# [r-t] ringing the Mathieu group

Tue May 24 21:35:26 UTC 2011

```Here's my approach:

We'll try to join together half leads of a plain method using a system
of hunts. For a right place method, there is onl one place within the
half lead where there is a choice of PNs, i.e. when the treble is in
6-7, the two methods being the reverse of each other. Arbitrarily fix
can choose either PN without upsetting the treble. For now, we'll use 1T
for each, and say that we are going to have a call every hl and lh. The
method we finish up with has 0 and E as secondary hunts and a 9 lead
plain course. We can get an 11 lead course by using e.g. PN 25 when the
treble is in 3-4, but that gives four consecutive blows.

Now all we need is to find the Q-sets that will let us join the 440
courses together.

First I try to keep the tenor unaffected as much as possible. We can
then fix most of the Q-sets with a skeleton course

1234567890ET
bb 13T54986E702
ax 186E5T943702
ba 12E65349T708
ab 149T6E320578
xb 1380T92E7654
aa 1E02789T3654
bx 104723T9865E
ab 1T987430526E
bx 19E78503426T

where a=25, b=8E and x=1T

We now try to extend this to a block where 1,E and T occupy each
combination of positions. We must have b at the lh where they are in 0E,
or we will repeat (and they cross in 67 at the hl, so are unaffected).
Using this Q-set and allowing E to be unaffected wherever possible we
join all but two of the leads (underlines denote the course heads or the
extra Q-set)

1234567890ET
------------
bb 13T54986E702
ax 186E5T943702
ba 12E65349T708
ab 149T6E320578
xb 1380T92E7654
aa 1E02789T3654
bx 104723T9865E
ab 1T987430526E
bx 19E78503426T
------------
bb 1ET874356029
ax 13568T47E029
ba 19658E74T023
ab 174T56E92803
xb 1E32T4960587
ab 1960234E8T57
------------
ba 170628E43T59
aa 146E30827T59
xa 19E876230T54
aa 13820E679T54
ba 14280976ET53
aa 1687E2904T53
ab 190478265ET3
------------
ab 12654089T7E3
------------
ab 189T5602E473
xb 103ET9267548
aa 16E2739T0548
bx 1E8720T93546
ab 1T93780E4256
bx 196734E0825T
------------
bb 16T378045E29
ax 10453T876E29
ba 19543678TE20
ab 178T456923E0
xb 1602T895E437
aa 1529E08T6437
bx 127E96T80435
ab 1T80E7623945
bx 185E0326794T
------------
bb 15T0E7634298
ax 16340T7E5298
ba 184305E7T296
ab 1E7T34589026
xb 1569T784230E
aa 1498267T530E
bx 19E285T76304
ab 1T762E590834
bx 17426095E83T
------------
bb 14T62E503987
ax 15036TE24987
bb 1076342T8E95
------------
bx 175368T24E90
ab 1T24358796E0
bx 1203497856ET
------------

The missing two leads can be joined by using the Q-set where E is behind
and T is in 890E, to give a block of 2640 rows wher 1ET occupy all
possible positions once each at hand and back. Using x wherever the hl
or lh is not restricted by the position of E or T gives a nine-part
palindromic block, where we now have 10ET in each position twice in a
23760 (a quarter peal). There are ten places in each part where we now
have an x; for each of these, the PN to use is determined by the
position of 0 in all but one part. If we choose a or b consistently for
each of the remaining ones, or for each symmetrically disposed pair, we
finish up with a round block.

I am sure there must be Q-sets with one element in each quarter (but not
the x's) - I leave this for someone else to find.

regards
Philip

Here is the gsiril code where each x is replaced by a:

12 bells
h=+-8E-8E-25-25-25-
l=+-25-25-25-8E-8E-
ha=h,+25
hb=h,+8E
la=l,+25
lb=l,+8E
aa=ha,la,"aa @"
ab=ha,lb,"ab @"
ax=ha,la,"ax @"
ba=hb,la,"ba @"
bb=hb,lb,"bb @"
bx=hb,la,"bx @"
xa=ha,la,"xa @"
xb=ha,lb,"xb @"

u="   ------------"
uu="   ============"

p1=bb,ab,ba,ab,bb,aa,ba,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,ba,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,aa,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,bb,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,ab,bb,u,ba,ab,ba,
uu

p2=bb,ab,ba,ab,bb,aa,ba,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,aa,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,ba,u,
bb,aa,ba,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,ba,ab,ba,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,ba,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,ba,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,ax,bb,u,ba,ab,ba,
uu

p3=bb,ab,ba,ab,bb,aa,ba,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,aa,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,xb,aa,ba,ab,ba,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,ba,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,ba,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,aa,bb,u,ba,ab,ba,
uu

p4=bb,ab,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,ba,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,ba,u,
bb,aa,ba,ab,xb,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,aa,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,aa,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,bx,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,aa,bb,u,ba,ab,ba,
uu

p5=bb,ab,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,ba,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,ba,u,
bb,aa,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,aa,ba,ab,ba,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,aa,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,aa,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,bb,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,ab,bb,u,ba,ab,ba,
uu

p6=bb,ab,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,aa,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,ba,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,aa,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,bb,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,aa,bb,u,ba,ab,ba,
uu

p7=bb,ab,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,aa,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,ba,u,
bb,aa,ba,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,ba,ab,bx,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,xa,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,aa,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,ba,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,ab,bb,u,ba,ab,ba,
uu

p8=bb,ab,ba,ab,bb,aa,bx,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,ba,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,ba,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,ax,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,bb,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,aa,bb,u,ba,ab,ba,
uu

p9=bb,ab,ba,ab,bb,aa,ba,ab,bb,u,
bb,aa,ba,ab,ab,ab,u,
ba,aa,xa,aa,ba,aa,ab,u,
ab,u,ab,ab,aa,bb,ab,bx,u,
bb,aa,ba,ab,bb,aa,bb,ab,bb,u,
bb,aa,ba,ab,bb,ab,u,
ba,bb,u,ba,bb,u,
ab,bb,aa,ba,ab,bb,u,
bb,ab,ba,ab,ab,aa,ba,ab,ba,u,
bb,ab,ba,ab,ab,ab,u,ab,u,
aa,aa,ba,aa,aa,aa,bb,u,
ab,ab,aa,ba,ab,bb,u,
bb,ab,ba,ab,bb,aa,bb,ab,ba,u,
bb,aa,bb,u,ba,ab,ba,u,bb,ab,bb,u,
bb,ab,ba,u,bb,ab,bb,u,ba,ab,ba,
uu

prove p1,p2,p3,p4,p5,p6,p7,p8,p9

Alexander Holroyd said  on 26/04/2011 20:03:
> 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
>

```