[r-t] winkie extents

[Matthew Frye]

On 24 Oct 2010, at 23:41, Alexander Holroyd wrote:

> On Sat, 23 Oct 2010, Matthew Frye wrote:
> 
>> In other news: I have an answer to Phil's "winking-up" question from months back (it's not possible [For certain cases. Within some limits. I think]). Full email to follow sometime this weekend.
> 
> I also came to the conclusion that this is impossible a while ago (assuming you are referring to an extent of major based on winked up minimus), but have forgotten some of the details.  I'd be happy to see your analysis...


Yes, I was talking about a major extent of winked up minimus, I'd been told no one else was looking at this and presumed a -ve result would have been at least mentioned. Well it's done now, so here's (some of) my analysis:


Quick answer: No, it’s not possible.
This statement needs a few qualifications. Most importantly I required using entire courses of the winked-up method. Doing anything else seems to ruin the entire point of extending an extent on n to an extent on 2n.
Secondly, I’ve proved this only for plain bob and double canterbury minimus as the base methods, these proofs represent two extremes but will extend to many if not all minimus extents in-between.

Some observations: each change in the minimus method corresponds to 4 rows in the winked-up. Now, look at the first two of these, in the “plain course” these all have all pairs (12, 34, 56, 78) remaining together following the path of a single bell at the lower stage. The ordering of these pair is well-defined as pairs are swapped for ever position they move, the ordering in the second row of each block of 4 is reversed (as the change here is always an x ).
These 48 rows form a sub-group of the extent that we’ll represent using [12345678] = {12345678, 21436587, 43215678, …}

Clearly, all courses starting from any member of [12345678] contain all of [12345678], so we can only have 1 course starting from each group. There are 840 sub-groups (40320 / 48), we need 420 courses (40320 / 96). ie we must select 50% of the groups to work with. This is all true for any base extent we can choose.

Focusing on the “plain course” of winked-up plain bob, specifically at the 3rd + 4th rows of sections arising from the x in the (minimus) pn we find that all these rows come from 3 groups: [13245768], [17645328], [16745238], thus excluding all courses from any of the members of any of those 3 groups. Here we can only use 1 group of these 4, half of what we need for the extent.

Now: {[12345678], [13245768], [17645328], [16745238]} form another sub-group.
(NB it’s not enough that {12345678, 13245768, 17645328, 16745238} form a group as multiplication of these [ ] groups is NOT well defined how you might naively expect, however the 192 changes included above do indeed form a group). So this maximum of 1 group in 4 is repeated across the entire extent, giving an upper bound of 20160 changes (I did not pursue this further to see if even this was possible, though I suspect not).

This obviously generalises to all right place base minimus extents, but further to all minimus starting methods where the treble crosses with each of the other bells at at least one x in the pn. This includes all 3-part extents with at least one x in the pn. ie 9 of the 11 standard minimus methods.

As x is clearly a bit of a problem, I'll now look at an extent without any: Reverse Canterbury. It's a little more involved and I'll just give a brief outline of what I did here.

There are 6 false groups, however these don't conveniently fall into a group, they do fix which 420 of our [ ] groups we must use, but this isn't relevant now. So we try a different approach.
Consider the case when you *don't* use a course from [12345678], each of [12345678] must be included in alternative courses from the 6 "false" groups. I won't go into detail unless anyone particularly want it, but if you work out which courses bring up which rows, it's not possible to get to all 48 without repetition of some.

As far as generalising this goes, I haven't. A quick look at Double Court suggests it will probably be the same and I suspect all other minimus extents would suffer from this problem (although it remains to prove this). Another little note: winking-up original singles to minor also cannot provide an extent for similar reasons.

Sorry, that's probably a bit more than I meant to write. Now, back to the 147 TDMM, it's starting to get exciting!

Matthew Frye
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