Infinite SeriesDate: 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/ |
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