[r-t] Grandsire Minor

[Robert Bennett]

	I believe that Rankine's theorem (from my reading of his 1948 paper)
applies only to odd numbers of bells. 
Grandsire Minor bob courses are reversable, which complicates it
further compared to Triples. 

	In Grandsire triples the Q-sets have 5 members and the courses get
added on 5 at a time (or 3 at a time is also possible). 

	Obviously 5+4a+2b cannot add to 72 plain courses or 120 bob
courses. 

	Grandsire minor the Q-sets have 4 members and the courses added on
3 at a time (and 1 at a time should also be possible?). If so, then 12
bob courses should be possible 

	4+3a+b could add up to 12.   (a=2 and b=2). 

	Since no bobs-only 720 exists, then perhaps there is a flaw in this
argument.  

	As far as Grandsire Triples goes, I think that Shipway's 5-part comes
close to Richard Pullin's desired composition. That composition uses a
composite Q set of (Bob,Hic) x 5. Starting with a 10-course block, it
is easy in principle to join the other 62 courses on using bobbed Q
sets.  

	  
----- Original Message -----
 From:ringing-theory at bellringers.net
To:
Cc:
Sent:Sun, 17 Feb 2013 14:14:00 -0800 (PST)
Subject:Re: [r-t] Grandsire Minor

On Sun, 17 Feb 2013, Richard Pullin wrote:

> On a similar wavelength, I'd be interested to know if anybody has
ever
> come up with a 5040 of Grandsire Triples made up of five unjoinable
> bobs-only blocks. Or perhaps as the Q-sets are wholly In-Course in
> Triples, the anti bobs-only law doesn't allow events to get this
> 'near' to a bobs-only extent, unlike in Minor where the feature of
the
> Q-set altering the nature of the rows can cause the added obstacle
of
> needing to arrive in the blocks 'in the right direction'.

That isn't possible. Any partition of the extent into mutually true 
in-course blocks of Grandsire Triples has an even number of blocks.
This 
was proved by W H Thompson in 1880 or so. Professional mathematicians 
caught up with ringers in 1948 when Rankin published a formal proof of

this and its natural generalizations. This proof was simplified by
Swan 
in 1999.

I guess this is what you are referring to as "the anti bobs-only law"!

RANKIN, RA. "A CAMPANOLOGICAL PROBLEM IN GROUP THEORY." Proceedings of
the 
Cambridge Philosophical Society. Vol. 44. 1948.

Swan, Richard G. "A simple proof of Rankin's campanological theorem."
The 
American mathematical monthly 106.2 (1999): 159-161.

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