An Explanation...

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 tetrahedron. 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 tetrahedron, 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 tetrahedron.

When you do a little calculation (let's say the grid-unit is centimeters), the tetrahedron (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 hole (let's call this A) is obviously 1 cm2.

This can and may be presented with a little use of trigonometry:

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