[r-t] Composition challenge (without html)

[John M Goldthorpe]

Hi Roddy,

I think there are some flaws in your reasoning.  If you start with a lead of say Glasgow, there are then 240 leads out of 20160 that you cannot then ring.  (London=234, Belfast=232, Bristol=216).  However, with each subsequent true lead you add to the composition, there is an increasing chance that the false leads will overlap.  It seems from your calculations that the average comes down to around 100 for a peal length.  I did try a search a while ago which tried to maximize the overlaps in order to produce better results, but I don't think it helped much.

I just started a search for 7 parts that meet your criteria.  The longest so far is 9408, so I think it quite likely that 10000 is possible.

John


-----"ringing-theory" <ringing-theory-bounces at bellringers.net> wrote: -----
To: <ringing-theory at bellringers.net>
From: Roddy Horton 
Sent by: "ringing-theory" 
Date: 29-04-2013 11:10AM
Subject: [r-t] Composition challenge

Sometime last year Martin Whiteley asked me if I could produce a composition of 10000 changes of Bristol, Belfast, London and 
Glasgow with similar properties to the 5024 I produced some years ago.

I am struggling to get past 6500 changes and have done some thinking on why this might be.

There are, with the treble fixed, 5040 lead heads and for 4 methods there are then 20160 possibilities.

Taking my own 2 one-part compositions and those of John Goldthorpe(JMG) and Glenn Taylor(GAAT) plus 2 of my own five-part 
compositions plus one of John Goldthorpe's seven-part compositions I have computed the number of false lead heads used to be as 
follows:-

5024 - 16438
5122 - 16632
GAAT - 16776
JMG  - 15810
RRH5 - 16120
RRH5 - 16095
JMG7 - 17059

This is approximately ratio of 100 false leads used for each 1 lead in the peal. A peal being approx 160 leads and all of these 
compositions using about 16000 false leads.

If this is anything like accurate then the theoretical maximum number of leads to be added to an existing peal will be
(20160-16000)/100 = 42 which gives another (42*32) = 1344 changes ie about 6500 changes maximum.

Can anybody see any flaw in my thinking or, even better, can anybody produce 10000 or more changes in these 4 methods, all the 
work with at least 2000 changes of each method?

Roddy


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