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Date: Wed, 25 Jun 1997 22:14:32 BST
From: David Singmaster Computing & Maths South Bank Univ
To: cube-lovers@ai.mit.edu
CC: notation@vax.sbu.ac.uk, for@vax.sbu.ac.uk, 4^3@vax.sbu.ac.uk,
and@vax.sbu.ac.uk, 5^3@vax.sbu.ac.uk
Message-ID: <009B653D.9EE57540.331@vax.sbu.ac.uk>
Subject: notation for 4^3 and 5^3
David Barr has described some moves for the 5^3 using his own notation.
In the early 1980s, I extended my 3^3 notation to the 4^3 and this can be used
on the 5^3 as well. I described this in my Cubic Circular but perhaps it would
be useful to give it here.
I will describe the situation for the 4^3. On the 5^3, we don't ever
have to make a slice move of just a central layer. Further, a combination of
4^3 and 3^3 processes will solve the 5^3, so we don't really need to label the
central layers.
Consider the four layers from L to R. I denote the inner layers
by l and r. So the four layers are: LlrR. Similarly we have four layers:
FfbB and UudD. To describe a piece requires more effort than before, but
each piec lies in three layers and we can describe the piece by these layers.
E.g. FUR is a corner piece; FUr is an edge piece, lying on the FU edge -
but there are two of these and they are distinguished as being FUr and FUl
(I usually give the layers in clockwise order, but it is not essential and
there are times when it is more informative to use the other order.); Fur is
a face-centre piece, the one in the upper right corner of the inner four cells
of the F face. If you have been paying attention, you will ask about fur.
This is one of the body-centre cells, invisible to you unless you make a
transparent cube!
Using the standard notation of [F, R] = FRF'R', we find a number
of easy 3-cycles.
[[F,R],L] = (FLU,ULB,RFU)
[[F,R],l] = (FlU,UlB,DfR) (I've copied this from my Circular, but
I wonder if it's right as I thought there'd be some symmetry with the
preceding??)
[[F,r],l] = (Flu,Ulb,Drb)
In theory, these and a careful consideration of parity are sufficient to
solve the 4^3 and the 5^3. However, the parity problem is a bit awkward.
In my original approach, I got the corners in place and then all edges
except leaving the four edges along the FU and BU edges. Examine the
parity of these carefully. If they are in an odd permutation, apply
r^2 D^2 l' D^2 r^2 which 4-cycles these four edges and moves some centres.
Once the parity is corrected, there is little difficulty restoring the
rest of the cube.
For the 5^3, once you have paired up the edges, one can solve the
central edges by treating the 5^3 as a 3^3 with fat slices.
To correct a single pair of edges, one can use the following.
rrDDl'DDrr rrD'RR [[R,U],l'] RRDrr = rrDDl'DR'UR'U'l'URU'lRDrr =
(UBl,UBr) (Ful,Ubl,Bdl,Ufr) (Fdl,Ufl,Bul,Ubr). This messes up some centres,
but they are not too hard to restore. Indeed applying
rrUUr (uurrll)^2 r'UUrr wil correct the F and B centres disturbed by the
above, leaving a 180 degree rotation of the four U centres.
After I had developed the notation and solution, a Peter Lees pointed
out an unexpected feature. The exchange of upper and lower case letters is
a duality. The dual of URF is urf, while the dual of URf is urF.
This gives us a Pricniple of Duality: The dual of a sequence of moves is
the same process on the dual pieces. E.g., we had
[[F,R],l] = (FUl,UBl,DRf), so [[f,r],L] = (fuL,ubL,drF).
This duality allows one to transfer a number of 3^3 processes to
4^3 processes and to solve the invisible interior part of the cube!
By always moving an outer layer with its inner layer, one is
obviously simulating the 2^3. However, if one always moves, say R and l
together, one is also simulating the 2^3 in eight copies! Ooops, one
wants to move R and l' together. If one moves R and l together,
I think you get eight versions of the 2^3, but each is a reflection of
its neighbours!
If you are tired of thinking about God's Algorithm on the 3^3,
try the 4^3. I'm not even sure how to count moves. E.g., to do r, one
normally does Rr and then R', so does r count as one move or two?
Likewise, does Rr count as one move or two?
Enough for now.
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