Date: Fri, 28 Oct 94 22:13:00 -0700 (PST)
From: Martin Schoenert <Martin.Schoenert@math.rwth-aachen.de >
~~~ Subject: Re: Shift Invariant Part 2
```Mark Longridge writes in his e-mail message of 1994/01/27
```
```...or (U1 R1)^35 ? And indeed, (U1 R1)^(35 * 40) is shift invariant.
```
```Mark  kindly points out, that my process (UR)^140 for the ``odd'' element
is a strange choice, given that (UR)^140 = (UR)^35.
```

I can't recall how I arrived at this process. Somehow I simply missed
that (UR)^140 = (UR)^35, which is especially strange since I know that
(UR) has order 105 since 1982.

Mark continues

```Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant
UR11 = U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 U2 R1 U1 R1 U1 R3 U1 R3 U1 R3
(22 q  or  20 h  moves)
```

Is UR11 the shortest process effecting the ``odd'' element in <U,R>?

Mark continues

Is this odd due to ( U1 R1 )^35?
Actually everything about the above description appears even.
It is an even number of quarter turns...

The ``odd'' element o has odd order as element of the cube group,
i.e., o^3 = id. All other shift invariant elements e have even order,
i.e., either e^2 = id or e^4 = id (for some ``abelian'' elements).

Mark continues

I actually did use GAP on the < U, R > group but I couldn't resolve
the resulting position (can GAP use letters? I should have used
letters).

I assume you wonder whether GAP can find a process for a given element.
In fact GAP can do this (you define a homomorphism from the free group
on U,D,L,R,F,B to the cube group and then ask for a preimage of the
element). But the process is usually extremly long, e.g., for
the ``central'' element GAP finds a process that has length > 2*10^6
(don't try this at home ;-).

There is an improved algorithm by Philip Osterlund, which is a lot
better, but still not good enough to help in the quest for god's
algorithm. For example it finds a process for the ``central''
element of length 228.

Mark continues

Martin, you will be pleased to hear that I like GAP,
but I need a bigger hard drive for that beast!

Look at it this way:
The system costs you \$200, and you even get a hard drive for free!

Seriously, you don't need the full distribution (32 MByte),
but only the executable and the library (5 MByte).
However, you should have enough real memory;
8 Mbyte is the minimum, 16 MByte is better,
and the 64 MByte that I have in my workstation don't hurt.

Have a nice day.

Martin.

```-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,   +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany
```