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Squaring rectangles

Consider the following innocent sounding problem. Can you draw a rectangle that can be cut into two or more squares of different sizes?

Give it a try, then come back…


This turns out to be remarkably difficult. It is even more difficult if the rectangle you are cutting up is itself a square.

A solution to the problem is called a perfect squared rectangle. Perfect here means that all of the squares are different sizes. If the rectangle is not composed of smaller perfect squared rectangles, then it is a simple perfect squared rectangle.

Archimedes wrote about dissecting squares over 2000 years ago. Yet the first perfect squared rectangles were not discovered until 1925. One of those rectangles (pictured below) is 33×32 and uses nine squares, which is the fewest possible. Can you determine the size of the smallest square?

There is an excellent Numberphile video describing this problem and its history, which is known as Squaring the Square. Most of the methods used to find perfect squared rectangles were developed at Cambridge University by four students in the Trinity Mathematical Society. They found a useful way to represent the rectangle and its squares as an electrical circuit, and applied Kirchhoff’s circuit laws to help find solutions. For a more detailed description of this and related problems, take a look at squaring.net.

We are creating a puzzle (to be introduced at the 2019 JMM) that uses squares with side lengths from 1×1 to 50×50. There are 64 ways you can assemble a subset of these squares into simple perfect squared rectangles. Below are some shots of the prototype set. The first version is rather large: the 50×50 is almost 10 inches on the side. Producing a smaller scale version would mean the smallest pieces are pretty tiny. But perhaps we will just omit the 1×1, 2×2, and the 3×3, and leave it to the imagination to recognize negative spaces in a solved puzzle.