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

Sun Feb 21 03:11:59 UTC 2016

I've made some more progress on this, and I now believe that all 147
methods can be achieved in 20 round extents but no fewer.

For methods with 2nd and 6th place variants, I require a plain lead of
either method in order to count it as having appeared in an extent. (So an
extent with e.g. only bobbed leads of Cambridge/Primrose would be allowed,
but wouldn't contribute either Cambridge or Primrose to the total count.)

With that constraint the minimum number of extents increases to 19. But
every such combination of 19 extents depends on some extent containing all
of Bg Bo Cj Ey Fg Hu Nl. Unfortunately no such extent has room for plain
leads of Fg and its three lead-splicers (Ba Cs Sk). There's also no way to
fit an extra lead of Fg elsewhere in the composition, so 19 extents can't
cover all of the lead-splicers.

But as it turns out, covering all of the lead-splicers requires only one
more extent, bringing the total to 20. One example is below. Very little
judgment went into selecting it; it's just the first one that came up, so
there are undoubtedly many ways to improve on the choice of extents.

Calls:
- 14
$ 1234
# 1456
@ 1256
% 1236

# 1
Bl Bl Bl Wk Wk-Bl Wk Bl Bl-Wk-
Bl Bl Bl Wk Wk-Bl Wk Bl Bl-Wk-
Bl Bl Bl Wk Wk-Bl Wk Bl Bl-Wk-

# 2
Do Fr Ey Cg Do-Do Fr Ey Fr-Do-
Do Fr Ey Cg Do-Do Fr Ey Fr-Do-
Do Fr Ey Cg Do-Do Fr Ey Fr-Do-

# 3
Pn Sg-Sg Pn-Le-Pn-Bt-Pn-Pv Li-
Pn Sg-Sg Pn-Le-Pn-Bt-Pn-Pv Co-
Pn Sg-Sg Pn-Le-Pn-Bt-Pn-Pv Li-

# 4
Nm Pn-Av Pn-Mp Pn Pn-So-So Pn
Nm Pn-Av Pn-Mp Pn Pn-So-So Pn
Nm Pn-Av Pn-Mp Pn Pn-So-So Pn

# 5
Nb Cu-Cu-Cu Cl Nb-Wo-Nb-Nb-Wo-
Nb Cl Cl-Cl-Cl Nb-Wo Nb-Wo-Nb-
Nb-Wo-Wh Wo-Nb Cl-Cl Cl-Cl Wh

# 6
Gl-Ch Ch Ch Ch Gl-Ak Te-Mu Mu
Nw-Ak Ch Ch-Sa Nw-Gl-Ak Te Sa
Nw Gl-Sa-Gl-Te Gl-Mu Mu Mu Mu-

# 7
Bs Bs Bu-Bs Bs-Bu Bs-Bv Ta-Ta
Bk Bs Bu Bs-Su Ta-Ta He Bs-Bu-
Bs Bs Bu-Wa Bv Ta-Ta Su-Bs Bu-

# 8
Cx#Du#We Lo$Yo#Bn#Du#Lo Lo$Yo@
Cx#Du#We Lo$Yo#Bn#Du#Lo Lo$Yo@
Cx#Du#We Lo$Yo#Bn#Du#Lo Lo$Yo@

# 9
Fo Ml Sh#Ox#Sh Fo Fo#Bz#Kt-Sh
Fo Fo#Bz Bz-Bz Bz#Fo Fo#Bz Bz#
Fo Fo#Bz Bz Ox#Sh-Kt-Sh-Kt#Bz#

# 10
Gl Ce So So-Ca-Mp Ca Sw Ke-Ke
Gl Ce So So-Ca-Mp Ca Cd Ke-Ke
Gl Ce So So-Ca-Mp Ca Va Ke-Ke

# 11
Cu#Cm Cm Ip-Cm Ip Ne Ip Cm-Nf
Ip-Cm#Cu#Ab Pr Ip-Cm-Ip Cm Cm
Ip Ne-Ne#Cu#Cm Cm Ip-Pr Ne-Ne#

# 12
Ti Tr-Tr$Be#Km Sn Tr-Tr#Be$Tr-
Tr Ti$Di%Ti$Be#Qu Tr$Be$Tr-Tr
Ti$Di%Ti$Be%Tr Qu Tr Ti$Be Di$

# 13
Cj Ey Bg Bg-Ey-Ey Sk Nl Bo-Ey-
Ey Hu Nl Kn Ey-Ey-Ba Nl Wl Ey-
Ey-Rs Bo Ey Cj Ey-Cs Nl Bg-Ey

# 14
Cn Dn$Cc$Cn-Ri Cf-Cf Bg Cn$Cc$
Bg-Fg Cf-Dk Dn Bg$Cc#Dn-Pe Bg-
Bg$Cc%Cn-Cn#Cc%Dn-Dn#Cc$Dk-Cf

# 15
C2 Le-Pm-C2 Os Pm C1-C1-C1 Pm-
Le-Mp-Bh Le Mp-Bh-Le-Mp-Pm Md
Mp Mp Bh-C2 Ed Bh-Mp-Le Os Pm-

# 16
Ta Sl Ns No-No-No Cr Bm Hm Cb-
Bm Cb Ol El-Ng El Wi Bm Wi-Cr
Bm-Cw El Cb Bc El-Wr Ta-Cb-Bc

# 17
C3 Pm%Oc Rc$By Bw%Cf#Kh Bw-C3#
Cf%Mp Mp-By#Ls Rc$Kh Bw C3-By
Mp Mp#Cf%Bw Mp Mp-Bw%Rc Cf$Wf

# 18
Es Ws Di-Di Cv Es$Ev Dt-Dt%Es$
Ck Sa-Wt Dt$Ms$Wo Dt Sa-Sa-Sa$
Ad$Po Po$Es Ad$Sa Sa Po%Es$Ev$

# 19
Wm Wm$Nw$Wm Lf$Cz$Lf Wm$Cz-Lu
Ct#Wm%Cz Ww-Ww Nw-Nw Cz$Lf#Mo
Cy Ww#St Wm#Lu Ct Cz#Ro Wm#Ww%

# 20
Ki Ma-Ma Me-Sl Ma-Br Ki-Wv Me
Bp Sl Ma-Br Ki-Wv Me Bp-Ki Cf-
Sd Ci Ki-Bp Cb Bp-Lv Ks Cf-Ny

On Sun, Jan 10, 2016 at 9:59 PM Alexander Holroyd <holroyd at math.ubc.ca>
wrote:

> Great work!  When I looked into this I find it was often quite hard to get
> both 2nd and 6th place variants together, and sometimes even to get 6ths
> place variants at all.  So I would not assume that it actually can be done
> using these combinations.  Nonetheless, fantastically awesome job!
>
> On Sun, 10 Jan 2016, John Danaher wrote:
>
> > 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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> >>>> ringing-theory at bellringers.net
> >>>>
> http://bellringers.net/mailman/listinfo/ringing-theory_bellringers.net
> >>>>
> >>>
> >>
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> >>
> >
>
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