A brief introduction to the old mathematical problem of squaring the square.

Squared squares

The pattern on the home page is a squared square.

The problem of whether it was possible to divide a square into smaller squares, no two of which have the same size, was thought about by various mathematicians in the early part of the 20th century, but it wasn't until 1939 that the first rather unwieldy examples were published. The strongly interlocking nature of the solutions means that you are not likely to find one just by using pencil and paper. With the advent of computers, the search speeded up, and the simplest possible squared square was discovered in 1978; it is the only one with 21 component squares ("of order 21") and is depicted at right. There are eight different solutions of order 22, and the one on the home page is one of these.

The term "squared square" is used here for what are known more precisely as simple perfect squared squares. A squared square is imperfect if not all of its component squares are of different sizes; it is compound if some of its component squares form a rectangle (or even a square) within it. A compound squared square loses a bit of its uniqueness in that it comes in at least four versions, according to how the rectangular part is rotated or flipped. These types of squared square are not considered here.

The solution used on the home page was chosen for purely visual reasons. It has the biggest smallest square relative to its size of any solution below order 24. Most solutions of these orders or higher have a smallest component more like a hundredth of the size of the main square, so do not present well on a computer monitor. Also, the sizes of the component squares here are reasonably distinct, so that it is obvious to the viewer that it really is a solution.

The squares have been coloured in so that no two with the same colour touch. Of course, only four colours were needed to achieve this according to the Four Colour Theorem famously proved in 1976, again with the aid of computers. It's not hard to find such a colouring... there are 4988 different ones for this particular squared square. The colouring here is one of two with the most equitable distribution of colours, the colour with the greatest area (the one used in the lower left hand corner square) having a mere 3.5% more than that with the least (used in the middle square along the bottom edge).

Another advantage of this particular squared square solution is that it is the only one of any order with a side of 192 units. At five pixels per unit, this makes a page width of 960 pixels. As well as being displayable on all but the oldest monitors. 960 has 28 aliquot parts, an exceptionally large number. This means a great choice of column layouts, should they be needed. There are many proponents of a width of 960 for grid layouts applied to web pages; see the 960 Grid System, for example.

There's plenty more to say about squared squares.