Square of Squares

Copyright © Karl Dahlke, 2023

It's easy to make a square out of smaller squares when some of those squares are the same size. An example is the tic-tac-toe board,or perhaps an arrangement like this.

Is it possible to build a square of squares where all the squares have different sizes? It is, but it's not easy. I'm willing to spoil here, because you need a computer to find a solution. The smallest example has 21 different squares that combine to form a larger square, 112 units on a side. When I was 18 years old I asked my grandfather to make me a wooden version of this puzzle. I wanted to hold it in my hands and physically put the pieces together. I gave him the dimensions of each of the 21 squares, and he cut them out on his jigsaw with his usual precision. (He was a skilled carpenter and architect.) I will always cherish this puzzle, for its mathematical beauty, and as a memento of all the good times I had with my grandfather. You might want to make a similar puzzle for yourself, out of paper or stiff cardboard. Let each unit be an eighth of an inch. Thus the largest square, 50 units on a side, is 6 and a quarter inches on a side. The smallest square, 2 units on a side, is a quarter inch on a side. Here are the sizes of the 21 squares.

50 42 37 35 33 29 27 25 24 19 18 17 16 15 11 9 8 7 6 4 2

If you took the time to cut out these squares, try to assemble them into one large square, 14 inches on a side. You might want to build a frame to make it easier. You don't need a computer for this part. It's a fun 21-piece puzzle.

I will present the solution algebraically, then geometrically. I'm not sure how the pictures will come out on a small cell phone screen; that was not a consideration when I first wrote this article. In the following table, the size of the square is given first, ranging from 50 units to 2 units. Then the at sign separates the size from the coordinates. If you want to be consistent with the pictures to follow, use the matrix convention. In other words, 7 @ 3,5 means the square of size 7 has its upper left corner at row 3 column 5, with 0,0 at the upper left. This is different from the xy plane, where 3,5 starts at the origin, moves 3 to the right, and then 5 up. It is different still from the internet ImageMagick drawing convention, where 3,5 starts at the upper left of the frame, moves 3 pixels to the right, and then 5 down. As my boss once said to me in 1993, “The great thing about standards is, there are so many of them.” Here are the 21 pieces, moving roughly from top to bottom.

50 @ 0,0
29 @ 0,50
33 @ 0,79
25 @ 29,50
4 @ 29,75
37 @ 33,75
35 @ 50,0
15 @ 50,35
9 @ 54,50
16 @ 54,59
2 @ 63,50
7 @ 63,52.
17 @ 65,35
18 @ 70,52
42 @ 70,70
11 @ 82,35
6 @ 82,46
27 @ 85,0
8 @ 85,27
24 @ 88,46
19 @ 93,27

The first picture is black & white, showing the borders of the squares with their sizes inside. Run across any row or any column, and note that the total is 112.

50 29 33
25 4
35 15
9 16
2 7
18 42
11 6
27 8

The second representation is in living color. Each square is assigned a color, red, yellow, green, or blue, so that adjacent squares always have different colors. The Four Color Theorem, proved in 1976, says that any map can be so colored. That is a fascinating topic that I may address in the future.

Cube of Cubes

It is natural to ask if there is a cube consisting of smaller cubes, each of a different size. There is not, and the proof is easy once you see the trick.

A cube of cubes implies a square of squares on the floor, much like the above. Within this square of squares is a smallest square, corresponding to the smallest cube sitting on the floor. This would be the square of size 2 in our example. Let's call it 2 meters, just to get a handle on it. Thus the square of squares is somewhat larger than a football field. The smallest square can't be on an edge. Try putting three larger squares around it and you get into trouble. Nor can it be in a corner. Thus the smallest square is somewhere in the middle of our square of squares, as shown above.

The smallest square is really a cube, 2 meters on a side. Four larger cubes surround this cube. If you sit on top of this cube, there are taller cubes all around you, as though you were standing in the middle of New York City. This acts as a frame around the 2 meter cube, a frame for another puzzle. Place another layer of cubes on top of this 2 meter cube. These cubes are smaller of course, perhaps measured in centimeters, but they cover the 2 meter cube, and create a square of squares. There is a smallest square, i.e. a smallest cube sitting on top of the 2 meter cube. Again this cube is in the interior, not by an edge. Perhaps this cube is 3 centimeters on a side. It is surrounded by four larger cubes. On top of this 3 cm cube is another layer of cubes, perhaps measured in millimeters. This has a smallest cube and on top of that cube is another layer of cubes, now requiring a microscope to see. This continues forever. Thus a cube cannot be built from a finite set of smaller cubes.