8×8 13×5 (aka 64=65) Geometry Problem Illustrated

I consider geometry to be a fairly important aspect of design, so when I came across the classic 8×8 13×5 “optical illusion” (it’s actually more of geometry problem) over the weekend, I decided to put together a little video illustrating the hole in the 8×8 rearranged into 13×5 problem.

The idea is that an 8×8 square can be divided into four shapes (2 quadrilaterals, 2 right-triangles) which can then be re-positioned to form a 13×5 rectangle (never mind for a moment that this would defy the laws of physics). Each quadrilateral is combined with one of the right triangles, and then end result is what appears to be two right triangles with a base of 13 and a height of 5. The thought is that these two triangles can then be combined into a 13 x 5 rectangle.

The problem with this line of thinking is that the two “triangles” (created by combining one of the quadrilaterals with one of the triangles) are not actually triangles at all, rather they a quadrilaterals with interior angles of 20.55°, 90°, 68.20°, and 181.25°. The 181.25° is close enough to 180° that at casual glance the composite shape appears to only have three sides, when, in fact, it has four.

When the two composite supposed “triangles” are joined together in an attempt to form a rectangle, a hole in the shape of another quadrilateral (a parallelogram) is formed. This hole has an area of 1, which accounts for the missing difference between 8×8 (or 64) and 13×5 (or 65).

Sadly, you can’t create an extra square-unit of area, by simply rearranging an 8×8 square into a 13×5. If you could, I’d buy myself an 8×8 sheet of gold and retire soon after.

You can also check out a fully-scalable Flash version of the 8×8 13×5 video (or download the .swf file by right-clicking or ctrl-clicking the link and doing a Save Target).


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