[r-t] Easter challenge

[Philip Saddleton]

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.
>