Squaring the Square

Today I turned 26234 days old and one of the properties of the number 26234 is that it is a member of OEIS A217156 with \(n\)=30.

A217156

Number of perfect squared squares of order \(n\) up to symmetries of the square.

The comments in the OEIS entry go on to say that:

a(\(n\)) is the number of solutions to the classic problem of 'squaring the square' by \(n\) unequal squares. A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does.

For \(n\)<21, it's not possible to put unequal squares together to form a larger square but for \(n\)=21, one such square is possible (see Figure 1):

Figure 1: source
The tiny square in the middle has a side of 2 units

This site puts this square (with a side of 112) in a historical context:

"The dissection was found in the night of March 22, 1978 with the aid of the DEC-10 computer of the Technological University Twente, The Netherlands. Since no simple perfect squared squares were found of orders less than 21, it is a simple perfect squared square of lowest order . Also, it is the only simple perfect squared square of order 21."

The acronym SPSS is often used in referring to these sorts of squares and the letters stand for Simple Perfect Squared Squares. The entries in OEIS A217156 are for both simple and compound squared squares. It is in fact the sequence represents the sum of two separate sequences so that a(\(n\)) = A006983(\(n\)) + A217155(\(n\)) representing numbers of simple and compound squares respectively. The progression from 21 up to 30 is shown in Figure 2:

Figure 2: A006983 + A217155 = A217156

The site squaring.net has PDF file for all the configurations of orders up to 236. The eight configurations for \(n\)=22 have sides of 110, 110, 139, 147, 154, 172, 172 and 192. The SPSS for the side of 192 is shown in Figure 3.

Figure 3: source

Here is a Numberphile video that delves into the history of these squares:

There is an auxiliary video to this which shows the first square discovered and another where the differences in sizes between smallest and largest are least extreme.

Perfect rectangles consist of squares of different sizes that fit together to form a rectangle. No square can be broken down further. The minimum required is 9 and there are two possible configurations. See Figures 4 and 5.

Figure 4: source
33 x 32 rectangle composed of 9 squares
where the tiny square has a side of 1 unit

Figure 5: source
69 x 61 rectangle composed of 9 squares
where the tiny square has a side of 2 units