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From: lambert@boring.UUCP
Newsgroups: net.math
Subject: Re: list of points problem
Message-ID: <6268@boring.UUCP>
Date: Sat, 29-Dec-84 22:59:53 EST
Article-I.D.: boring.6268
Posted: Sat Dec 29 22:59:53 1984
Date-Received: Mon, 31-Dec-84 02:57:21 EST
References: <203@cmu-cs-g.ARPA> <25@epsilon.UUCP> <208@cmu-cs-g.ARPA>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 32
Summary: Apparently-To: rnews@mcvax.LOCAL

Like Ed Sheppard (25@epsilon.UUCP) and Peter Monta (208@cmu-cs-g.ARPA),
I found, by exhaustive search, solutions up to 17 points and none above.
Since these programs were developed independently, this is strong evidence
that the maximum number of points is indeed 17.
    However, my program gave a different result for the lexicographically
first solution, namely:
        p1:[0,1/17)     p2:[4/7,7/12)    p3:[6/7,13/15)  p4:[2/7,5/17)
        p5:[8/11,11/15) p6:[5/11,6/13)   p7:[1/7,2/13)   p8:[13/14,14/15)
        p9:[3/8,5/13)   p10:[9/14,11/17) p11:[3/14,3/13) p12:[11/14,4/5)
        p13:[1/2,9/17)  p14:[1/14,2/17)  p15:[16/17,1)   p16:[5/16,6/17)
        p17:[11/17,12/17)
(``pi:[lo,hi)'' means, of course, ``lo <= p[i] < hi''.)  The solution
reported by Peter Monta came sixth.

Why the maximum is 17, I do not know either.  I take it that ``Why'' means
here: ``Give a short proof''.  The program could easily be made to output a
proof, but that would be rather long.  But maybe all proofs of the theorem
``max length = 17'' are long.

Some more data that may help to find a shorter proof: There are 1536
(= 2^9*3) solutions, half of which are the mirror images of the other half.
It is possible to identify a solution by the permutation formed by p1-p17.
For example, for the first solution (given above) that permutation is
1af5d83g7b4e92h6c, where a-h stand for 10-17.  Now in all 1536 solutions,
p13 gets the code 9, so it is the median point.  Furthermore, p5-p6 are
always either 5a or its mirror image, d8.  Also, p9 is either 7 or b, p11
is either 4 or e, and p16 is either 6 or c.
-- 

     Lambert Meertens
     ...!{seismo,philabs,decvax}!mcvax!lambert@boring.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam