[r-t] Extension of TDMMs

Philip Saddleton pabs at cantab.net
Sat Aug 14 15:28:48 UTC 2004

Richard Smith <richard at ex-parrot.com> wrote at 15:34:15 on Sat, 14 Aug 
>> 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.

I suppose I was slightly imprecise there with the use of "methods". What 
I meant was the proportion of the combinations of pairs that could cross 
at the half-lead that gave a PB lead end. This is simply the number of 
possible lead ends over n-1, and seems an appropriate measure of the 
chance that a random candidate extension of a random method with 
appropriate characteristics (i.e. symmetrical and with the treble and 
one other bell pivoting at the half-lead) would have a PB lead end. I'm 
ignoring the fact that it might come round, or give a short course, but 
adjustment could be made for this.

>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.)

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.

>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.


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