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

[Alexander Holroyd]

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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>>>
>>
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>