Date: 26 Jul 1980 1315-PDT
From: Davis at OFFICE-3
Subject: Some (perhaps well-known) Transformations
To: cube-lovers at MIT-MC
I have just recently joined the cube-hackers mailing list, but I have read over
all the old mail. I suspect, however, that some of what I have to say will be
well-known already; on the other hand, I suspect that some of it is new.
When I first got my cube, I had the advantage and disadvantage of working in a
vacuum. Hence, my original solution was more complicated than it should have
been, but I did go about it another way, and discovered a few interesting facts
along the way. Having only seen one cube, I foolishly assumed that the color
pattern was the same on all cubes, and my notation for a move was simply a
color. A R (or Red) move meant that you look at the red face, and turn it 90
degrees courter-clockwise. I wrote a little hack based on that notation, and
since all my notes are in that form, I will use it here. It should be trivial
enough to convert it to the Left, Right, ... form. For reference, here is my
color pattern in the "solved" condition:
GGG
GGG
GGG
OOO RRR YYY WWW
OOO RRR YYY WWW
OOO RRR YYY WWW
BBB
BBB
BBB
When I purchased my cube, the store was out of all but the demonstration model,
so I never got a chance to mess around with a virgin cube, and I approached the
problem as follows: My program would simply take moves and print out the
condition after so many moves. It would also, given a sequence of moves, find
the order of the move as a group element. I then tried a number of patterns,
some from my experience with the cube, and some completely at random, and found
the orders of those moves. If the order came out to be, say, 90, then I would
do 30 and 45 moves to see what the cube looked like after that many moves.
These half- and third- way patterns were often fairly simple, and after trying
aout about 20 or 30 likely candidates, I generated most of the primitive
transformations I used to solve the cube originally. I also generated (as half
way positions) a large number of nice patterns, which are pretty to look at,
but not of much use for solving a cube.
In particular, laughter, mentioned by ALAN, is gotten by:
3(OYRW)
I find 3(RYYR) a useful solving transformation, as well as 3(RYRRRYYY) and
3(RYYYRRRY) (these last two being commutators). I have not yet gotten a look
at any of Singmaster's stuff, but these last two may be well known. Believee
it or not, one of the transformations I used to solvee the cube originally was
45(RGWY), which has the net effect (after 180 moves) of flipping the R-Y and
G-Y side cubies in place. I'm glad I found a better one than that.
In fact, if you haul out any book on group theory, it is interesting to find
group equations, and plug in random cube moves as the group elements to see
what happens. Needless to say, things which group theorists find interesting
usually tend to do interesting things to the cube.
One other thing I discovered which also does interesting things to the cube is
the following transformation: Leave the body-slicing center slice in place,
and rotate the other two sides, one toward and one away from you. Then rotate
the top and bottom 180 degrees, and rotate the sides back. This transformation
also gives a cube configuration which looks like something in the center-slice
group, but is not.
-- Tom Davis
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