Virginia Mathematics Teacher Fall 2016

Unsolved Mathematical Mysteries: Rectangles Covering Rectangles Matt Harvey

For a rectangle R , there are three restrictions that limit how small x can be (see figure 2 on the left). 1. In order for the left two rectangles to cover R from top to bottom, , so . 2. In order for the right rectangle to cover R from top to bottom, . 3. In order for all three rectangles to cover R from left to right, , so . The three conditions are graphed in figure 2 on the right. To cover R , all three conditions must be met, so the minimum value of x that the arrangement will permit, in terms of h , is

A rectangle with dimensions can be covered by three slightly reduced copies of itself, rectangles of sizes where x <1. For a square, one such covering was identified by the mathematician Henry Dudeney and published in his book Puzzles and Curious Problems in 1931. It is shown in figure 1 on the left. In his construction, the scaling factor x is the square root of , so .

Figure 1. (left) Dudeney's covering of a square by three smaller squares; (right) a covering of a non-square rectangle by three smaller copies of itself. For any non-square rectangle, a covering such as the one shown in figure 1 on the right, with one rectangle turned , works. It is natural to ask: how small can be in this construction? Since similar rectangles will have the same minimum value, and since every non-square rectangle is similar to a rectangle (with ), we may restrict our attention to rectangles, where

The graph of x (as well as the point corresponding to Dudeney's square construction) is shown in figure 3 on the left. Note that the point on the graph corresponds to a golden rectangle, for which , so restrictions 2 and 3 are met exactly, as shown in figure 3 on the right.

Figure 3 . (left) The graph of x as a function of h ; (right) covering the golden rectangle.

Figure 2 . (left) To cover a rectangle, x must satisfy three conditions; (right) graphs of the equations that govern the three conditions.

This function describes only one possible arrangement, however, the one with a rectangle turned . Other arrangements might give covers with smaller x values. The significant drop from the graph to the Dudeney point is evidence that there is room for improvement for rectangles

Virginia Mathematics Teacher vol. 43, no. 1

51