[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):
34.23.45.23.34.12.23.34.23.45.23.34.23.45.34.23.34.12.45.23.34.23.45.23
The same trick works for some other in-course extents - Carter:
23.34.45.23.34.12.23.34.23.45.23.34.23.45.34.23.34.12.45.23.34.23.12.34
Gransire (p-*3):
34.23.45.23.34.12.45.23.34.12.45.23.12.34.23.45.12.34.23.45.
23.34.23.45.12.34.45.23.12.34.45.23.34.12.45.23.34.23.45.23
or
23.34.23.45.12.34.23.45.12.34.23.45.34.12.45.23.34.12.45.23.
34.23.45.23.34.12.23.45.34.12.23.45.12.34.23.45.23.34.23.45
Spliced grandsire and double grandsire :
34.23.45.23.34.12.23.45.12.34.23.34.12.34.23.45.23.34.23.45.
23.34.23.45.12.34.23.45.34.12.23.45.34.12.45.23.34.23.45.23
or
23.34.23.45.12.34.45.23.34.12.34.23.34.12.45.23.34.23.45.23.
34.23.45.23.34.12.45.23.12.34.45.23.12.34.23.45.23.34.23.45
(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.....
Ander
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