[r-t] ringing the Mathieu group

Alexander Holroyd holroyd at math.ubc.ca
Sun May 22 20:43:09 UTC 2011

Hi Colin,

Thanks for these interesting suggestions!  This was just the sort of thing 
I was hoping to provoke...

Regarding, e.g. suggestion 1 for the caters extent: what makes you so 
confident this can be done?  (I think I share this confidence, but I'm not 
sure how to justify it).  Certainly there are "plenty of Q-sets" (only a 
few are missing, arising from the "magic block"), and there is no 
(obvious) parity barrier.  But why does this imply that it is possible to 
join everything to one block?  A general argument (rigorous or heuristic) 
would be much more interesting than just checking it by giving an example 
in this particular case.  Ditto for the second caters suggestion...

I think the call you mention for the Mathieu group (125T in place of 18ET) 
actually gives Q-sets linking 4 courses, not 8 (alternating these two pns 
amounts to two lots of plain hunt on 4).  In any case, no parity barrier, 
as you say.  Again, the question is: how can we tell whether it is 
possible, other than a very tedious verification?

In both cases, another natural approach is the usual subgroup-based 
"composition of peals in parts" method.  Given the large size of the 
problem, it mighjt be helpful to do this in several "levels", starting 
with a subgroup of a subgroup.  This might even yield methods that someone 
would consider ringing(!)  Any ideas for a specific plan in this 
direction, anyone?


On Sat, 30 Apr 2011, Wyld Family e-mail wrote:

> 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. 
> 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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