[r-t] Fwd: bobs-only Grandsire Triples
pje24 at cantab.net
Tue Jan 31 15:11:09 UTC 2017
The following is a message from Roy Dyckhoff, who is having technical
difficulties sending messages to this list himself.
I have a paper of about 8 pages on Thompson's famous 1886 paper about
bobs-only Grandsire Triples. It begins with (as background) a short
proof of his main result, eschewing both group theory and case analysis.
The novelty is that I reject his argument that there is no touch of
greater than 5000 changes; I don't claim that there is such a touch, but
his argument is evidently fallacious. Perhaps the search for such a
touch ceased in 1886; but it seems to me that someone might have another
look. Modern computers might ease this tedious task.
The crucial point is to consider Q-sets as Q-cycles, and to accept his
point that if a row in such a cycle is Plained then the next cannot be
Bobbed. If one sees a touch as determining for each row in a Q-cycle
whether it is Plained, Bobbed, or Absent, then there is the possibility
of patterns such as ABPPP, which has no P immediately followed by a B.
Some of the Q-sets in a long touch will be complete, and so either fully
Plained or fully Bobbed; but some will include changes not in the touch,
and here the argument about "all are Plained or all are Bobbed" falls
down. There is a counterexample of 280 changes to the argument.
Happy to send the paper to those interested.
University of St Andrews
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