[r-t] Extents of doubles

[Richard Smith]

Philip Earis wrote:

> Here are some words you don't often hear: Barry Peachey has written an
> article in this week's RW with some interesting bits in it.  Amongst the
> usual bluster, misplaced grandiosity and misguided sentiments he talks a bit
> about Crambo doubles, which was apparently 'first published in Fabian
> Stedman's Campanalogia in 1677'.

Crambo is one of only eleven doubles extents (up to
rotation, reflection, etc.) with the following properties:

  - the extent forms a perfect five part (i.e. it is a
    principle);

  - no more than four consecutive blows in any one place;
    and

  - uses only single changes.

All eleven are totally asymmetric and seemingly devoid of
any pattern.

Using only 17th century technology (i.e. not using a
computer), how easy would it be to find one of these?  At a
guess, not very easy; in fact, I find it astonishing that
Stedman managed to find such an extent (though as it
definitely is in Campanophile, he, or one of his
contemporaries, must have).

> Crambo is a doubles principle with 24 changes per lead; ie the plain course
> generates the extent. There is no RW reference next to it on the Methods
> Committee website (http://www.methods.org.uk/online/prin5.htm#76),
> suggesting that it hasn't been rung in recent years.

Well, until August 2004, it wasn't listed at all.

> The noation for Crambo is:
> 345.125.145.123.345.123.125.345.123.345.125.345.123.345.145.123.345.125.345.123.345.125.345.123
> = 45123 (you can even call it cyclic!)
>
> As Barry points out, it would be a serious challenge to ring Crambo.  It
> puzzles me a bit that many ringers assume you have to go to higher numbers
> of bells for complexity:

I think the assumption is more that you have to go to higher
numbers for *interesting* complexity.  And once you've
introduced the subjectivity of what is or isn't interesting,
it's hard to dispute that ;-)

But just looking at the variety and complexity of
compositions of spliced minor available, it's hard to
dispute that minor offers plenty in the way of complexity.

> There are other interesting methods on the same webpage, from the familiar
> Orpheus, to Mermaid (1.3.1.3.1.5.1.3.1.3.1.125.3.5.3.5.3.1.3.5.3.5.3.145),
> which has some beautiful symmetry.

OK.  What am I missing?  Mermaid doesn't have any symmetry,
does it?

> Can anyone generate an exhaustive list of all 24 change-per-lead doubles
> principles whose plain course generates the extent?

It ought to be feasible.

> Or at least estimate a lower bound for the number that
> exist?

An interesting question.  Without generating an exhaustive
list, can anyone see how to find a good lower bound (or,
indeed, a good upper bound) on the number?

Clearly there are F(5)-1 = 7 changes available, where F(n)
is the Fibonacci series, F(0)=F(1)=1, F(n)=F(n-1)+F(n-2).
Equally clearly, we cannot have the same change twice in
succession as this is immediately false, meaning that
everything other than the first change is limited to a
choice of 6.  Finally, we can exclude rotations, which, as
a perfect 5n part is impossible unless n=1, is as simple as
dividing by 24.  This gives an upper bound of 7.6^23/24, or
about 2.3 x 10^17.

But what about a lower bound?  Now I'm stuck.  I have
absolutely no idea how to do this beyond counting those that
I know about.

> It sounds like the kind of thing RAS might have done before, but I can't
> find the relevant email if he has!

No, not yet, though I'll look into it.

RAS