Squaring the Square

What’s going on

Cutting a square into smaller squares is easy: draw a grid. Cutting it into squares that are all different sizes is so hard that many mathematicians once thought it impossible. Every square you place leaves a step that must be filled by a new, different square, and in the end everything has to meet in a perfectly flat edge.

Such a tiling is called a perfect squared square. The scene shows the smallest one possible: 21 squares, with sides 2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42 and 50, and together they fill a square of side 112. Press Build it square by square to watch it fit together.

You can check it like an accountant:

• Along every edge the sides add up. The top row is 50 + 35 + 27 = 112. The left side is 50 + 29 + 33 = 112.
• The areas add up. 2² + 4² + … + 50² = 12,544, and 112 × 112 = 12,544. No gaps, no overlaps.

You can even check it cell by cell: each of the 12,544 little unit squares is covered exactly once.

A surprise from electricity

In the late 1930s four students at Cambridge, Rowland Brooks, Cedric Smith, Arthur Stone and William Tutte, found a way to stop guessing. They turned the tiling into an electric circuit. Press Show as a circuit:

• every horizontal edge in the tiling becomes a junction (blue);
• every square becomes a wire with the same resistance, running from the junction along its top to the junction along its bottom;
• the top of the big square is connected to +112 volts and the bottom to 0.

Then the current in each wire equals the side of its square, and the voltage drop across it equals its height. Two laws of electricity become two facts about squares. Current in = current out at every junction: the squares resting on an edge are exactly as wide in total as the squares hanging below it. And for wires of resistance 1, Ohm’s law says voltage drop = current: height = width, which is what makes each piece a square.

So instead of guessing tilings, they could study circuits, where there were already methods for working out every current. In 1939 the German mathematician Roland Sprague published the first perfect squared square, with 55 squares; the Cambridge group found one with 69 at almost the same time. Smaller ones followed, and in 1978 the Dutch mathematician A. J. W. Duijvestijn used a computer to find the 21-square tiling in the scene. He also showed that no perfect squared square has fewer than 21 squares, and that with 21 this is the only one, apart from turning or flipping it.

Rectangles are easier, but not by much

Press Puzzle: 33 × 32 and fill a 33 by 32 rectangle with nine squares of sides 1, 4, 7, 8, 9, 10, 14, 15 and 18. Their areas add up to 1,056 = 33 × 32: just enough room. No rectangle can be cut into fewer than nine different squares, and with nine there are only two: this one, and a 69 by 61.

Try it yourself

1. One junction

In the scene, the 35 and the 8 both rest on the same horizontal edge, and the 15, the 17 and the 11 hang below it. Without measuring, how long is that edge?

Answer

43. The squares above cover it exactly: 35 + 8 = 43. So do the squares below: 15 + 17 + 11 = 43. In the circuit, this is the rule that current flowing into a junction equals current flowing out.

2. Where is the smallest square?

Can the smallest square of a perfect squared square (or rectangle) touch the outer edge?

Answer

No. Suppose it lies along the bottom edge. Its neighbours on each side are bigger (or are the outer wall), so they stick up higher, leaving a well exactly as wide as the smallest square. A square of the same width is forbidden, so the well needs smaller squares, and there are none. In the scene, the 2 sits deep inside.

3. Cubing the cube

Can a cube be cut into smaller cubes that are all different sizes?

Answer

No. The cubes on the bottom face tile it as a perfect squared square, so the smallest of them is not at the edge: it sits in a well, with bigger cubes all round. Its top face must be covered by smaller cubes, which tile it as a perfect squared square, with an even smaller cube in a well, and so on forever. But there are only finitely many cubes.

Go deeper

• The four Cambridge students published their circuit method in 1940 in a paper called “The dissection of rectangles into squares”. During the Second World War, William Tutte worked out how the German Lorenz cipher machine was built, without ever seeing one; he later became a leading expert on graphs and networks.
• Arthur Stone, one of the four, is also known as the inventor of flexagons, paper toys that turn inside out to show hidden faces.