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Date: Fri, 30 Apr 1999 12:33:43 +0100
From: David Singmaster
To: whuang@ugcs.caltech.edu
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D768E.2615E470.8@ice.sbu.ac.uk>
Subject: RE: Reinventing (and some edge-flipping techniques)
I've just spent three days being a Rubik's Cube demonstrator at a trade
fair, using the method given the my Step by Step solution in the middle of the
fifth edition of my Notes. The problems that Wei-Hwa Huang has described are
ones that I found in trying to develop an easy method. Basically there is no
'simple' way to flip edges, where 'simple' means easily understood and
remembered. There are simple ways to move edges, move corners and twist
corners. Consequently I decided to get the edge orientations correct at the
beginning of work on the last face, so that I wouldn't have to worry about what
happened to the rest of the face. In case you haven't got my notes at hand, I
used BLUL'U'B' which is a simple conjugate of the commutator [L,U]. This
exchanges the four U corners as two pairs of exchanges and cycles three U
edge, but effectively flips two of them on the way. Using the inverse process
BULU'L'B' does the same thing, but one of them effectively flips two adjacent
edges and the other flips two opposite edges. Two applications will flip all
four edges. Then we know simple processes which do 3-cycles or pairs of
2-cycles of edges, preserving orientation, or of corners and we have simple
processes for twisting corners. So we could carry out these three steps in any
order giving six possible algorithms and I'm pretty sure I received examples of
all of these. Indeed, of one considers doing the last face as having a fourth
stage of orienting edges, there are 24 possible algorithms and at one point I
was classifying algorithms into these 24 cases. I don't think I kept up with
this long enough to have all 24 cases, but I expect all of them exist!
Some personal comments and recollections, much of which is in my Notes.
The ideas of monotwist and monoflip, though blindingly obvious, took
well over a year to emerge! Despite the fact that lots of quite bright
mathematicians were working on the cube (e.g. Conway, Penrose, Rubik), I
remember first hearing about the idea in Jan 1980 (?) from Peter McMullen who
said they were using the idea at Cambridge. At first it seemed unreasonable as
we generally were looking for moves that only affected the U face and the
mono-moves are almost all elsewhere. However once I realised that the idea
gives a way of building simple moves, I realised that the commutator [F,R]
was a mono-move in the L face and its square was a mono-twist in the L
face. The Cambridge group had been using shorter, but less simple, moves.
With these mono-moves, it was now pretty easy to build the algorithm that is my
Step by Step solution and I think I did it within a few weeks as I recall the
5th ed. of my Notes was produced by March. (Remebering dates from 20 years
ago is always a bit dodgy - check what's in the Notes, which I don't have a
copy of here.)
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