[r-t] Smith's Theorem

[Robin Woolley]

  Hi All,

I named the result after Tony Smith who, as far as I know, was the first 
to mention its use in a ringing context at RW71/974. This naming is not 
unknown in the mathematical litereature - not after the originator which 
Tony did not claime - but after its populariser. (I know of another much 
more famous example - but can't find the reference).

As it happens, we were discussing in the car on Saturday night that it 
is necessary, but not sufficient. All the theorem does is to give a 
quick test for one class of methods not having extents. As Richard says 
it does not mean an extent is possible. This asym. Doubles method - 
3.14.5.3.5.123.125.1.345 - 'passes the test' but seems to have only one 
possible 'touch' - a 4-lead course using 125 as the lead end. Others 
have a set of 'required leads' but these cannot be joined up into an 
extent, such as 3.1.5.3.345.3.5.14.3 which has a maximum length of 100 
with any combination of lead-end change.

The only 4-lead doubles methods which seem to have extents are those for 
which the plain lead is plain hunt. Another way (most likely better) of 
looking at this is that an ‘omit’ is a ‘required call’ for the method. 
This requires thinking that a method is defined up to the first lead-end 
and the ‘plain course’ is just identically generated leads joined up by 
an agreed ‘silent’ call.

Best wishes
R
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