LETTERS TO THE EDITORS
Building a Better Mattress Flip
To the Editors:
Could I suggest that Brian Hayes in "Group Theory in the
Bedroom" (Computing Science, September-October)
consider mattresses in the form of Möbius strips, maybe
arranged as trihexaflexagons for compactness? These would certainly
have a simple golden rule for flipping. If they were futons instead,
you could even curl up in them a bit. Obviously this design would be
useful as a "self righting" raft for emergency use, since
it could be straightened out easily even if it ever were flipped.
This is only as serious as the spherical cow
analogy—ridiculous in itself, but food for thought.
Peter M. Lawrence
Melbourne, Victoria, Australia
To the Editors:
I believe there's a flaw in Brian Hayes's logic for the proof that
there can be no golden rule for flipping mattresses. I'm more of a
hands-on experimentalist, so I used my copy of American
Scientist as an analogue of a mattress. It quickly became
obvious that an alternating sequence of any two of your basic
flipping operations does indeed yield a golden rule. A roll followed
by a pitch followed by another roll then another pitch cycles you
through all four orientations and returns you to the original state.
Every other pair of operations yields the same result: A golden rule
simply consists of alternating any two of the three flipping moves.
Beach Park, IL
Mr. Hayes responds:
I had thought I was the only person nerdy enough to seek a
mathematical solution to the problem of mattress flipping, but my
column on the subject brought an outpouring of ingenious—and
in some cases even practical—suggestions. Mr. Lawrence's
proposal of a one-sided Möbius mattress is deeply intriguing;
if only I could sleep on such a mattress, I would not only eliminate
the flipping problem but also I would never get out of bed on the
wrong side in the morning.
As for Mr. Roberts's algorithm, I agree that it works but not that
it's golden. In the column I defined a golden rule as "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." Mr. Roberts's rule requires alternating between two
different maneuvers and thus is not golden; it's more of a silver rule.
A number of other readers wrote in with methods that qualify as
silver rules for mattress flipping, as shown above.