[r-t] Some stedman focus

Philip Saddleton pabs at cantab.net
Sun Mar 23 22:37:39 UTC 2008

Philip Saddleton said  on 23/03/2008 12:26:
> Stephen Penney said  on 23/03/2008 12:19:
>> A well known bobbed-course of Stedman Triples is S,L,Q.
>> A few weeks ago I realised this also works for Cinques if you call the
>> 11th S,L,Q but add in singles at 1 and 14 (with the 11th unaffected in
>> 0E).
>> Why does this extension work? Is it entirely obvious? If so how does it
>> extend to 15, 19 etc?
>> Percy
> Change the bobs to singles, and this may give you a clue.
The two bells affected by a single both go in the same way, so there 
will be another place in the same course where they can also be 
affected. On n = 3 (mod 4), after a single at an odd-numbered six they 
go in quick, and the other position is n-2 sixes later (or n+2 sixes 
earlier) and after a single at an even-numbered six they go in slow, and 
the other poition is n+2 sixes later (or n-2 earlier). A pair of bobs 
has the same effect as a pair of singles.

Now, if n = 3 (mod 4), the calling postions are:
S: (n-1)/2, (n+1)/2
L: (n+7)/2, (n+9)/2
Q: (3n+3)/2, (3n+5)/2

Now notice that the first call of the Q is n-2 sixes after the first of 
the L, and the second call of the Q is n+2 sixes after the second of the 
S. If we replace the bobs by singles these will cancel out, leaving us 
with singles at (n-1)/2, (n+9)/2.

The first of these is cancelled by a single at (3n-5)/2, which for 
triples is the same as the second. The second is cancelled by a single 
at (3n+13)/2 (mod 2n). These singles are thus four sixes before the 
first call of the Q, and four sixes after the second call. On fifteen 
and above these are when the observation bell is in 12-13 down and 10-11 
up respectively.

A similar relationship holds for n = 1 (mod 4), except now the first 
call of the Q matches the first of the S, and the second the L, and the 
singles are three sixes either side of the Q (10-11 down and 8-9 up).


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