[r-t] bobs-only Grandsire Triples
Roy Dyckhoff
roy.dyckhoff at googlemail.com
Fri Feb 3 13:02:55 UTC 2017
I submitted details of a draft paper on bobs-only Grandsire Triples,
which I am revising in light of comments. Here is my response to the
comments (for which I am most grateful):
Alexander Holroyd (3 February) says
the "modern" way to establish the non-existence of a bobs-only
extent is simply that it is an immediate special case of Rankin's
theorem (of which Swan's is the "book proof"). Of course, Thompson
still gets full credit for the first proof. Rankin's and Swan's can
be seen as generalizations that closely follow Thompson's approach.
I agree; but would also (as a mathematical logician, specialising in
proof theory) argue that non-modern ways can also be of value,
especially if they are short. The point of my paper, however, is not so
much to offer a splendid (ancient or modern) proof of Thompson's result
about extents but to argue that Thompson's 'proof' of the result that no
bobs-only touch can exceed 4998 changes is incorrect. For this purpose
it is helpful to have, as background, a direct proof of the result about
extents rather than to have it as a special case of a more general result.
Incidentally, I now suspect the proof (of Thompson's Theorem) that I
wrote out on my typewriter about 45 years ago and only recently have
bothered to typeset properly is just an adaptation of that of Dickinson
(1957).
He also says
I thought Thompson was the senior wrangler at Cambridge? That
wouldn't square too well with the assertion that he was "not a
mathematician".
https://en.wikipedia.org/wiki/Senior_Wrangler_(University_of_Cambridge)
lists the Senior Wranglers from 1748-1909; I don't see Thompson's name.
I cannot vouch for the accuracy of this list; see
https://en.wikipedia.org/wiki/Talk:Senior_Wrangler_(University_of_Cambridge)
for discussions thereon. However, many distinguished mathematicians did
the Tripos but were not Senior Wranglers.
I have no information or strong opinion on whether or not Thompson was a
mathematician (e.g. in the sense of having a mathematics degree) or was
aware of group-theoretic tools. His name does not appear in
http://www-history.mcs.st-and.ac.uk/Indexes/T.html. I am aware of many
tools that I choose not to use, or that I use only for discovery rather
than for public presentation.
Martin Bright (1 February) reports that
A student here in Leiden wrote a nice undergraduate thesis on this
topic with me last year. He also noticed that the argument about
the maximum length of a bobs-only touch doesn't work, and came up
with a new argument along similar lines.
This is indeed (so far as I know) a new argument, which I believe to be
correct, in a very nice thesis, introducing the new concept of an
extended Q-set. Well done the student (G.L. van der Sluijs)! My own
paper gives, I believe, more detail of just how Thompson's proof is
incorrect. But van der Sluijs deserves, so far as I can see, the credit
for the first, indeed only, correct proof of the Theorem that there is
no touch of length greater than 4998.
Andrew Johnson (1 February) correctly criticises my use of the word
"impossible" to describe a certain case analysis, exponential in the
number of bells. I stand rebuked. The analysis can certainly be reduced
substantially to a lower complexity: the Leiden thesis does just that,
along lines also identified by Johnson; I now do this too, but by a
different approach. What I would like to see, however, is a short proof
(that bobs and plains generate the relevant group, isomorphic to
Alt(4n+2)) that is of ***constant*** (rather than exponential, quadratic
or even linear) complexity in the number of bells (we are considering
the method on 4n+3 bells, for n = 1,2,3, etc). But maybe there is no
such proof; and maybe the lack thereof is of no importance or interest.
Thanks again for comments!
R
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