[r-t] Double extent of major

[Philip Saddleton]

Simon Humphrey said  on 05/12/2007 09:22:
>> Come to think of it, is it possible to construct a true 40.320 of
>> plain major using just bobs? 
>>     
> I'd be interested in seeing a proof of the answer to this.  Ander?
> I remember arguing with Denis Carlisle about it, some 30-odd years ago.  I
> had produced a 7-part 40320 of Plain Bob Major and asked him if he'd prove
> it; but he said he didn't need to because it was bound to be false.
> My feeling now is that Denis was right.  But I wonder . . . now, where did I
> put that composition . . .
> SH
>
>   
The proof is precisely analogous to that for Grandsire Triples. Here's 
my version:

We require to produce a cycle of all of the in-course lead heads where 
each one maps to the one obtained by a plain or bobbed lead, according 
to the compostion. Think of this as a permutation of the 2520 lead 
heads. It is an even cycle, hence an odd permutation, A.

Construct another permutation that maps each lead head to the previous 
one in the same course. This consists of odd (7-) cycles, so is an even 
permutation, B.

Now consider B followed by A. This maps each plain lead head to itself, 
and each bobbed lead head to the one that would be obtained by calling a 
bob a course later.

Since it is a 1-1 mapping, each bobbed lead-head must map to another 
bobbed lead-head (the Q-set rule). The combined mapping consists of odd 
(3-) cycles, so is even. But B is even and A odd, so BA is odd.

Hence A does not exist.

-- 
Regards
Philip
http://myweb.tiscali.co.uk/saddleton/