[r-t] # methods

[Richard Smith]

Chris Poole wrote:

> Now that we've moved up from 6 bells, here's a question - what is an order
> of magnitude estimate for the number of (regular) methods where the
> treble, say, treble bobs, on stage N?

We covered some of this back in August in the thread on
extension of TDMMs.

First we need to define "regular".  Do you just mean regular
lead heads?  Or are you also adding additional requirements,
such as no more than two consecutive blows in one place, no
single changes, no 5ths above the treble, etc.?

For minor, it is usual also to require that an extent is
possible.  The obvious way of doing this is by requiring
palindromic methods with specific parity structures in each
dodging position of the treble: ++--, +--+, --++ or -++-.

If this restriction is continued to major, then Cambridge is
disallowed because it has a double change during the
treble's 3-4 dodge giving the parity structure ++++.

Ignoring all of these issues, we can certainly derive an
expression for the number of place notation sequences
consistent with a treble-dodging treble on n bells.  This is

  F(n-2)^6 F(n-3)^2 F(n-4)^6 F(n-3)^2 ...

where F(n) is the n-th term in the Fibonacci series.

(Consider over and underworks separately; by symmetry, there
must be equal numbers of each.  For overworks, there are
three changes on (n-2) bells when the treble dodges in 1-2,
followed by one change on (n-3) bells as the treble moves
dodging position.)

Richard