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Date: Thu, 21 Aug 1997 16:07:30 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: isoglyphs
In-Reply-To: <199708210406.AAA21524@life.ai.mit.edu>
To: michael reid
Cc: cube-lovers@ai.mit.edu
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On Thu, 21 Aug 1997, michael reid wrote:
> here are several other applications that your pattern finder should
> be able to handle easily:
>
> 2. confirm my results on the "czech check" problem, i.e. classify all
> patterns that have exactly eight squares correct on each face.
If I understand your question correctly, and if I am using Herbert's
program correctly, there are 54 such isoglyphs unique up to M-conjugacy.
3 of them involve only corners as the incorrect facelet, and 51 of them
involve only edges as the incorrect facelet.
(I am assuming that by definition the center facelet is always correct,
thus eliminating the 6-spot from consideration. If you count the
6-spot, then there are of course 2 such isoglyphs unique up to
M-conjugacy.)
If you tell Herbert's program to consider only continuous isoglyphs with
exactly eight squares correct on each face, there are 3 such isoglyphs
unique up to M-conjugacy. 1 of them involves only corners as the
incorrect facelet, and 2 of them involve only edges as the incorrect
facelet. (I suppose you would say that the other 51 are discontinuous.)
Herbert's program only lists positions which are unique up to
M-conjugacy. Here's a modest suggestion. It might be nice for the
program to list the size of the M-conjugacy class for each such
position. That way, you could count both "positions" and "patterns",
where I am using "position" to mean any element of G and "pattern" to
mean an M-conjugacy class (or a representative thereof).
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990