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From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Hamiltonian circuits on the cube
Date: 20 Apr 1998 15:55:44 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <6hfr60$lfq@gap.cco.caltech.edu>
References: <cube-lovers.009C4C21.E208C3B3.8@ice.sbu.ac.uk>

David Singmaster <david.singmaster@sbu.ac.uk> writes:
>	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.

The answer to B is "Yes"!!

I was pretty surprised to come up with this within ten minutes of reading
the question:

+--+--+--+
|42|43|44|
+--+--+--+
|47|46|45|
+--+--+--+
|54| 3| 4|
+--+--+--+--+--+--+--+--+--+--+--+--+
| 1| 2| 5| 6| 7| 8|26|27|40|41|48|53|
+--+--+--+--+--+--+--+--+--+--+--+--+
|14|13|12|11|10| 9|25|28|39|38|49|52|
+--+--+--+--+--+--+--+--+--+--+--+--+
|15|16|17|18|21|22|24|29|36|37|50|51|
+--+--+--+--+--+--+--+--+--+--+--+--+
|33|32|19|
+--+--+--+
|34|31|20|
+--+--+--+
|35|30|23|
+--+--+--+

X=====X=====X=====X
H     H     H     H
---------------+  H
H     H     H  |  H
X=====X=====X==|==X
H     H     H  |  H
---------------+  H
H     H     H     H
X=====X=====X=====X
H     H     H     H
---+  H  +-----+  H
H  |  H  |  H  |  H
X==|==X==|==X==|==X=====X=====X=====X=====X=====X=====X==|==X==|==X==|==X
H  |  H  |  H  |  H     H     H     H     H     H     H  |  H  |  H  |  H
H  +-----+  H  +-----------------+  H  +-----+  H  +-----+  H  |  H  |  H
H     H     H     H     H     H  |  H  |  H  |  H  |  H     H  |  H  |  H
X=====X=====X=====X=====X=====X==|==X==|==X==|==X==|==X=====X==|==X==|==X
H     H     H     H     H     H  |  H  |  H  |  H  |  H     H  |  H  |  H
H  +-----------------------------+  H  |  H  |  H  +-----+  H  |  H  |  H
H  |  H     H     H     H     H     H  |  H  |  H     H  |  H  |  H  |  H
X==|==X=====X=====X=====X=====X=====X==|==X==|==X=====X==|==X==|==X==|==X
H  |  H     H     H     H     H     H  |  H  |  H     H  |  H  |  H  |  H
H  +-----------------+  H  +-----+  H  |  H  |  H  +-----+  H  +-----+  H
H     H     H     H  |  H  |  H  |  H  |  H  |  H  |  H     H     H     H
X=====X=====X=====X==|==X==|==X==|==X==|==X==|==X==|==X=====X=====X=====X
H     H     H     H
H  +-----+  H  +---
H  |  H  |  H  |  H
X==|==X==|==X==|==X
H  |  H  |  H  |  H
H  |  H  |  H  +---
H  |  H  |  H     H
X==|==X==|==X=====X
H  |  H  |  H     H
H  |  H  |  H  +---
H  |  H  |  H  |  H
X==|==X==|==X==|==X

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
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