[r-t] Grandsire Triples

[Mark Davies]

Percy:

> I was told once that Grandsire Triples can be called round from any
> change using three calls or less.
> I've failed to find a proof for this, in terms of logical reasoning.

I'm not sure you can prove this without checking all cases, because it
probably depends on that mysterious feature "linkage", however you can
certainly demonstrate a weaker premise, namely that it isn't impossible.

Consider all the possible arrangements of working bells. There are 6 working
bells and the plain course is of length 5, so there are 6x24 = 144 courses.
Half the courses are the reverses of the other half, so we only need to
consider 72 of them (remember there are two ways of coming round,
handstroke or backstroke).

Now with just one bob you have 6 possible callings (depending on how many
plain leads before the bob), with two bobs 6x6=36 callings, and with three,
6x6x6 = 216. If you take singles into account too you have 12, 144 and 1728.
Add these up and there are 1884 possible different ways of using three
calls. Some of these might give the same results, and we haven't shown that
there isn't some "rogue" course which isn't brought round by any of the
1884, but since 1884 is quite a lot larger than 72 it certainly shows it
might be possible.

MBD