# [r-t] Grandsire Minor

Alexander Holroyd holroyd at math.ubc.ca
Mon Feb 18 19:43:28 UTC 2013

```On Mon, 18 Feb 2013, Robert Bennett wrote:
>
> 	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.

No, it's nothing to do with number of bells.  Rankin's theorem applies to
abstract groups without reference to the order of the symmetric group that
they are subgroups of.  It states that in the directed Cayley graph
generated by a and b, if a^-1 b has odd order then in any partition of the
vertices into disjoint oriented cycles, the number of cycles has the same
parity as the index of the cyclic subgroup generated by a.

As I said, this does indeed apply to grandsire triples, showing that no
partition into an odd number of bobs-only blocks is possible.  This is
because (plain)^-1 bob has order 5, i.e. Q-sets have 5 elements.  In
grandsire minor, Q-sets have 4 elements, which is why it does not apply
here.

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

The only flaw here is a misunderstanding on the statement of Rankin's
theorem.  It implies that under certain conditions an extent is NOT
possible, but it says nothing about what happens if those onditions fail.

> 	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.

Not really - the question as I understood it was about ordinary bobs.
With other calls it is of course possible to get any number of round
blocks.

>
> 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.

```