[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 

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

lists the Senior Wranglers from 1748-1909; I don't see Thompson's name. 
I cannot vouch for the accuracy of this list; see 
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!

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