From:

~~~ Subject:

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