[r-t] Principles

[Mark Davies]

MR writes,

> I think the question you're asking is equivalent to asking if we can increase
 > the offset between the 2 lines to more than 2 changes.

Yes, you are right I think, for n=2 examples anyway. That's a simpler 
way to look at it.

Ander write,

> I still don't understand what the reduction is.  Your description of
> "dewinking" does not fit either of the examples (e.g. in mine, 78 only
> ring together near the lead end).

Yes, it covers both of the examples. No reducible pair of bells gets 
further than 2 blows apart, and only do this when "running through" 
another reducible pair of bells. (The reducible pairs of bells in your 
exmaple are 12, 34, 56 and 78). Consider the back four bells in the 
first four changes:

  5678
  6587 x
  6857 58
  8675 x

Here the pairs (56) and (78) are crossing with each other, so equivalent to

  ab
  ba x

In the next set of four changes the (56) is effectively on its own at 
the make, so making a place in the Minimus reduction, whereas the (78) 
pair is now crossing with (34) in the four places between 3rds and 6ths:

  3847
  8374 x

Following which they are crossing on the front with the (12) pair:

  2817
  8271 x
  8721 14

> However, I suspect that no construction
> like this (involving somehow "contiguous" bells) can account for all
> possible unprinciples...

Well, that's my question. I can't decide either way! Can you come up 
with a counter-example, or a proof of non-existence?

MBD