[r-t] Crambo - mystery solved(?)

Alexander Holroyd holroyd at math.ubc.ca
Fri Sep 8 17:21:51 UTC 2006


On Fri, 8 Sep 2006, Richard Smith wrote:

> Alexander Holroyd wrote:
>> It _is_ possible on 6 (and I would guess on all higher
>> numbers).
> Yes, it is possible on all higher numbers.  The proof for
> n>6 is below, which together with your example for n=6 and
> my earlier negative proof for n<6 covers everything.
>> Here is an example (which I think it is safe to
>> say no-one in their right mind would ring).
> Cool.  Unfortunately it's not clear to me that the obvious
> generalisation works; however, this has given me enough of
> an insight into the problem to constuct a similar 7 bell
> method that I *can* generalise to n bells.
>> p = 234.345.
> This is Erin Minimus with 5,6 covering that has been
> 'Orpheused' by inserting 1234 between each change meaning
> that the plain course contains all 48 rows with 5,6 in 5-6
> in either order.
> (Tying in with the earlier part of the thread, it is also a
> crambo based on Erin Minimus; but instead of Erin with 5,6
> covering, it is a crambo of Erin with 5,6 dodging behind.)
> Clearly there are 5x6/2 = 15 mutually-true courses that
> between them include the whole extent.
> This can be readily generalised.  Take any (n-2)-bell
> extent, add two covering bells, and 'Orpheus' it by
> inserting the change 123..(n-3) between each pair of actual
> changes.  The plain course of this method has the property
> that any one row can be removed leaving a legal touch.
> We now have n(n-1)/2 mutually-true courses including the
> whole extent.  Unfortunately it is not obvious if and how
> these can be joined.  On five bells, for example, we know
> they cannot be joined whilst retaining the property that any
> row can be removed leaving a legal touch (because no touches
> with this property exist).
>> - = 234.345.
> Ander's touch works having a 1234->1236 bob.  In order that
> both the plain and the bob leads produce legal touches when
> an arbitrary row is omitted, the bob must occur between two
> changes with place made in 4-5-6 -- in Ander's example,
> between a 3456 and a 1456.  This could easily be generalise
> on higher numbers to a 1...(n-3)(n-2) -> 1...(n-3)n bob
> inserted between two changes with places in (n-2)-(n-1)-n.
> (And we can easily prove that for n>5, there exists an
> (n-2)-bell extent with three or more blows behind.)
> Unfortunately, that's as far as I can take a bob-based
> solution as I cannot prove that in general we can find
> suitable Q-sets to join the courses.  (I imagine this is
> why Ander has just asked the circumstances under which an
> odd number of courses can be joined by bobs, though on 4n
> and 4n+1 bells, we have an even number of courses.)
> Let's instead start with a 7-bell method based on Erin
> Minimus.
>  2347.3456.2345.456.2347.3456.2345.456.2347.3456.2345.56
> The front four bells ring a whole-pull extent of Erin whilst
> the back three ring whole-pull plain hunt.
> If we introduce a 12345->12367 single between changes with
> places in 4-5-6-7.  E.g.
>  p = 2347.3456.2345.456.2347.3456.2345.456.2347.3456.2345.56
>  s = 2347.3456.2367.456.2347.3456.2345.456.2347.3456.2345.56
> As this Q-set has order 2, we can definitely join all the
> courses together.
> This generalises to n bells.  Start with any (n-3)-part
> extent (that has at least three blows behind) and ring this
> whole-pull on the front (n-3) bells.  Add whole-pull Plain
> Hunt on the back three.  As there are suitable starting
> extents on four or more bells (no three bell extents have
> three blows behind), this proves the existence of extents on
> any n > 6 that are still legal touches if an arbitrary row
> is omitted.
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