[r-t] Out of course treble-dodging minor

Ian Fielding Ian.Fielding at nbt.nhs.uk
Mon Dec 7 16:23:00 UTC 2015

Hi, following an e-mail discussion with Peter Ellis, his suggestion was:

in answer to the question about what to look at next, my suggestion would be to see if there are any splices of methods within each of the four groups that we don't know about yet. For instance, are there any extents of Surprise Minor which have not so far been discovered.

Don’t know if this helps the search of the haystack or not………

Ian Fielding

Chief Pharmacy Technician
North Bristol NHS Trust
0117 414 2277 (Telephone)
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From: ringing-theory [mailto:ringing-theory-bounces at bellringers.net] On Behalf Of John Danaher
Sent: 06 December 2015 23:15
To: ringing-theory at bellringers.net
Subject: [r-t] Out of course treble-dodging minor

Inspired by Ander's post last fall, I've been working (on and off) on an exhaustive search for extents of minor. If my code is correct, I've been able to enumerate every composition of the 147 regular treble-dodging methods, spliced at the lead end, using bobs or any/all of the four single changes as calls.

My approach starts similarly to Ander's: a backtracking search on all of the possible leads, using Knuth's "dancing links" data structure to solve the complete-cover problem.

Where my approach differs is that rather than producing plans that need to be joined at a later stage, I've included the calls in the search itself. So rather than choosing 30 leads to cover 720 rows, I choose 30 leads and 30 calls to cover 840 "rows": the 600 rows without the treble leading, plus each treble-lead row twice, once in a method and once at a call (or lack thereof). Then each call covers 2 rows and each lead covers 26: the 24 you'd expect, plus the two "call" rows with the treble leading at the snap again, since those two rows won't appear at either blow of an actual call.

In theory, that permits pruning to eliminate unjoinable plans early, at the cost of additional bookkeeping to avoid cycles short of the full extent. That said, I haven't implemented a plan-based solution for comparison, so I don't know how much, if any, my solution improves upon a plan-based strategy.

I found 419,388,705 total compositions of the 147 methods excluding lead-splices, rotations, and reflections. The list includes 32330 combinations of methods, again excluding lead-splices, though that number is a bit fuzzier since e.g. a composition with only bobbed leads of Cambridge might also be considered to include Primrose instead.

I've been focused on just getting to this point, so I'm honestly not sure what sort of thing to look for in that haystack. My favorite discovery so far is that (again supposing my code is correct) if you try choosing four methods from the four quadrants (one each of S, 3D, 4D, and TB), there are only three sets that can produce an all-the-work composition:

Wo Di Te Ms
Sa Di Ev Ms
No Cv Br Kt

The first two form a tidy bundle of intersecting work above and below the treble, but the third has more variety and also only uses bobs. One of its three compositions:

No No No No No-
Br No-Kt-No Br
No-Kt-No Br No-
Kt-No Br No-Kt
Kt-No-No No Br-

I know bobs-only treble-dodging minor has been investigated in the past. Was that composition (or one like it) previously known?

 - John

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