[r-t] ringing the Mathieu group
Alexander Holroyd
holroyd at math.ubc.ca
Tue Apr 26 19:03:40 UTC 2011
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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