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From: wolit@rabbit.UUCP (Jan Wolitzky)
Newsgroups: net.math
Subject: Re: Rinsing Puzzle
Message-ID: <2554@rabbit.UUCP>
Date: Wed, 29-Feb-84 10:14:54 EST
Article-I.D.: rabbit.2554
Posted: Wed Feb 29 10:14:54 1984
Date-Received: Fri, 2-Mar-84 07:24:12 EST
Organization: AT&T Bell Laboratories, Murray Hill
Lines: 20

Let X = the original volume of water in the water glass, 
Y = the volume remaining in the milk glass after dumping it,
and Z = the amount of water transferred in each rinse.  The algorithm 
is:  dump the milk glass, add Z units of water from the water glass,
dump the mixture, and repeat until you run out of rinse water.  
After the initial dump, there are Y units of milk in the glass.  
After the first rinse, there are (Y) * (Y / (Y+Z)) units; 
after two rinses, (Y) * (Y / (Y+Z)) * (Y / (Y+Z)), etc.  
You can rinse (X/Z) times before you run out of rinse water, 
so the total volume remaining when done is (Y) * ((Y / (Y+Z)) ** (X/Z)).  
Since X, Y, and Z are all positive numbers, with Y and X constant, 
this expression is minimized when Z is minimized;  
i.e., when an infinitesimal amount of rinse water is used 
an infinite number of times.
Note that this is equivalent to rinsing with a continuous stream of
water (at a rate that allows complete mixing), which may (but probably
doesn't) explain the presence of a faucet, rather than a series of 
buckets, in most kitchen sinks.

	Jan Wolitzky, AT&T Bell Labs, Murray Hill, NJ