[r-t] 3 and 4

[Richard Smith]

Samuel M. Austin wrote:

> How many different methods could be rung using 3 bells, not having more than
> 3 blows in one place and the method was to be say 18 changes long with no
> null changes?

I assume you also want the method to be true in the sense of
each row appearing three times?  I also assume that you want
to include "methods" where each bell does something
different (e.g. Shipping Forcast).  I have excluded
rotations, reflection, reversals and shorter method repeated
(e.g. Plain Hunt twice is not counted as a double-extent);
this leaves four extents:

  &3.3.1.3.1.3.1.1.3,1  (3 blows)  (PGR)
  3.3.1.3.1.3.1.3.1.3.1.3.1.1.3.1.3.1  (3 blows)  (R)
  3.3.1.3.1.3.1.1.3.3.1.3.1.3.1.1.3.1  (3 blows)  (R)
  3.3.1.3.1.1  (x3)  (3 blows)  (R)

(The symbols in the final set of brackets are the methods'
symetries: P = palindromic, G = glide, R = rotational.)

> What would be the formula to work out the following question and how would
> it apply to 12, 24 and 30 changes?

It is unlikely that there will be a simple formula for this.
The figures are:

   6         1
  12         1
  18         4
  24        18
  30       109
  36       864
  42      8720

> How many 48s of Minimus are there with not more than 6 blows in one place
> and no null changes?

Probably millions, going by the number of single extents --
see

  http://www.math.ubc.ca/~holroyd/minimus.html

I don't have code to hand that will do this search in a
sensible length of time.  If you want to restrict it
further, I might be able to give you an answer.

Richard