[r-t] Spliced Doubles puzzle

Wed Mar 5 00:43:39 UTC 2008

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> > 5 bells> > f85=" 85F",&3.1.5.1.145,+1,"@ \"> u93=" 93U",&3.1.5.1.145,+123,"@ \"> f87=" 87F",&3.1.5.123.145,+1,"@ \"> u95=" 95U",&3.1.5.123.145,+123,"@ \"> f89=" 89F",&3.1.125.1.345,+1,"@ \"> u98=" 98U",&3.1.125.1.345,+123,"@ \"> f90=" 90F",&3.1.125.3.145,+1,"@ \"> u99=" 99U",&3.1.125.3.145,+123,"@ \"> c116="116C",&3.145.5.1.145,+1,"@ \"> v124="124V",&3.145.5.1.145,+123,"@ \"> c118="118C",&3.145.5.123.145,+1,"@ \"> v126="126V",&3.145.5.123.145,+123,"@ \"> c120="120C",&3.145.125.1.345,+1,"@ \"> v129="129V",&3.145.125.1.345,+123,"@ \"> c121="121C",&3.145.125.3.145,+1,"@ \"> v130="130V",&3.145.125.3.145,+123,"@ \"> c147="147C",&345.1.5.1.145,+1,"@ \"> v155="155V",&345.1.5.1.145,+123,"@ \"> c149="149C",&345.1.5.123.145,+1,"@ \"> v157="157V",&345.1.5.123.145,+123,"@ \"> c151="151C",&345.1.125.1.345,+1,"@ \"> v160="160V",&345.1.125.1.345,+123,"@ \"> c152="152C",&345.1.125.3.145,+1,"@ \"> v161="161V",&345.1.125.3.145,+123,"@ \"> f163="163F",&345.145.5.1.145,+1,"@ \"> u171="171U",&345.145.5.1.145,+123,"@ \"> f165="165F",&345.145.5.123.145,+1,"@ \"> u173="173U",&345.145.5.123.145,+123,"@ \"> f167="167F",&345.145.125.1.345,+1,"@ \"> u176="176U",&345.145.125.1.345,+123,"@ \"> f168="168F",&345.145.125.3.145,+1,"@ \"> u177="177U",&345.145.125.3.145,+123,"@ \"> > peal="@ \",u98,f90,v161,v155,c149,f87,v126,c116,u171,u177,f167,c120,""> prove peal> > peal="@ \",u99,f89,v160,v157,c147,f85,v124,c118,u173,u176,f168,c121,""> prove peal> > peal="@ \",u95,f85,c147,u171,u173,f167,v160,c152,f90,v130,v129,c118,""> prove peal> > 12345 98U> 14352 90F> 13524 161V> 14532 155V> 12543 149C> 15324 87F> 13245 126V> 15234 116C> 12453 171U> 15432 177U> 13425 167F> 14253 120C> 12345> 120 rows ending in 12345> Touch is true> 12345 99U> 14352 89F> 13524 160V> 14532 157V> 12543 147C> 15324 85F> 13245 124V> 15234 118C> 12453 173U> 15432 176U> 13425 168F> 14253 121C> 12345> 120 rows ending in 12345> Touch is true> 12345 95U> 14352 85F> 13524 147C> 15432 171U> 13425 173U> 12453 167F> 14532 160V> 12543 152C> 15324 90F> 13245 130V> 15234 129V> 14253 118C> 12345> 120 rows ending in 12345> Touch is true> > > > > _______________________________________________> ringing-theory mailing list> ringing-theory at bellringers.net> http://bellringers.net/mailman/listinfo/ringing-theory_bellringers.net
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