Date: Wed, 18 Jan 95 10:51:39 -0500
From: michael reid <mreid@ptc.com >
~~~ ~~~ Subject: searching for superflip in quarter turn metric

here's my approach to searching for superflip in the quarter turn metric.
i gave a maneuver of length 24q for superflip on january 10. suppose
there is a maneuver of length 22q (or shorter). consider three cases:

case 1. there is a minimal maneuver which contains a half-turn.

case 2. no minimal maneuver contains a half-turn, but there is a
minimal maneuver which contains consecutive turns of
opposite faces.

case 3. neither case 1 nor case 2 hold.

in case 1, we may find a minimal sequence of the form

sequence_1 sequence_2,

where sequence_2 is at least 3q long. as in the face turn metric,
we may also suppose that sequence_1 starts with one of

```R1 F1,   R1 F2,   R1 F3,   R1 U1,   R1 U2,   R1 U3,
R1 L1 U1,   R1 L1 U2,   R1 L1 U3,   R1 L1 F1,   R1 L1 F2,
R1 L1 F3,   R1 L3 U1,   R1 L3 U2,   R1 L3 F1,   R1 L3 F2.
```

furthermore, the case starting with R1 F2 may be included in the
case starting with R1 F1, and similarly for other cases. thus we
may suppose that sequence_1 starts with one of

```R1 F1,   R1 F3,   R1 U1,   R1 U3,
R1 L1 U1,   R1 L1 U3,   R1 L1 F1,   R1 L1 F3,
R1 L3 U1,   R1 L3 F1.
```

in case 2, we may find a minimal sequence of the form

sequence_1 sequence_2,

where sequence_2 is at least 2q long. as in case 1, we may suppose
that sequence_1 starts with one of the ten sequences above.

in case 3, the best we can do is 1q in stage 2. however, i claim
that we can find three consecutive turns of mutual adjacent faces.
otherwise, we'd have a maneuver for superflip using only the four faces
F, R, B, L, (for example) which is ridiculous, because edges can't
change orientation using only these turns.

therefore, we may suppose that a minimal sequence starts with three
consecutive turns of mutual adjacent faces. up to symmetry, there
are eight cases for these turns:

```U1 R1 F1,   U1 R1 F3,   U3 R1 F1,   U3 R1 F3,
D1 R1 F1,   D1 R1 F3,   D3 R1 F1,   D3 R1 F3.
```

replace U1 R1 F1 sequence by R1 F1 sequence U1 , and
similarly for the other seven cases. thus we have a minimal
maneuver in the form sequence_1 sequence_2 , where sequence_2
is 1q long and sequence_1 starts with either R1 F1 or R1 F3.

combining all the above cases, a maneuver for superflip in 22q or less
(assuming one exists) may be found in one of the forms:

```R1 L1 U1  sequence_1  sequence_2,
R1 L1 U3  sequence_1  sequence_2,
R1 L1 F1  sequence_1  sequence_2,
R1 L1 F3  sequence_1  sequence_2,
R1 L3 U1  sequence_1  sequence_2,
R1 L3 F1  sequence_1  sequence_2,
```
```where  sequence_1  is at most  17q  long,
```
```R1 U1  sequence_1  sequence_2,
R1 U3  sequence_1  sequence_2,
```
```where  sequence_1  is at most  18q  long,
```
```R1 F1  sequence_1  sequence_2,
R1 F3  sequence_1  sequence_2,
```
```where  sequence_1  is at most  19q  long.
```

i don't know how feasible this is (but it sure looks formidable).
to get some idea, first i'll test for 20q or less.

mike