[r-t] Extension of TDMMs

[Richard Smith]

Philip Saddleton wrote:

> >So, it appears the numbers don't increase as fast as might
> >be expected.  (This is no doubt due in part to the
> >indefinite extensions.)
>
> Yes - what my calculation was meant to represent was the chance of
> sporadic occurrences of regular methods, so the indefinite extensions
> ought to be excluded. I would expect similar probabilities from
> extending the irregular ones as well.

In which case they over-estimate the probabilty of it
working -- 1529 out of 103533 extensions sporadically work
on 8 bells and 505 on 10 bells.

> >Similarly, the five methods
> >that worked on 6, 28 and 52 may actually have stopped there.
>
> Presumably these extensions have something in common, or this seems a
> remarkable coincidence. My guess is that they do stop - I would expect
> the stages at which an indefinite extension has PB lead heads to be an
> arithmetic progression.

They're all mode-4 AB over mode-4 FG extensions.  The
methods in question are

  &-3-45-5-3.2-4.5,2  Eminent Nick Lawrence S.
  &-3-1-5-3.4-34.5,2  Ripple D.
  &-3-1-5-3.4-2.5,2   Droitwich D.
  &-3-4-2-3.4-34.5,2  Beverley S.
  &-3-4-2-3.4-2.5,2   Surfleet S.

I checked Beverley and Surfleet further -- up to 100 bells
-- and it does appear to be an indefinite series: 6, 28, 52,
76, 100.  If you ignore the first step, it does form an
arithmetic progression.

This behaviour is surprisingly common.  Of the extensions I
looked at, I found 27 different seemingly indefinite
extension series.  Some of these were simple arithmetic
series: when expressed an+6, the series with a=2, 4, 6, 8,
12, 16, but most are more complicated:

  Freq  Series
  ------------------------------------------
   918  4n + 6
   464  2n + 6
   136  4n + 6 (n != 1 mod 3)
   100  6n + 6
    76  2n + 6 (n != 2 mod 3)

    58  12n + 6
    49  6, 4n + 8
    36  8n + 4
    26  6, 12n + 16
    18  6, 12n + 8

     8  6, 14n + 10 ??
     7  6, 6n + 10
     6  6, 26, 50 ??
     5  6, 24n + 28
     4  4n + 6 (n != 4 mod 7) ??

     3  6, 8n + 12
     2  6, 24n + 20 ??
     2  8n + 6 (n != 2 mod 3) ??
     2  6, 12n + 14
     2  6, 12n + 10

     2  6, 8n + 10
     1  6, 6n + 8
     1  6, 4n + 8 (n != 2 mod 3)
     1  16n + 6 ??
     1  6, 8n + 12 (n != 2 mod 3)

     1  4n + 6 (n != 7 (mod something?)) ??
     1  2n + 6 (n != 1, n != 22) ??
  ------------------------------------------

(Those marked '??' are more guesses than anything else.)

Richard