[r-t] Place methods
holroyd at math.ubc.ca
Tue Oct 13 18:16:54 UTC 2009
I think I can prove:
plain hunt is the only round block such that:
- some bell is plain hunting (all the way from front to back) throughout
- there are no points or dodges
- there are no adjacent places
There must be some internal place made (otherwise either it's plain hunt
or there are obviously points). Suppose k is an internal place made at
some change. Then 1<k<n (because k=2 or k=n-1 would necessitate adjacent
places). Neither place k-1 nor k+1 is made at this same change, so
this place has swaps either side of it in the change:
k-2 k-1 k k+1 k+2
X | X
Now what can happen at the next change? Neither of the bells in k-1 , k+1
can make a point, and they can't both make a place (otherwise we'd have
adj places), so one must swap with k and the other must make a place:
k-2 k-1 k k+1 k+2
X | X
So _at the next change there is an internal place at either k-1 or k+1_.
Hence at every change there must be an internal place, and furthermore
there exists a sequence of internal places snaking around, moving by one
position left or right at each change:
1 2 3 4 5 6 7 8 ...
(there may be other internal places as well). But it's impossible for a
plain hunting bell to ever cross over this snake, so there can't be one.
As Graham points out, little methods and treble place methods are
presumably another matter.
On Mon, 12 Oct 2009, Richard Smith wrote:
> I believe that, except Plain Hunt, there are no plain methods which do not
> contain any points or dodges (i.e. are Place methods) and which do not
> contain any adjacent places.
> For example, Reverse Canterbury Doubles does not contain either points or
> dodges, but the 345 place notation at the start contains adjacent places.
> I have done an exhaustive search on up to 7 bells and also checked the case
> of symmetric methods on 8, 9 and 10 bells, and I have found nothing, so I am
> moderately convinced that Plain Hunt is the only example.
> But I cannot see any obvious reason why this should be so. Can anyone else?
> I stumbled across this oddity while trying to prove that all Place methods
> are trivial variants of non-Place methods made by changes dodges into pairs
> of places and similar. It turns out that this isn't true, though such
> 'intrisically' Place methods are rare.
> On four bells, Single, Reverse and Double Court all intrinsic Place methods,
> as is Grandsire. A bobbed lead of Plain Bob Doubles is an example on five
> bells, but that's not a legal method as it has more hunt bells than working
> bells. On five, if you limit it to legal methods (i.e. a true plain course,
> no more than four blows in one place and more working bells than hunt bells),
> there are only ten such methods:
> &18.104.22.168.5,1 Untitled Differential Place Doubles
> &22.214.171.124.5,4 Oake Place Doubles
> 126.96.36.199.188.8.131.52.5.4 Untitled Place Doubles
> 184.108.40.206.220.127.116.11.5.1 Untitled Place Doubles
> 18.104.22.168.22.214.171.124.5.4 Untitled Place Doubles
> 126.96.36.199.188.8.131.52.5.1 Untitled Differential Place Doubles
> 184.108.40.206.220.127.116.11.5.1 Untitled Differential Place Doubles
> 18.104.22.168.22.214.171.124.5.1 Untitled Place Doubles
> &126.96.36.199.2,1 Dunston Place Doubles
> &188.8.131.52.5,1 Untitled Differential Place Doubles
> On six there are 250 legal methods, but all are either differential hunters
> or asymmetric. On seven symmetric non-differential hunters methods exist
> (152 to be precise), and, unsurprisingly, none have ever been rung. On
> eight, as on six, there are no symmetric non-differential hunters.
> I had thought that all of the methods contained four blows at lead or lie
> with an adjacent place in the middle of it, but this is not actually so -- on
> seven bells you can have something like &184.108.40.206.34.1.34,1, which feels very
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