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From: pmontgom@sdcrdcf.UUCP (Peter Montgomery)
Newsgroups: net.math,net.puzzle
Subject: Re: A problem about lists of points
Message-ID: <1558@sdcrdcf.UUCP>
Date: Sat, 15-Dec-84 18:15:14 EST
Article-I.D.: sdcrdcf.1558
Posted: Sat Dec 15 18:15:14 1984
Date-Received: Sat, 26-Jan-85 06:10:16 EST
References: <203@cmu-cs-g.ARPA>
Reply-To: pmontgom@sdcrdcf.UUCP (Peter Montgomery)
Organization: System Development Corp. R+D, Santa Monica
Lines: 22
Keywords: golden ratio
Summary: 

> Consider a (bounded) line-segment.  Choose point 1 anywhere on the segment.
> Then choose point 2 so that the first two points lie in different halves of
> the segment; choose point 3 so that the first three points all lie in
> different thirds of the segment; etc.  What is the maximum number of points
> you can choose (before further choice becomes impossible)?

> It is convenient to consider a half-open segment, say [0,1[, and to regard
> 1/2 as belonging to the second half, 2/3 to the third third, 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.  I recall a picture in one of Donald Knuth's volumes
of "The Art of Computer Programming" showing this sequence, but cannot
locate that now.
-- 
			Peter Montgomery

	{aero,allegra,bmcg,burdvax,hplabs,
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