Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83 (MC840302); site mcvax.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!philabs!cmcl2!seismo!mcvax!lambert
From: lambert@mcvax.UUCP (Lambert Meertens)
Newsgroups: net.math,net.puzzle
Subject: Re: A problem about lists of points
Message-ID: <6259@mcvax.UUCP>
Date: Tue, 18-Dec-84 04:17:46 EST
Article-I.D.: mcvax.6259
Posted: Tue Dec 18 04:17:46 1984
Date-Received: Thu, 20-Dec-84 05:35:37 EST
References: <203@cmu-cs-g.ARPA> <1558@sdcrdcf.UUCP>
Reply-To: lambert@mcvax.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 43
Xref: watmath net.math:1663 net.puzzle:499
Summary: 

>> Consider a (bounded) line-segment.  Choose point 1 anywhere on the segment.
   [etc.]
> I believe you can go as far as you want.  Use the fractional parts of
> multiples of the golden ratio (SQRT(5)-1)/2, or about 0.618.  The sequence
> 0.618, 0.236, 0.854, 0.472, 0.090, 0.708, 0.326, ... seems to satisfy the
> requirements.

Not so.  Take n = 7, and consider p[i] = (i*phi) mod 1 --> floor(p[i]*n):

        p1 = 0.618 --> 4
        p2 = 0.236 --> 1
        p3 = 0.854 --> 5
        p4 = 0.472 --> 3
        p5 = 0.090 --> 0
        p6 = 0.708 --> 4
        p7 = 0.326 --> 2

So both p1 and p6 fall in segment 4, and the last segment, 6, is not
represented.

I have not (yet) tried to prove this, but it appears extremely unlikely
to me that an infinite sequence could exist for which every initial
segment is evenly distributed.
Here is as far as I came using a backtracking method:

         0   <= p1  < 1/10
        1/2  <= p2  < 5/9
        3/4  <= p3  < 7/9
        1/4  <= p4  < 2/7
        7/8  <= p5  < 8/9
        3/8  <= p6  < 2/5
        5/8  <= p7  < 2/3
        1/8  <= p8  < 1/5
        9/10 <= p9  <  1
        2/5  <= p10 < 1/2

I do not claim that this is the only solution up to 10, nor
that it cannot be extended.
-- 

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