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Date: Mon, 30 Nov 1998 23:13:51 -0500
From: michael reid
Message-Id: <199812010413.XAA11878@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: new types of cyclic shifters
a few months ago, i introduced the position superflip composed with
four spot and showed that it had a new type of cyclic shifting
property. i've now found some new ways to generalize cyclic shifting,
and this in turn suggests some new positions to consider.
first, some brief review.
the position superflip is central, so it commutes with all turns.
therefore, if x y produces superflip, so does y x . we can shift
one turn at a time, removing the first turn of the sequence and
shifting it to the end. in other words, if m produces superflip,
then x m = m x for all turns x . clearly, any position m with
the property that x m = m x for all turns x , is central, so the
only such positions are superflip and start.
i showed in an earlier message, that if x is any turn and m is
superflip composed with four spot, then x m = m y , where y is
x conjugated by the cube rotation C_U2 . more generally, we can
ask for positions m such that for any turn x , there is another
turn y satisfying x m = m y . for such a position, we can
cyclically shift any maneuver, one turn at a time, by replacing
the turn x^(-1) at the beginning with the corresponding y^(-1)
at the end. some other positions with this property are: four
spot, six spot, six spot composed with superflip.
a new way to generalize this is to consider positions m such that
for any turn x , we have x m = n x , where n is the same
pattern as m , but perhaps in a different orientation. for such
a position, we can cyclically shift any maneuver, by shifting the
first turn to the end, and then conjugating by the appropriate cube
symmetry. for example, consider the position in which the UFR corner
is twisted clockwise, and the other seven corners are twisted
counterclockwise. (i'll call this "1-7-twist" for now, but this
pattern needs a better name.) this position is created by
U F2 B' U B U D2 R2 U2 B U' D2 B U F2 B L2 B2
now, cyclically shift the U at the beginning to the end to get
F2 B' U B U D2 R2 U2 B U' D2 B U F2 B L2 B2 U
which produces a different orientation of the same position; this
time, the ULF corner is twisted clockwise. now conjugate this
maneuver by C_U to get
R2 L' U L U D2 B2 U2 L U' D2 L U R2 L F2 L2 U
which produces the original position, in its original orientation.
actually, there are 3 cube symmetries by which one could conjugate,
since the position has 3-fold symmetry. another position with
this type of cyclic shifting property is 1-7-twist composed with
superflip.
we can combine both types of generalizations, and ask for positions
m that have the property that for any turn x , we have x m = n y ,
where y is another turn, and n is the same pattern as m , but
perhaps in a different orientation. for such positions, we can
cyclically shift any maneuver by replacing x^(-1) at the beginning
of the maneuver by the corresponding y^(-1) at the end, and then
conjugating the whole maneuver by the appropriate cube symmetry.
two such positions are: 1-7-twist composed with four spot, and
1-7-twist composed with four spot composed with superflip.
here's all the examples of cyclic shifters that i know, along with
minimal maneuvers:
1. central positions
start (0q*, 0f*)
superflip
R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q*, 22f)
F' B' D2 L' B2 L2 F2 U' D B' D2 R L D' F2 U' L2 D' F2 D' (20f*, 28q)
2.
four spot F2 B2 U D' R2 L2 U D' (12q*, 8f*)
six spot F B' U D' R L' F B' (8q*, 8f*)
four spot composed with superflip
U2 D2 L F2 U' D R2 B U' D' R L F2 R U D' R' L U F' B' (26q*, 21f)
F U2 R L D F2 U R2 D F2 D F' B' U2 L F2 R2 B2 U' D (20f*, 28q)
six spot composed with superflip
R' U D R' U F' D R' B U' L' U' F' D F' B' D' R' F D F D' R2
(24q*, 23f)
U2 F B' R F L2 F2 D B2 D2 R2 B' L2 F' D2 R2 D' B R B2 (20f*, 30q)
3.
1-7-twist
F R' U' L' F' U' B' L' U' R2 F L' D' R' F' D' B' R' D' L2 (22q*, 20f)
F R2 L' F L F B2 U2 F2 L F' B2 L F R2 L D2 L2 (18f*, 26q)
1-7-twist composed with superflip
F R F' R U B L D B D' L' D L F B' R B D F R' (20q*, 20f)
F R L' B L D F' L2 B D2 B' L' F2 B' D B U L B (19f*, 22q)
4.
1-7-twist composed with four spot
F B2 L' U' B' L U' D L2 U R' U' R F' U' L' F D B' U' (22q*, 20f*)
1-7-twist composed with four spot composed with superflip
F U' R' L F' U R L' U2 D' B R L F' R D' R' F2 L U' (22q*, 20f*)
as usual, i give a maneuver which is minimal in both metrics whenever
this is possible. i don't claim that i've found all positions in
these categories, but these are all that i know. if you find any others,
they'd be good candidates for positions far from start.
mike