Something I found interesting today….

Squared Squares (original research from Philipp Kleppmann)

Figure 1 shows a rectangle dissected into smaller squares all of which have different side lengths. These are called squared rectangles.

The challenge of producing a squared rectangle came about in a book called the Canterbury Puzzles and was discovered by R. Sprague in 1939 (more than 30 years after the puzzle was published).

In the late 1930s some Cambridge University undergrads used electrical network theory to produce more squared rectangles.

Low Tech

See Figure 2. Draw a rectangle cut up into smaller rectangles. Assign values x and y to the side lengths of the two ‘squares’. From these, determine all the other side lengths. You then need to ensure that the two vertical sides of the big rectangle have the same length.

(5x + 3y) + (8x + 4y) = (4x + 4y) = (4x + 5y) i.e. 5x = 2y. So, taking x = 2 and y = 5 we get the squared rectangle from Figure 1.

This method was developed by Arthur Stone, but systematic analysis was made difficult because the some equations created negative side lengths.

High Tech

Suppose you have a squared rectangle like Figure 1. Construct a directed graph with a vertex for each horizontal line segment and an edge for each square. There is an arrow from vertex (v) to vertex (w) only if the corresponding horizontal line segments V and W in the rectangle are connected by a square and V is above W. The arrow can then be labelled with the side length of the square as in Figure 3.

The graph in figure 4 is constructed this way from the rectangle in figure 1. This is a Smith diagram.

Now call P and Q the poles of the network, and interpret the labels of the edges as currents.

Things to notice:

1. For any vertex that isn’t a pole the sum of the currents entering it is equal to the sum of the currents flowing out of it: The sum of the side lengths of squares lying directly above one horizontal line segment in the squared rectangle is the same as the sum of side lengths of squares lying directly below it.

2. The sum of currents around any circuit is zero. The length of any straight vertical path from one horizontal line segment to another one is the same.

3. The sum of the currents leaving P is equal to the sum of the currents entering Q, since the lengths of the two horizontal sides of the rectangle are equal.

The construction works in either direction. If an electrical network satisfying these three conditions is constructed which has three different currents along all of its wires, then it is the blueprint of any squared rectangle.

Now how about a squared square? (Figure 5)

Read a full description in Eureka from the Cambridge University Mathematical Society