[r-t] Bobs only Stedman Triples

[Andrew Johnson]

> From: Alexander Holroyd <holroyd at math.ubc.ca>
> Date: 28/06/2012 13:16
> Subject: Re: [r-t] Bobs only Stedman Triples
> Sent by: ringing-theory-bounces at bellringers.net
> 
> On Mon, 25 Jun 2012, edward martin wrote:
> 
> > They certainly are an amazing discovery;but  the achievement was
> > surely figuring out how to programme a computer to look for them and
> > then having the expertise in knowing how to make use of what the
> > computer had found...no perhaps you are right an amazing achievement
> 
> I don't know where this idea came from, but if I've understood correctly 

> it is quite wrong.
> 
> As Philip pointed out, the computer-assisted part was finding the two 
> different 10-part structures (the two blocks of Johnson's 10-part and 
the 
> single 10-part block that cannot be joined by ordinary q-sets).  This 
was 
> done 17 years ago, and the blocks have been public knowledge ever since 
> (and in any case it is a very standard search taking only minutes on 
> modern hardware).
> 
> Given that knowledge, the rest was pure brain power (and very innovative 

> and interesting).
> 
> Possibly Andrew used a computer to confirm that the list of 148 extents 
> was exhaustive, but this could also be done by hand - the computer 
search 
> would be just a slightly quicker and less error-prone check.
> 
> Ander
There was more computer work than that. I was trying to find twin-bob
style linkages for the single 10-part block, and a new program I was 
writing
didn't seem to be working correctly, though it was finding ordinary Q set
linkages. I wanted to give it some more raw material to work with, so
tried seeding some tree searches. I tried feeding a tree search
with some of single 10-part blocks as a starting configuration. One time
I happened to chose a group generated by 1543267 and seeded with two of
the blocks.
One of the results was:
2520!3 
2314567QSPP---PPP--P-P----PPPP--P--P----PP-P--P-PP--------PPPPP---P-P--PP--P-PPPP-P-PPP-P--P-PP---PPP--P-P----PPPP--P--P----PP-P--P-PP--------PPPPP---P-P--PP--P-PPPP-P-PPP-P--P-*1(2)3425167QSPP---PPP--P-P----PPPP--P--P----PP-P--P-PP---P--P--PP--P-PPPP-P-PPP-P--P-PP---PPP--P-P----PPPP--P--P----PP-P--P-PP-PP-----PPPPP---P-P--PP--P-PPPP-P---PPP-P--P-PP---PPP--P-P----PPPP--P--P----PP-P--P-PP--------PPPPP---P-P--PP--P-PPPP--PP--P----PP-PP-P--P-*2(1)


which immediately looked interesting as there were only a 3 blocks to link
but had P-PP-----PPPPP which I had never seen occur before - runs of 5 
bobs
normally only occurred towards the end of a part (with 76 behind).

I fed it into the linkage program which produced rearrangements so I 
started
examining the blocks starting PP---PPP--P-P and saw the twin bob/omit 
pattern.
The linkage program only applied one Q set at a time. I could have
extended it but instead just chose to extract all the blocks found, 
generalized them with the added or removed twin bobs, and 2n+1 bobs where
76 was behind and put them into a search trying any 10 of the blocks.
Several peals resulted which was great, including some exact 2-part peals.

I then had a bit more work to do to check I'd found all the part types,
including looking at what the Q set linkage search found and trying
9 of the blocks and any touch for the last block just in case there were
any similar peals.

The Q sets and twin bob linkages generate round blocks as follows:

Number of touches dividing into   1 round blocks: 1445
Number of touches dividing into   3 round blocks: 12252
Number of touches dividing into   5 round blocks: 12751
Number of touches dividing into   7 round blocks: 4729
Number of touches dividing into   9 round blocks: 1220
Number of touches dividing into  11 round blocks: 295
Number of touches dividing into  13 round blocks: 65
Number of touches dividing into  15 round blocks: 10
Number of touches dividing into  17 round blocks: 1

Total touches: 32768

The 1445 peals reduce under rotation - there are 75 2-parts reducing 
to 15 and 1370 1-parts reducing to 137 giving the 152 peals listed
in a previous message.

If you compare the Upham peal Ds G3 C1 C3 D Ds G3 C1 C3 D 
with the Cholsey peal          B C3 C1 C3 D B C9 D D D
then there happen to be 3 + 2 partial blocks in common. That isn't enough
to seed a 1-part search (which was something I tried ages ago with longer
seeds), but is more than enough to seed a 2-part search.

Andrew Johnson


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