[r-t] Spliced Courses

[Robert Bennett]

 The reason that Royal behaves differently from Minor, Major and Maximus
with this sort of splice is that the plain course of Royal (with a single
hunt bell) has 9 leads, and nine is not a prime number; but 5,7 and 11 are
prime numbers. 

Another example of Royal behaving differently is that there is a spliced
course of Plain and Little Bob Royal PLLPLLPLL which has all 9 leads. 

This is not possible for Minor, Major and Maximus. 

The 9 leads of Royal can be separated into 3 subgroups of 3 leads each,
and this gives extra possibilities. Usually the options available for
Major, maximus etc work for Royal in the same way; for example the courses
PLP and PLL...L work in the same way. The special Royal options do not have
any equivalents on 6,8, 1nd 12 bells. 

Single hunt 16-bell and 22-bell methods should also follow the Royal
pattern, as should double hunt Cinques methods. 

Congruence (Clock) arithmetic works well when working out this sort of
splice.
 The easiest way to use it is to count 1 for each lead of Plain bob, 2 for
each lead of Double... up to nlc-1 for each lead of St Clements. (NLC being
the number of leads in the plain course.) 

For the course to come round, it must add up to a multiple of NLC, or in
algebraic terms a+2b+...+(nlc-2)y +(nlc-1)z = 0 mod (NLC)
 where a,b and z are the number of leads of plain, double bob, little
bob and st. clements respectively. (Or other methods with these lead
heads). 

  

Robert Bennett 

  

 On Sun 19/12/10 7:47 AM , "Robin Woolley" robin at robinw.org.uk sent:
  As you may remember, a couple of years ago I did some work on spliced
 courses. I found 72 distinct splices (on eight bells) for either
completely
 2nds place or 8ths place methods.

 Looking at these results further, it seems that if one particular lead
end
 group appears an odd number of times and it is the only such 'odd' group,
 then the splice is palindromic. So, CEEFFEE is palindromic but BCFDEDD is
 not.

 Letting an 'odd' group to be a lead-end group appearing an odd number of
 times, (and similarly for 'even' groups), then putting this in more
 mathematical language:

 "A splice is palindromic if and ony if it has just one odd lead end
group."

 This also works for 6-bell splices. There are only four of them, all
 palindromic, so can be easily republished: AABFB ABAEE AEFFE BBFEF

 However, this does not work for Royal. Some splices obeying the above
 condition are palindromic and some are not. ABDFFDBAC is whilst ABFDFBDAC
is
 not.

 Would I be right in guessing that the condition is true for Maximus, 
 Fourteen and Eighteen but not for Sixteen?

 Best wishes
 Robin

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