[r-t] ringing the Mathieu group

[Wyld Family e-mail]

I think the following strategies could work.  If they do not it may take 
some tedious arithmetic to find out.

I have two suggestions for the caters problem.  Both are based on organising 
the required changes in to courses of original and linking them together 
using 5ths place calls (bobs).  In the first I suggest joining, alternately, 
forward and backward hunting courses with a 5ths place i.e. 
9.1.9.1.9.1.9.1.9.1.9.1.9.1.9.1.9.5.1.9.1.9.1.9.1.9.1.9.1.9.1.9.1.9.1.5 
repeated 8 times.  This joins together 18 courses leaving an even number to 
be introduced by Q sets of 5ths place bobs.  There are 18 Q sets that can 
not be bobbed having been used in the basic block but that leaves at least 8 
routes into the courses affected.

The second strategy is simply to use the 5th place bobs instead of either 9 
or 1 with forward hunting throughout.  The usual constraints on Q sets of 
bobs apply i.e. that the number of blocks into which the changes are 
organised is changed by an even number.  Since the extent requires 10080 
courses it might at first glance seem that you would inevitably end up with 
the extent in two parts.  There are however 20160 courses to choose from 
since each course can be rung in both directions.  There is no obvious 
reason why adhering to the Q set rule you should not be able to reduce this 
larger collection to two parts each containing the required changes once. 
The two parts would be mirror images of each other and some of the courses 
would be rung partly in one direction and partly in the other.  Since this 
is how 720s of treble bob minor and 5040s of plain bob triples work it seems 
worth trying here.

The Mathieu group may be easier to handle.  Again I suggest using x and one 
of the other place notations alternately to organise the changes into 
courses.  Use the other place notation as a call leaving the xs unaffected. 
If I have worked it out correctly a Q set of such calls links in 7 courses 
i.e. an odd number so it may be possible simply to join together the 
required courses.  If not the option to use parts of some courses rung 
backwards leading, as with the caters, to two mirror image blocks may 
overcome the problem.

More interesting, at least from a ringing point of view, might be to 
constrain the treble to plain hunting and change the place notation, when it 
leads, to join courses together.

Colin


----- Original Message ----- 
From: "Alexander Holroyd" <holroyd at math.ubc.ca>
To: <ringing-theory at bellringers.net>
Sent: Wednesday, April 27, 2011 1:19 AM
Subject: Re: [r-t] ringing the Mathieu group


> In a similar vein, does anyone know how to get an in-course extent of 
> caters using only the place notations 1, 5, and 9?
>
> On Tue, 26 Apr 2011, Alexander Holroyd wrote:
>
>> Here is today's brain teaser.
>>
>> Consider the three 12-bell place notations
>> x 125T 18ET
>>
>> The group generated by these pns (i.e. the set of all rows you can get to 
>> from rounds using only these pns) contains 95040 rows.  It is a very 
>> interesting group from a mathematical perspective, called the Mathieu 
>> Group M_12.  (It is the second smallest of the 26 "sporadic groups"). 
>> One interesting property is that it is "sharply 5-transitive", which 
>> means that any given 5 bells (e.g. 12345) ring exactly once of each of 
>> the possible places that 5 bells can occupy (counting different orders of 
>> 12345 as different), giving 12x11x10x9x8 = 95040 rows.
>>
>> According to the "Lovasz conjecture", it should be possible to ring a 
>> true round block of these 95040 rows using only these three pns.  Can 
>> anyone come up with an elegant way of doing this?  It would obviously be 
>> nice to do it right-place, ie without 3 consecutive blows.  I don't know 
>> whether that's possible.
>>
>> Ander
>>
>
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