# [r-t] Extents of doubles

Richard Smith richard at ex-parrot.com
Tue Aug 22 14:02:03 UTC 2006

```Richard Smith wrote:

> > 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.

Well, somewhat later than anticipated, I have a list of all
24 change-per-lead doubles principles.  Once you've factored
out reflections, rotations, translations and "Kent / Oxford
variants", there are 52,227,975 doubles principles
whose plain course generates the extent.

By "Kent / Oxford variant", I mean the same as Ander in his
list of minimus extents -- if a block of four rows can be
rung in both the orders (a,b,c,d) and (a,c,b,d) such that
both are joined by valid changes, then they are K/O
variants.  For example, 5.125.5 and 345.125.345 are a pair
of K/O variants.  All pairs are characterised by swapping a
dodge for places.  This particular example is precisely the
difference between Plain Bob and Reverse Canterbury.  (The
term "Kent / Oxford variant" is a bit of a misnomer as Kent
and Oxford do *not* differ by a K/O variant.)  On five
bells, there are 17 pairs of K/O variants.

These fifty-odd million principles can be categorised by
symmetry as follows:

52,214,650    Asymmetric
12,953    Palindromic
372    Glide-symmetric
----------
52,227,975

It would appear (unless there's a bug in my code) that there
are no rotationally-symmetric principles.

They can also be classified in terms of maximum consecutive
blows in one place:

101     2 blows  \
506,059     3 blows   | "Legal"
8,703,933     4 blows  /
13,055,579     5 blows
11,590,519     6 blows
7,733,797     7 blows
4,551,747     8 blows
2,599,963     9 blows
1,496,379    10 blows
855,875    11 blows
489,505    12 blows
271,352    13 blows
154,848    14 blows
89,658    15 blows
54,516    16 blows
31,830    17 blows
17,940    18 blows
10,434    19 blows
5,818    20 blows
3,832    21 blows
1,576    22 blows
2,714    24 blows
----------
52,227,975

Note that, unsurprisingly, no methods have 23 consecutive
blows in one place.  Unfortunately, this analysis is
somewhat spoilt by the fact that K/O variants do not
necessarily contain the same number blows in one place and
the choice of which K/O variant to discard is, to some
extent, arbitrary.

As the list runs to several gigabytes, I can't readily make
it available.  However, if anyone wants me to do any further
analysis on this data, let me know and I'll try.

RAS

```

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