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Date: Sun, 13 Jul 1997 19:37:58 -0400
From: michael reid <reid@math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: minimal maneuvers for composition of superflip and pons asinorum

i've finished calculating all minimal maneuvers (in the face turn metric)
for the composition of superflip and pons asinorum.  in my search for
maneuvers of length  <= 18f  for this position, i used symmetry to show
that we may suppose that the first face turn is either  U  or  U2 .
in fact, there is more symmetry available, and this time i will use it.

in the case beginning with  U , there are four symmetries, generated
by the cube rotation  C_U .  using these, we may suppose that the second
face turn is one of  D , D2 , D' , R , R2  or  R' .

in the case beginning with  U2 , there are eight symmetries.  these are
generated by the cube rotation  C_U  and reflection through the left-right
plane.  using these, we may assume that the second face turn is one of
D , D2 , R  or  R2 .

we can reduce these cases somewhat further.  the cases beginning with
U D2  and with  U2 D  are equivalent, so only one needs to be seacrhed.

the cases beginning with  U D'  and with  U2 D2  can also be eliminated.
both  U D'  and  U2 D2  commute with both pons asinorum and with superflip,
so we may cyclically shift these turns to the end of the maneuver.  our
position cannot be achieved only using these "slice" turns, so we'll
always be able to cyclically shift until we do not begin with a slice turn.
(alternatively, note that any maneuver of length  19f , or any odd length
cannot consist only of slice turns!)

that leaves seven cases to search.  my program took just less than one
day to search all through 17f.  it found 26 maneuvers, 16 for the case
beginning with  U D .  however, this case has 8 symmetries, so there are
just 2 different maneuvers, each in 8 different orientations.

this leaves 12 different maneuvers, which come in 6 pairs of inverses.
they are:

     U  R    F  D  R  U' D  L' U' D  F' B2 R  L' D' F' L' B' R'   (19f)
     U  D    F  R  L' F  B' L  D2 R  L  F' B' U' L2 F  B' U2 L'   (19f)
     U  D    F' B' L' U2 F' B  L2 U' R' L' F' U' D  F' B  D' L2   (19f)
     U2 R    F  U  F  B' L' D' F  B' L  B  R  L' U  D2 B' R' U2   (19f)
     U2 R    F  U2 D' R' L  F' L' F  B' U  L  F  B' D' B' R' U2   (19f)
     U2 R    U2 D2 R  U' L' U  B  R  F2 U' D  B' R' F' D  B' L2   (19f)

and their inverses.  the first of these is the maneuver found by
jerry bryan.  it's also the only of these that is  20 quarter turns
long, which is consistent with his findings.

mike