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From: ldenenbe@bbncc5.UUCP (Larry Denenberg)
Newsgroups: net.puzzle
Subject: Re: Find a set of positive real numbers whose sum is 100 and whose product is as large as possible.
Message-ID: <1692@bbncc5.UUCP>
Date: Thu, 13-Feb-86 12:01:16 EST
Article-I.D.: bbncc5.1692
Posted: Thu Feb 13 12:01:16 1986
Date-Received: Sun, 16-Feb-86 05:04:38 EST
References: <1057@decwrl.DEC.COM>
Reply-To: larry@bbncc5.UUCP (Larry Denenberg)
Organization: Bolt Beranek and Newman, Cambridge, MA
Lines: 26
Summary: Can't be done

It can't be done.

Let S be any set of positive real numbers whose sum is 100.  If S has one
element then it is {100} and the set {49,51} has greater product.  Otherwise
S has two elements x and y with x < y.  Choose any z such that 0 < z < y-x
and neither x+z nor y-z is in S---this is obviously possible if S is countable
(but if not then S has an infinite number of elements less than 1 and its
product is as close to zero as makes no never mind).  Replace x and y in S
with x+z and y-z.  The sum of S is still 100, and the product has increased
since (x+z)(y-z) = xy + (y-x)z - zz > xy.

OK, how close can we get?  Well, the argument above, stripped of formality,
just says that if two numbers in S are different you can improve the product
by bringing them closer together.  If S were not a set but a multiset, this
process could stop only when all the elements in S were equal.  If we look at
the function x**(100/x) we find (not surprisingly) that x = e gives the
global maximum.  So S should have 100/e copies of e.  But 100/e is not an
integer;  it's between 36 and 37.  The best multiset is thus either
36 copies of 100/36 or 37 copies of 100/37;  probably the latter, since
100/e is 36.78 or so.  To get a good set S, just take 37 copies of 100/37
and perturb them a hair:  pick z = 10**(-10000000000), say, and let
  S = { 100/37 - z, 100/37 + z, 100/37 - 2z, 100/37 + 2z, ... }
where S has 37 elements and the coefficients of the z's add up to 0.  This
gives a good solution, but picking a smaller z will always give a better one.

/Larry Denenberg
larry@{bbncc5.ARPA,harvard.harvard.EDU}