Infinite Series

Date: 03/12/2001 at 05:40:40
From: Alice Klein
Subject: Infinite Series/Number Theory

Suppose that we have two jugs, each containing one litre of water. 
Pour half of one jug into the other. Now we randomly choose any jug 
and pour half the water in it into the other jug. Can we ever reach a 
stage in which each jug again contains a litre of water? What if we 
pour exactly 1/2 of the water from any jug into the other jug?

This problem is from the Australian Maths Enrichment Series, and I 
have some problem trying to write a formal proof on it. Can you 
expain how I can tackle these kinds of problems of infinite series?

Thanks heaps!

Date: 03/12/2001 at 12:33:11
From: Doctor Rob
Subject: Re: Infinite Series/Number Theory

Thanks for writing to Ask Dr. Math, Alice.

I don't see the difference between your two scenarios.

You cannot get back to 1 litre in each jug in a finite number of
steps. You can prove by induction that after n pourings, the amount
of water in each jug has the form N/2^k, where N is an odd positive
integer. Certainly it is true for k = 0. If it is true for k, and
there is N1/2^k litres of water in the first jug, and N2/2^k litres
in the second jug, then when you divide the water in jug 1 in two
(for example), you will have N1/2^(k+1) litres, which, when added to
the water in jug 2, will give you (N1 + 2*N2)/2^(k+1) litres in
jug 2 (and similarly if you are pouring in the other direction).
Now if k > 0, N/2^k = 1 is impossible.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/