[r-t] Extension of TDMMs

[Richard Smith]

Philip Saddleton wrote:

> Of symmetrical methods with the treble and one other bell pivoting at
> the half-lead, the proportion that have plain bob lead ends at stage n
> (even) is 1 in (n-3)(n-5)...3

Are you saying that on n bells, the number of families of
lead head is 2(n-3)(n-5)...3?  E.g. for minor, two are the
regular ones (one 2nds place and one 6ths place), two are
the usual irregular ones (again, one 2nds place and one 6ths
place), and the remaining two are fourths place.

If so, this is wrong.  One eight bells, there are 32 (not
30) different lead head families of which two are regular.

Certainly, the number of different lead *ends* is


             (n-1)!
      -------------------  =  (n-1)(n-3)...3.
         (n-2)/2   n-2
       2           --- !
                    2

But I don't see how this simply relates to the proportion of
regular lead heads.

> giving
>
> Minimus 1
> Minor 3

Of the 2400 symmetrical TDMMs with five lead courses, no
more than two consecutive blows in one place and that have a
parity structure consistent with producing an extent, 778
are regular (or 1 in 3.08), so this ties in surprisingly
accurately.

Even for plain methods (with the corresponding criteria), 46
out of 134 (or 1 in 2.91) are regular.  Given that the
different lead ends occur with vastly different frequencies,
I'm rather surprised that it is this accurate.

> Major 15

(Or 16?)

And again this matches the data surprisingly well.  For
plain methods with same criteria, 5307 out of 77968 (or 1 in
14.6) are regular.

> Royal 105
> Max 945

(136 and 1152?)

> I suppose that given the number of possible Minor methods and potential
> extension paths, a chance of roughly 1 in 5000 that a random choice will
> give PB lead ends at three stages means that such an example is almost
> inevitable - but is there a sufficiently large sample for four or more?

I guess that depends what methods you allow.  I looked at
all the extensions of the 778 regular methods from the 2400.
This gave, in total, 103533 distinct extensions (ignoring
those that produced three blows in one place but including
those that extended Surprise to Delight (or similar)).  Of
these, 2138 (1 in 48) extended to 8 bells, 1912 (1 in 54)
to 10 bells and 719 (1 in 143) to 12.

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.)

> Does the number of additional possibilities as the stage increases
> compensate for the much smaller chance of a random occurrence?

Probably not in the sample I investigated.  I found 2065
extensions that covered a single stage and 9 that covered
two stages.  (This is actually a little subjective: if a
method had no extensions on between 30 and 60 bells, I
assumed it had stopped.  The three extensions that appeared
to work on 6 and 22 bells only may have had further
extensions beyond 60 bells.  Similarly, the five methods
that worked on 6, 28 and 52 may actually have stopped there.
With the Warkworth extension that I originally posted, I
proved that it did indeed terminate after 10, but, for
obvious reasons, I didn't do this with all the methods.)

Perhaps if I had allowed up to four consecutive blows in one
place (which gives 67115 regular methods) the sample space
might have been just about large enough.

Richard