Concentric Shapes

See the image below of a square inscribed inside a circle inscribed inside a square. If the outer square has an area of 100 square inches, is there a quick way of figuring out the area of the inner square? Click below for the answer.

Spin the inner square around so the corners touch the top, bottom, and sides of the outer square.

Now imagine taking each corner of the outer square and folding them in so they touch in the center of the image. It's easy to see that the area outside the inner square will perfectly cover the area inside, so the inner square has exactly half the area of the outer square, or 50 square inches.

Unsolved: The Perfect Cuboid Problem

In Tomorrow's Math, C. Stanley Ogilvy asks

Does there exist a rectangular parallelepiped (box) all of whose edges and face diagonals are of integral length, and the length of whose main diagonal is also an integer?

Put another way, can you construct a rectangular solid where all three edge lengths, all three face diagonals, and the long space diagonal have integer values (or prove that such a box cannot be constructed)?



This deceptively simple-sounding problem is known as the Perfect Cuboid Problem (among other names). It is closely related to the problem of finding an Euler Brick, which has many solutions, the smallest of which is a box with edges 125, 244, and 267.

No perfect cuboids were found during an exhaustive search of the integers up to 100 billion (10¹⁰). This lends evidence (but is far from a proof) to the suspicion that no perfect cuboids exist. One near miss is the cuboid that has edges with lengths 44, 117, and 240, and diagonals with lengths 125, 244, and 267.