Date: Sun, 10 Apr 94 17:39:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: More Shifting
```More Notes on Invariant Shifting
--------------------------------
```

Let us define a process as "Shift Invariant" if it results in the same
displacement even after a series of left or right shifts. That is,
from a process of length N we can generate N-1 processes which
result in the same displacement by shifting the process. Sometimes
the processes generated are not all unique!

```e.g. P8   2 x ORDER 2: (symmetry level 3)
D2 F2 T2 F2 B2 T2 F2 T2
(8)
```

Q: Is this the longest such process?

```A: No.
```

The following processes are also shift invariant:

```2 Swap                   D2 R2 D2 R2 D2 R2 (6)
(symmetry level 12, SI level 2)

p21  2 H                 L2 R2 B2 L2 R2 F2 (6)
(symmetry level 6, SI level 6)
```

Amazingly, the process p3 (found using Dik Winter's program) is actually
a series of 20 processes which all result in the same displacement!

```p3   12 flip             R1 L1 D2 B3 L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3
R2 U3 F2 D3 (20)
(symmetry level 1, SI level 20)

Since p3 is shift invariant, we can easily shift the 3 consecutive half
turns to the beginning without fear of altering the end result:
L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3 + R2 U3 F2 D3 R1 L1 D2 B3
```