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Group Theory in the Bedroom

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

Brian Hayes

Having run out of sheep the other night, I found myself counting the ways to flip a mattress. Earlier that day I had flipped the very mattress on which I was not sleeping, and the chore had left a residue of puzzled discontent. If you're going to bother at all with such a fussbudget bit of housekeeping, it seems like you ought to do it right, rotating the mattress to a different position each time, so as to pound down the lumps and fill in the sags on all the various surfaces. The trouble is, in the long interval between flips I always forget which way I flipped it last time. Lying awake that night, I was turning the problem over in my head, searching for a golden rule of mattress flipping.

The essential characteristic of a golden rule is universality: One rule works all the time, for everyone. The famous archetype of such rules—the one about doing unto others—certainly has this property. So does the rule of the road: "Drive to the right." ("Drive to the left" works just as well; what matters is not which side you choose but that everyone make the same choice.) Not all rules generalize so smoothly. A sign saying "Please use other door" is not helpful when it's posted on every door. A golden rule of mattress flipping would be some set of geometric maneuvers that you could perform in the same way every time in order to cycle through all the configurations of the mattress. Following this algorithm might entail more physical labor on each occasion—perhaps making multiple flips or turns—but at least it would eliminate the mental effort of remembering.

If you too spend sleepless nights fretting over this problem, I'm afraid I have some disappointing news. You will not find the golden rule of mattress flipping in this column. As a matter of fact, no such rule exists—at least not in the form I originally imagined. But please read on anyway: The search for a mattress-flipping algorithm leads to some diverting mathematics, not just in the bedroom but also in the garage and at the breakfast table. Furthermore, although I can offer no golden rule for mattress flipping, I do have some practical advice.

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