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Date: Wed, 15 Apr 1998 18:07:57 +0100
From: David Singmaster
To: cube-lovers@ai.mit.edu
Message-Id: <009C4C21.E208C3B3.8@ice.sbu.ac.uk>
Subject: Hamiltonian circuits on the cube
The discussion of isoglyphs, etc., has reminded me of a problem which I
worked on in the early 1980s but never resolved. I took an all white cube and
traced a Hamitonian circuit through all the 54 facelets. If you jumble this
up, it is essentially impossible to restore. Indeed there are probably many
solutions to the problem. This led me to ask some questions about such
Hamiltonian circuits through the 54 facelets.
A. How many are there?
B. Are there any such circuits where the pattern is the same on each
face? I thought I could prove that such did not exist, but I think I assumed
that the circuit entered and left each face once, but this need not be the
case.
I was able to find a circuit with two types of face pattern and the two
types were mirror images. If you index the facelets on a face by 11, 12, ...,
33, then the path on the face is: 11, 12, 22, 21, 31, 32, 33, 23, 13.
If the circuit enters and leaves each face just once, then the sequence
of faces visited forms a Hamiltonian circuit on the faces of the cube, which is
better viewed as the vertices of an octahedron. It is easy to see that there
are just two such circuits on the octahedron (up to isomorphism). One of these
circuits has two kinds of vertex behavior and hence is not suitable.
Does this question interest anyone? The reason for the second question
was that if just one type of face pattern could be used, then it would be easy
to print up stickers for sale - one would just do the same pattern six times!
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk