Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83 (MC840302); site boring.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!whuxlm!harpo!decvax!genrad!mit-eddie!godot!harvard!seismo!mcvax!boring!lambert 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