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Date: Fri, 16 Jan 1998 17:58:10 +0000 (GMT)
From: Jonathan Tuliani
To: Cube-Lovers
Subject: MEGAMINX
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The following is based on an email I sent recently to Kurt Endl. He says that
he does not have time to work on this at present, and, with a thesis to write,
nor do I! I expect that someone has heard of this and the answer is
known--I am new to this discussion group. Otherwise, hopefully
someone will find it sufficiently interesting to think it through.
I was delighted to be given a MEGAMINX this Christmas, together with
Kurt Endl's instruction booklet. I resolved to attempt the puzzle
without looking in the booklet, at least at first.
My approach was similar--I built a solution in layers, starting
at the bottom (which I shall call the north pole, consistent with the
notation in the latter part of the book) and proceeding, layer by
later towards the south pole. I was successful in my efforts until I
reached the south cap, at which point I became stuck. My problem was to
position and orient the south pole edges. Try as I might, the best I
could do was to reach a position where two south pole edges needed to be
exchanged. And try as I might, I could not find a way to do this.
After a week, I gave up and turned to the instructions. I was delighted
to see that their approach was similar to mine, and fascinated by the
simple moves L_{**}, L^{**}, R_{**} and R^{**} used. My
methods were, of course, far less elegant.
I was able to start at section 8, `Setting the South Pole
edges'. The procedure for setting the edges affects the southern
equatorial corners, which you then arrange later. My layer-by-layer
approach had, of course, already set these corners correctly before
attempting the south pole edges (and indeed before setting the southern
equatorial edges). Perhaps this was the key? Having the southern
equatorial corners set should not affect the validity of the book's method,
which should work with any arrangement of these corners. But following
the instructions, I was unable to position the last two south pole edges
correctly. The statement ``The remaining two South Pole edges will be
correctly placed again at the same time'' on page 21, section 8 of the
instructions must be incorrect--here after all was a counterexample! My
last two south pole edges needed to be exchanged.
I ignored the problem for the time being. I oriented the two edges
correctly, with them still in the wrong positions. I was then able to
complete the MEGAMINX, positioning and orienting the southern equatorial
and south pole corners as in the instructions. The result was a complete
MEGAMINX, except that it appeared that two little triangular stickers,
each on the border of the southern cap, had been exchanged.
After some thought, I have found a way out of this problem. I believe
that this is a detail that may be required for solution in some
circumstances that is not in this instruction booklet. I will try to
describe what I think went wrong.
The twelve faces of the MEGAMINX are coloured using only six colours, with
opposite faces bearing the same colour. Thus, each edge piece has an
`identical twin' on the opposite side of the completed MEGAMINX. When
solving the MEGAMINX from a totally jumbled position, these twins are
indistinguishable and may therefore be assigned either to their own
original position, or to their twin's position at random. In this sense
the solution of MEGAMINX is not unique. (Strictly, the solution may still
be unique but we have not proved conclusively that it is so, and have
demonstrated reason to believe that it may not be.)
Now, suppose we return to the two edges I wished to exchange. Imagine
them as south pole edges also in adjacent faces. Turn the MEGAMINX so
that the two faces concerned are in the left and right positions (using
your terminology). We need to exchange these edges, but the problem we
have is that every sequence of moves seems to rotate 3 edges cyclicly,
rather than exchange a pair! For example, the book uses L_{**} or R_{**} to
bring one of the two edges concerned down into the front southern
equatorial edge (``...the edge we have misused so brutally...''), and the
same moves to move it back up again. But these moves, together with any I
can find, cycle 3 edges. How can moves cycling 3 edges be used to
exchange just 2 edges? I was stuck.
The solution comes from the `identical twins' I talked about before.
Suppose one of the two edges I'm interested in is, say, yellow/blue, and
the other yellow/orange, so the southern pole is yellow. Tucked away on
the opposite side of MEGAMINX is *another* yellow/blue edge. By a simple
sequence of moves, this may be brought into the postion of the front
southern equatorial edge. Now consider cycling these three edges. As two
of the edges are identical, cycing these three edges *looks* like a swapping
of just two edges! Now we return to the other side of MEGAMINX the twin
of the piece that was originaly there. (Some fixing of the MEGAMINX is
required to repair the damage caused by bringing an edge from the opposite
side of the MEGAMINX to the front and sending it's twin back, but this
isn't too hard.) Now, having apparently `swapped' two adjacent edges, we
can proceed as per the instructions and complete the MEGAMINX. This I
have done.
Have any other people encountered this problem, or was I just
extremely unlucky? The question arising is, of course, just how many
solutions of MEGAMINX are there? There are 10 of these `identical
twin' edge pairs. I reckon swapping just one twin pair is not possible in
a complete solution, but that swapping any even number of twins may be
(unproven), and so there are 512 solutions, each of which would be
distinct if 12 different colours had been used on the original puzzle.
Does anyone fancy having a stab at this conjecture?
Jonathan Tuliani
Mathematics Department
Royal Holloway, University of London
Egham
Surrey TW20 0EX
U.K.
jont@dcs.rhbnc.ac.uk