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

[John Danaher]

Well, the combinatorics involved meant that the next step wasn't "very
short" after all, but I've got some results now: all 75 methods (the 147
minus lead-end variations and lead-splices) can be achieved in 17 extents.
One example:

Bl Wk
Bu Cj Do
C3 Nm Pn
Bp Bw Le Li
Cc Dn Fr Sg
Ch Cl Cu Gl
Cm Cu Ip Ne
Es Nb Sa Te
Bm Cb Ma No Ta
Fo Kt Ox Rc Ti
Av Ca Ce Ke Mp So
Be Cf Di Ms Qu Tr
Bg Bh C1 C2 Mp Pm
Bn Cx Du Lo We Yo
Bo Bs Bv Du Ki Yo
Di Dt El Ey Sa Ws
Ak Cn Cz Lf Nw Wm Ww

If I haven't made any mistakes, then that's as good as you can do: all 75
methods can't be achieved in fewer than 17 round extents.

Note that since lead-end variations are being ignored, it's possible that
the extent containing "Cm", for example, could actually contain only plain
leads of Primrose and no Cambridge at all. I also haven't (yet) verified
that each method appears enough times in its extent(s) to also cover all of
its lead splicers, so the full 147 methods might still require more than 17
extents.

 - John

On Tue, Dec 29, 2015 at 7:15 PM Alexander Holroyd <holroyd at math.ubc.ca>
wrote:

> Awesome stuff!  I think it is a very short step from here to finding the
> minimum number of extents required to get all the methods.  (Although
> taking account of lead splicers and lead end variants would complicate
> things).
>
> Ander
>
> On Tue, 29 Dec 2015, John Danaher wrote:
>
> > For the 147 methods the number's a bit larger. There end up being 1133
> > maximal sets that admit an extent.
> >
> > Out of the 11 methods {Bg Bh Bp Bw Cc Cf Cn Dn Le Mp Pm}, any 5 form a
> > maximal set, so that's 462 right there. That set of methods also shows up
> > frequently in combination with others. For example, another 260 sets
> take 4
> > of those 11 methods then add 1 of {C2 C3} and 1 of {Rc C1}, and another
> 162
> > sets take 3 of those 11 methods and add one of {Bo Cz Li Qu Tr}.
> >
> > The full list is too long for an email, so I posted it to
> > http://methodatlas.com/maximal-sets-147.txt . Extents achievable with
> only
> > bobs are marked with a *, and the few that have only a single work above
> > the treble are marked with a #.
> >
> > (All of the maximal same-above sets are also achievable with only bobs,
> but
> > that's not the case for all sub-maximal sets: {Es Ox} and {Cf Es Ox}
> share
> > work above the treble but can only achieve an extent with the aid of
> > singles.)
> >
> > - John
> >
> > On Mon, Dec 14, 2015 at 3:45 AM Alexander Holroyd <holroyd at math.ubc.ca>
> > wrote:
> >
> >> Here are the maximal sets of splicable methods from the standard 41
> >> (ignoring lead-end variants and lead splicers).  As I said, a very short
> >> list.  I would like to see the same thing for the 147.
> >>
> >> In course:
> >>
> >> Bm
> >> Ke
> >> Li
> >> Wk
> >> Bo Ne
> >> Ak Nb Cl
> >> Nb Sa Cu Cl
> >> Ak Ch Sa Cu Cl
> >> Nw Ak Ch Sa Cl
> >> Bv Bo Du Cm Yo
> >> Nw Sa Ch Cu Cl
> >>
> >> Out of course:
> >>
> >> Bm
> >> Lf Ak Wm Nw
> >> Ip Bo Cu Cm
> >> Sa Bo Ws No
> >> Bv We Cu Yo
> >> Lo Ip Cu Cl
> >> Lo Yo Cu Cl
> >> Ip Cu Cm Lo
> >> Ip Cu Cm Ne
> >> Ip Cu Cm Yo
> >> Ip We Cu Lo
> >> Sa Ws Ne No
> >> Lf Ak Wm Nb Ne
> >> Lo We Du Yo Cu
> >> We Yo Lo Bo Bv Du
> >>
> >> On Tue, 8 Dec 2015, Alexander Holroyd wrote:
> >>
> >>> That's awesome work John!
> >>>
> >>> Suggestion for what to look for: all _maximal_ sets of methods that
> >> admit an
> >>> extent.  It may take a bit of thought to figure out a good algorithm
> for
> >>> this, but it should be doable, and the list will probably not be that
> >> long.
> >>> I did it for the surprise only by a pretty naive algorithm.  There were
> >> only
> >>> about 20 sets IIRC.
> >>
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> >
>
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