[r-t] Extents of doubles
Alexander Holroyd
holroyd at math.ubc.ca
Tue Aug 22 16:45:26 UTC 2006
Well done Richard.
Can we see a list of the 101 2-blow ones?
And maybe the legal glide-symmetrics (if there are any)?
On Tue, 22 Aug 2006, Richard Smith wrote:
> 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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