[r-t] Out of Course Spliced S Minor

[Alexander Holroyd]

I've recently done an exhaustive search for all possible treble-fixed 
extents of spliced surprise minor from the standard 41 (indluding 
out-of-course splices, with singles), with some interesting results.

I'm still working on analysis of the results, and will write more messages 
as this progresses.

The method is an extension of what Richard Smith did a year or two ago for 
in-course extents of treble-dodging methods, following my suggestion. 
The basic idea is to do a backtracking search for sets of whole leads that 
together comprise the extent.  The key to making this feasible is to 
choose which step to try next in a special way, by finding a row that 
appears in the fewest possible leads.  This makes it billions of times 
faster.  I eventually realized that this is actually an instance of a 
known computer science problem, called the exact cover problem, and the 
algorithm is essentially Kunth's "dancing links" algorithm. Realizing this 
makes the coding much easier.

The result of this search is a list of over 2 million sets of leads, which 
I call "plans", including many that involve a variety of out-of-course 
splices, both simple ones and more complicated ones, and some that involve 
3 different overworks.

For any given plan, there is still a question whether the leads can be 
joined together to form an extent, using calls plus 2nd and 6th place lead 
ends.  Not all plans can be joined in this way, and I'm still working on 
seeing which ones can.  It is in principle a straightforward search 
problem.

One consequence is that it may be possible to get all 41 methods into 9 
extents (IIRC, 10 extents was the in-course record).  The required plans 
exist, but it remains to be seen whether they can be made into extents.

For now, here are a few examples of interesting out-of-course extents (new 
so far as I know). The first one was rung in Bristol this week; this is 
possibly the first extent of spliced s minor from the 41 to include 
Norwich.

-=14 bob;  $=1234 single;  #=1236 extreme

No No-No-No Hu$Sa Sa-Sa Sa$Hu; 3-part
Yo Yo Yo#Lo Lo Lo#; 5-part atw
Yo Du Yo#We Lo We#; 5-part atw
Ip Ip Ip#Cu Cu Cu#; 5-part atw
Bc-Bc-Bc-Bc$Bc$Bc Bc-Bc-Bc$Bc$; 3-part
Sa Sa-Sa-Sa Sa-Wo Wo$Ad-Ws Ws$; 3-part
Ws$Sa Sa$Ws$Ws$Ws Ws$Sa$Ws$Ws$; 3-part
We$Bv Bv Bv Bv Bv$We#Yo$Cu Cu Cu Cu Cu$Yo#...
...We We#Yo Yo#We We#Yo Yo#We We#Yo Yo#We We#Yo Yo#; atw

Ander