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

John Danaher jsd at alum.mit.edu
Sun Dec 6 23:14:57 UTC 2015

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

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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