Date: Sun, 19 Feb 95 23:20:22 -0500 (EST)
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ Subject: Qturn Lengths of M-Symmetric Positions

1. The length of Start is of course 0 qturns.

2. The length of Pons Asinorum is of course 12 qturns. This result
has been known since 1980. However, in the process of testing
out half-depth searches, I tested Pons Asinorum and encountered
a minor surprise. There are five "halfway" positions which
are unique up to M-conjugacy. I only expected three. They
are:

```a. (RRLL)(FF)          expected -- continue BB etc.
b. (RRLL)(FB)          expected -- continue FB etc.
c. (RRLL)(FB')         expected -- continue FB' etc.
d. (FB')(RRLL)         a surprise to me
e. (RL')(FB')(RL')     a surprise to me
```

3. The length of Pons Asinorum composed with Superflip is
20 qturns. Half-depth searches through level 9 found nothing.
A half-depth search at level 10 found ten "halfway" positions which
are unique up to M-conjugacy. I have my usual trouble of spinning
tapes containing representative elements in order to find the
processes, but I should have them in a couple of days or so.
I expect we will find that many (or all) of them are really
closely related, differing only by commuting in fairly trivial
ways, just as do the five half-way positions for Pons Asinorum.

4. The length of Superflip is 24 qturns. Half-depth searches
through level 11 found nothing, so the length is greater than
22. Mike Reid has found a Superflip process of length 24.
Hence, the length is 24. It would be more satisfying if I
could perform a half-depth search to level 12, but the
problem is just too big. Hence, I have no idea how
many "halfway" positions there are which are unique up to
M-conjugacy.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax
837 Chestnut Ridge Road                              BRYAN@WVNVM
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```