Date: Sun, 10 Apr 94 17:37:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: More Sq Notes
```Additional Notes on Squares Group Patterns
------------------------------------------

Note that p80a, p99a and p108a are 2 DOT patterns,
all of the form U1 (swap edges & corners in U and D faces)  D3
or   U1 (swap edges & corners in U and D faces)  D1
or   D1 (swap edges within U and edges within D) D3

P66a alternate method    F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U1 D1
(13)
p67a alternate method    F2 R2 F2 U3 D3 L2 B2 D2 L2 B2 U1 D1 B2
(13)

p80a alternate method    U1 F2 R2 L2 U2 D2 F2 U2 D3
(9)
p99a alternate method    U1 R2 F2 B2 U2 D2 R2 D1
(8)

P100a alternate method   F2 U2 D2 F2 R2 L2 D1 F2 R2 L2 B2 U1
(12)
p108a alternate method   R2 F2 B2 L2 D1 R2 U2 R2 L2 U2 R2 D1
(12)
p130a alternate method   F2 R2 F2 B2 U1 D1 F2 R2 D2 F2 L2 U3 D3
(13)
p133a alternate method   R2 U1 F2 R2 L2 U2 D2 F2 U2 D3 R2
(11)
```

A) In general, any sequence of half turns which swaps edges and
corners in the U and D faces can be sanwiched between a single quarter
turn of U and a single quarter turn of D. Such a process would lead to
a square's group position.

B) Furthermore, any sequence of half turns which swaps edges within U
and
edges within D can be sanwiched between a single quarter turn of U or D
and a single quarter turn of U or D. Once again, such a process would
lead to a square's group position.

Here is an example of a position which takes over twice as many half
turns as full group moves:

```L2 T2 L2 T2 L2 T2 F2 L2 T2 F2 T2 R2 B2     (13)
U1 F2 R2 L2 B2 D1                           (6)
```

As discussed in point A above, sequences which move all elements of
the U face to the D face and also move all elements of the D face
to the U face (excepting the centres naturally) appear as 2 DOT
patterns on the cube. This makes sense, as the initial quarter
turn in process p80a must be balanced by another quarter turn.
Since all of the elements subjected to the quarter turn are now
in the D face, we must turn that face a quarter turn to remain
in the squares group.