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# Group Theory in the Bedroom

An insomniac's guide to the curious mathematics of mattress flipping

A Silver Rule

The absence of a golden rule for mattress flipping is a disappointment, but it does not portend the demise of Western Civilization. We can adapt; we can learn to live with it.

Suppose you flip your mattress at regular intervals, but each time you choose an axis of rotation at random. This is clearly a less-than-optimal algorithm, but how large a penalty does it carry? Under the ideal rotation schedule, each orientation of the mattress would get 25 percent of the wear. A quick computer simulation shows that if you do random flips quarterly over a period of 10 years, the most-used orientation will get 31 percent of the wear and the least-used 19 percent. Except for those among us who suffer from a severe Princess-and-the-Pea complex, ±6 percent is probably good enough.

But we can do even better. We can cheat.

The no-golden-rule theorem for mattress flipping assumes that the surfaces of the mattress are unmarked, so that the four allowed configurations are indistinguishable. Everything changes if you label the surfaces. Specifically, suppose you go through the four possible orientations of the mattress and label each one with a number from the set 0, 1, 2, 3. You might place the labels so that in each configuration, one of these numbers is facing upward in the corner closest to the righthand side of the headboard. Given this labeling, the mattress-flipping algorithm calls for nothing more than counting. Each time you are ready to make a flip, you note the number that appears in the upper righthand corner, and calculate the successor of that number modulo 4. (In other words, you cycle through the sequence 0, 1, 2, 3 and then return to 0 again.) Turn the mattress in whatever way is necessary to bring the successor number into the upper-right position. The turn needed will not always be around the same axis, but the closure property of the group guarantees that you will always be able to bring the next number into position with a single flip around some axis.

The counting algorithm is not a golden rule, but perhaps it deserves to be called a silver one. As a practical matter, this solution is so simple that I would expect mattress makers to adopt it, by embossing numbers on their products. Some of the manufacturers require periodic flipping as a condition of maintaining a warranty, and they give complicated—often ambiguous—instructions on how to comply. Wouldn't it be easier just to count?

The algorithm can be simplified even further for those who can't count as high as 3. If a mattress had lengthwise stripes on one side and crosswise stripes on the other, you could cycle through the four states by always flipping parallel to the stripes. Another possibility is to somehow adapt to our purposes the label that reads "Do not remove this label." (Now there's a golden rule!)