[r-t] asymmetric treble-dodging minor
Alexander E Holroyd
holroyd at math.ubc.ca
Fri Jul 24 19:18:54 BST 2020
We rang a quite interesting asymmetric delight minor method this week:
https://bb.ringingworld.co.uk/view.php?id=1379403
Lobster Delight Minor: 34-34.1-2-3.2-2.3-2-3-2-4-34-1 m
There is a very simple 720: (sI sO I)*3, with single = 1456.
It is "properly asymmetrical", with (depending exactly how you measure)
3 separate differences between the two halves. I believe it is the
first asymmetric treble-dodging minor method with no single changes in
the plain course. In fact I think all others rung are straightforward
lead-splicers with a symmetric method. (I did not check this fully, and
I did not think through whether that necessitates a single change).
It's rather miraculous that it works. The process that led to it was
quite interesting.
Given a set of possible leads of a method (e.g. all possible in course
treble-fixed leads), the problem of finding the maximum possible set of
mutually leads is equivalent amounts to finding a maximum independent
set in the graph with a vertex for each possible lead, and an edge
between any two mutually false leads. In general that's a hard problem
to solve, although simulated annealing seems to work quite well for
smallish ringing related versions.
However, for the particular problem of in-course treble-fixed leads of a
TD minor method, the problem of existence of an extent is more specific
and simpler. We want to know whether there is a set of 30 mutually true
leads out of the 60 possible. Since 30 is half 60, it turns out that
this is equivalent to the graph being bipartite, which can be tested
very quickly.
So what I did was checked all possible methods (subject to certain
criteria) to find those that have 30 mutually true leads. There are not
so many, and quite a few have rather trivial asymmetry. Lobster (along
with a few others) is nice in that it has a regular 5 lead course, and
the true leads can be arranged as 6 mutually true courses.
There is still the question of joining the courses together. The usual
calling WHW or IOI makes essential use of reversing part of a course (to
break the q-set parity barrier), which will not work with an asymmetric
method (unless you want to splice with its reversal). Using singles in
general won't help, because the true leads are in course ones.
However, for lobster there is a miracle. Swapping 5 and 6 place bells
for a lead gives a lead splice with the corresponding reversed lead. So
the above calling magically works. There don't seem to be many other
examples that work like this.
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