[r-t] Crambo

Alexander Holroyd holroyd at math.ubc.ca
Fri Aug 4 18:07:55 UTC 2006

On Fri, 4 Aug 2006, edward martin wrote:

> I tried in vain to put myself in FS's shoes and figure out what
> percievable logic was behind the principle Crambo

I recently realised that there is a beautiful and mysterious logic behind 
Crambo.  Start with an course of Stedman - this has all 60 in-course rows. 
Now, to turn this into an extent, one option is to try to _insert_ the 60 
out-of-course rows, each one between two of the existing in-course rows. 
To put it another way, expand each double change in the stedman into two 
single changes that have the same effect:

1 becomes 123.145 or 145.123
3 becomes 123.345 or 345.123
5 becomes 125.345 or 345.125

It doesn't seem obvious that this can be done in a way that produces a 
true 120, but in fact it can, in exactly one way (up to symmetries, if we
require a 5-part).  The result is Crambo (reversed and started in a 
different place):

The same trick works for some other in-course extents - Carter:

Gransire (p-*3):

Spliced grandsire and double grandsire :

(unfortunately these 3-part ones all have 5 blows in one place).

All of them are strange and unsymmetrical, so what is really going one 
here?  I don't know.  E.g. can you prove or disprove that any in-course 
extent of doubles can be treated in this way?

Did FS have this in mind?  Who can say?  Can you ring Crambo by thinking 
of it as slowed-down stedman?  Errrr.....


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