How to divide an L shape into five equal parts of the same shape and size

Here's a proof that it can't be done -

The total area of the shape given is 3/4 * L^2, where L is the length of one of the longer sides.

The area of each of the 5 shapes you divide it into must then be:

3/20 * L^2

Since these smaller shapes are just a scaled-down version of the larger shape, we can say they will have an area of 3/4 * a^2, where a is the length of one of the "longer" sides on the smaller figure.

Using some algebra,

[math]3/4 * a^2 = 3/20 * L^2 → a^2 = 1/5 * L^2[/math]

[math]a^2 = 1/5 * L^2 → a = 1/\sqrt {5} * L[/math]

Since the ratio between "a" and "L" is an irrational number, and the lengths of the sides of the smaller figure are either "a" or "a/2", there's no way you can evenly divide up any one of the longer sides (length L) of the larger figure, using sides of the smaller figure.

In fact, you could string together as many L-length sides in a row as you want, and put any combination of a-length and (a/2)-length sides in a row next to it, and the joints would never line up, no matter how far you went.


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