[r-t] Easter challenge

[Wyld Family e-mail]

Sorry not to have contributed earlier which might have saved some 
duplication of effort.  I drew a Cayley colour graph for hunt dominated 
doubles about 25 years ago.  This was published in the RW August 1985 along 
with graphs for doubles principles and double major principles.

----- Original Message ----- 
From: "Philip Saddleton" <pabs at cantab.net>
To: <ringing-theory at bellringers.net>
Sent: Sunday, April 12, 2009 9:42 PM
Subject: Re: [r-t] Easter challenge


> This is a problem that is small enough to analyse in depth with pencil and 
> paper (as I did in my hotel room last night). I came up with the following 
> graph which allows me to find all symmetrical single-hunt methods whose 
> plain course is the extent.
>
> 1A - 1B - 1C
>  | x  | x  |
> 2A - 2B - 2C
>  | x  |    |
> 3A - 3B   3C
>  | x  |    |
> 4A - 4B - 4C
>  | x  | x  |
> 5A - 5B - 5C
>
> Each [number,letter] pair represents the rows occupying the same position 
> in a half-lead, with the lines connecting them possible transitions. The 
> numbers are the position of the treble and the letters the order of the 
> other bells. If our method has Plain Bob lead heads these are represented 
> by 1A. To find a method we need a path in the graph that visits each 
> vertex precisely once (some possible paths do not give a method - the 
> half-leads cannot be joined into a touch, and most will contain more than 
> four blows in one place).
>
> To answer the question, the given treble path is not possible - any path 
> from 1? to 5? must contain the sequence 2C, 3C, 4C or its reverse, and 
> there is no link to another 3 at either end. There are methods where the 
> treble path is 112321234543455, e.g.
>
> 123.3.145.1.5.345.1.5.123.1.125.5.3.345.5 lh 125
>
> -- 
> Regards
> Philip
> http://myweb.tiscali.co.uk/saddleton/
>
>
> Philip Earis said  on 11/04/2009 19:21:
>> I like the concept of the plain course of a method generating the extent.
>>
>> On 5 bells, with a fixed treble and four working bells,  this gives 30
>> changes per lead to play with.
>>
>> There are some rung examples, shown on
>> <http://www.methods.org.uk/online/tpl5.htm>, eg Daedalus Doubles.
>>
>> One treble-work I like the look of is where the treble dodges in 1-2, 
>> 2-3,
>> 3-4, 4-5 and does four blows at front and back.  Eg treble path:
>>
>> 1121232343454555 (treble rings six blows in each position in the lead)
>>
>> Is a plain course extent on such a plan possible?  I don't mind about 
>> more
>> than four consecutive blows.
>>
>> The possible changes are:
>>
>> 1|125|145
>> 3|345|5
>> 3|345|5
>> 3|345|5
>> 1|145
>> 1|145
>> 1|145
>> 125|5
>> 125|5
>> 125|5
>> 1|3|123
>> 1|3|123
>> 1|3|123
>> 345|5
>> 345|5
>>
>> It's easy to put together a simple false examples on this plan, but a 
>> good
>> true one would be nice.
>>
>
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