

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. 