[r-t] pure triples

Philip Saddleton pabs at cantab.net
Thu Oct 12 19:02:32 UTC 2006

Alexander Holroyd said  on 12/10/2006 17:45:
> What is the simplest extent of pure triples?
>
> Obviously the meaning of "simplest" is subjective, but what I am driving
> at is that the construction should be straightforward, and should not
> require an extensive computer or hand search, and it should be
> self-evident that it works.
>
A six-part of spliced twin-hunt methods is probably the most
self-evidently true. There are examples on RW93 p163, including one of
2m by Pitman - it may be possible with fewer calls.

I think that the most straightforward construction is Quick Six Triples:

3.1.7.1.3.1.3.1.7.1.3.1.3.1.7.1.3.1.3.1.7.1.3.1.3.1.7.1.3.7

The coset graph for the Scientific group using these three PNs consists
of five hexagons with other links and this Hamiltonian cycle is easily
found. The blocks can be linked by replacing two quick sixes (the last
two for the composition below) by two slow sixes, traversing the
hexagons in reverse, and cunningly joining two blocks without
introducing any false rows.

5040 Quick Six Triples (PABS)
123456   4  6  7
----------------
415263   -  -  -
642315   -  -
465312      -
514623   -     -
256314   -  -
524316      -
351264   -  -  -
632451   -  -
361452      -
153624   -     -
216453   -  -
321546      -  -
----------------
Repeat

> Also, how many fundamentally different extents of pure triples are
> popularly known?  Not many it would seem...
>
> Ander
It depends on what you mean by fundamentally different.

A) By joining an even number of blocks:

Twin-hunt peals using a mixed q-set of 3,5,7. (Spliced, or Grandsire
with 3rds & 5ths place calls).
Swapping pairs of quick and slow sixes (Quick Six, or Nigel Newton's "Erin")
Magic Block Stedman (Wylde or Johnson/Saddleton)

B) Finding an odd number of blocks in the first place:

Johnson's 10-part of Stedman.
Brian Price's Spliced Stedman and Erin.

--
Regards
Philip