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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):<br>
<br>
Alexander Holroyd (3 February) says<br>
<blockquote>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.<br>
</blockquote>
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.<br>
<br>
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). <br>
<br>
He also says<br>
<blockquote>I thought Thompson was the senior wrangler at
Cambridge? That wouldn't square too well with the assertion that
he was "not a mathematician".<br>
</blockquote>
<a class="moz-txt-link-freetext" href="https://en.wikipedia.org/wiki/Senior_Wrangler_(University_of_Cambridge)">https://en.wikipedia.org/wiki/Senior_Wrangler_(University_of_Cambridge)</a>
lists the Senior Wranglers from 1748-1909; I don't see Thompson's
name. I cannot vouch for the accuracy of this list; see
<a class="moz-txt-link-freetext" href="https://en.wikipedia.org/wiki/Talk:Senior_Wrangler_(University_of_Cambridge)">https://en.wikipedia.org/wiki/Talk:Senior_Wrangler_(University_of_Cambridge)</a>
for discussions thereon. However, many distinguished mathematicians
did the Tripos but were not Senior Wranglers. <br>
<br>
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 <a class="moz-txt-link-freetext" href="http://www-history.mcs.st-and.ac.uk/Indexes/T.html">http://www-history.mcs.st-and.ac.uk/Indexes/T.html</a>. I am
aware of many tools that I choose not to use, or that I use only for
discovery rather than for public presentation.<br>
<br>
<br>
Martin Bright (1 February) reports that <br>
<blockquote>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.<br>
</blockquote>
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. <br>
<br>
<br>
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. <br>
<br>
Thanks again for comments!<br>
<br>
R
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