to be left over.

Hope you found this vaguely interesting,

JB

Hello ebaum!
This is Barb from Phiilly.
Referring to the "hole" in the
"cool" page....
You know that configuration of the 4 colored "triangle" that when the
portions are switched,are still the "same" size...
leaving an empty box? Well,in ln less than a minute I figured it out! If
you look carefully,at the top one,the red area has 14 section
blocks,some are cut off though. They still count as blocks if you find
them.The one on the bottom has just 13 section blocks.Hense,leaving
enough space for an empty box! And people think webtv folks are stupid!
haha Cheers! Barb

OK. The thing is - the big triangle is not really a triangle. The smaller
ones - red and green - are triangles, but the angle at the bottom left of
each triangle is not the same - the red triangle's is a little narrower than
the green. So, in the top picture, when you put the red triangle in the
lower left corner, then the big shape really has a bit of a "dent" where the
two smaller triangles meet - i.e., the line going from the lower left corner
to the upper right corner is not really a straight line. And in the bottom
picture, there is a bit of a "bump" where the two triangles meet. Not much,
but enough to give you room to squeeze another square in there.

How do I know this - well, if you count the squares as if the big shape was
really a triangle, you'll see that the area should be 1/2 of 5 (height) x
13 (width), or 32 1/2 square units. If you add up the areas of all the
smaller shapes, you get 32 square units on top and 33 square units on the
bottom (including the white "hole"). So, the big shape can't really be a
triangle.

And to get even more pedantic, for the big shape to be a triangle, the red
and green smaller triangles would have to have the same angle in their lower
left corner, and if we calculate that angle, we find that they are not the
same, so the line in both shapes going from lower left to upper right is not
really a straight line, but a "bent" one.

Red Angle = arctangent(3/8) = 20.556 degrees
Green Angle = arctangent(2/5) = 21.801 degrees

Close, but no cigar!

Still a GREAT illusion - I'm sure it drives most people crazy!

Steve

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