[r-t] Spliced Doubles puzzle

[Philip Saddleton]

Very good. Concentrating on the more difficult frontworks in two extents 
seems a good idea. The arrangement does have the slightly inelegant 
feature that each each extent contains repeated leads. Can you find an 
arrangement where each contains 12 methods? No.3 is easily fixed with 
lead splices, but for the first two you will need to interchange leads 
of C and D between them.

Philip

Matthew Frye wrote:
> Interesting problem, it took a bit of thinking about but I think that 
> i may've solved this one. Just a slight warning, i've been working 
> with pencil + paper and haven't run this through any proving software, 
> so there's a chance that everything im about to say is nonsense, but i 
> think that im right.
>  
> Firstly, i picked a backwork for each bell and assigned each frontwork 
> a letter:
> 2 = 3.1
> 3 = 3.145
> 4 = 345.1
> 5 = 345.145
>  
> A = 5.1.145
> B = 5.123.145
> C = 125.1.345
> D = 125.3.145
>  
> The methods are referenced using this number and letter (followed by 
> the pn of the lead end), so "3C 123" would be 3.145.125.1.345 with 123 
> at the lead end which you called 129V. Sorry if this system isn't very 
> clear (especially with the different lead ends) but it's what i've used.
>  
> The three extents each focus on a different frontwork, with the 
> methods with the remaining frontwork scattered throughout the 3 extents.
>  
> The first extent has frontwork C throughout, except when the 2 is 3 
> pb, when it has frontwork B. With careful selection of lead ends, this 
> wraps up all the "C" methods.
>  
> 12345  3C
> _15342_  123
> 15324  3B
> _14325_  123
> 14352  3C
> _12354_  1
> 13245  2C
> _14235_  1
> 12453  4C
> _13452_  123
> 13425  4B
> _15423_  1
> 14532  5B
> _13542_  123
> 13524  5C
> _12534_  123
> 12543  5C
> _14523_  1
> 15432  4C
> _12435_  1
> 14253  2C
> _15243_  123
> 15234  2C
> _13254_  1
> 12345
>  
> There will almost certainly be other arrangements, but i couldn't find 
> anything resembling a regular 2- or 3-part extent that provides the 
> required methods.
>  
> The second extent is basically the same as the first one, with a few 
> swaps: frontwork D is used instead of C, backworks 2 and 5 swap, 
> backworks 3 and 4 swap. This gives 2 extents including all the C and D 
> frontworks and 6/8 of the B frontwork.
>  
> The last extent needs a bit more tweaking than just swaping in 
> frontwork A, as this leaves us 1 method short, this can be solved by 
> ringing backwork 4 when the 5 is pivot and backwork 5 when 4 is pivot, 
> this mixes up the extent from what's shown above, but provides the 
> required lead ends to fit all the methods using lead splices between A 
> and B.
>  
> 12345  3A
> _15342_  123
> 15324  3A
> _14325_  123
> 14352  3A
> _12354_  1
> 13245  2B
> _14235_  1
> 12453  5B
> _15423_  1
> 14532  4A
> _12534_  123
> 12543  4A
> _13542_  123
> 13524  4A
> _14523_  1
> 15432  5A
> _13452_  123
> 13425  5A
> _12435_  1
> 14253  2A
> _15243_  123
> 15234  2A
> _13254_  1
> 12345
>  
>  
>
>
> > Date: Tue, 4 Mar 2008 17:55:16 +0000
> > From: pabs at cantab.net
> > To: ringing-theory at bellringers.net
> > Subject: [r-t] Spliced Doubles puzzle
> >
> > Consider the symmetrical single-hunt doubles methods with
> >
> > - 3pb making 3rd's to start with and pivoting at the half-lead
> > - 4th's at the half-lead
> > - PN 1 or 123 at the lead-head
> >
> > (e.g. the reverses of St Simons/St Martins etc.)
> >
> > There are
> > 4 backworks (3.1, 3.145, 345.1, 345.145)
> > 4 frontworks (5.1.145, 5.123.145, 125.1.345, 125.3.145)
> > 2 lead-heads (1, 123)
> >
> > Any combination gives a valid method with a 3 or 4 lead course, so 
> 4x4x2
> > = 32 methods.
> >
> > It is easy enough to splice 12 methods into an extent: pick a different
> > backwork for each 3pb, and a different frontwork for each bell making
> > 4th's (either 4pb or 5pb, depending on the backwork), and choose the
> > lead-heads to join the bits together. 24 methods in two extents is also
> > straightforward. But can you fit all 32 into three extents? It seems
> > that there ought to be enough freedom to manage it, particularly as the
> > first two frontworks are lead splices.
> >
> > If anyone wants to take up the challenge, I attach the methods and some
> > sample extents in siril.
> >
> > regards,
> > Philip
> >