A teacher gave her class a question that sounded harmless. A driver goes from one town to the next at thirty kilometres an hour. He wants his average speed for the whole journey, there and back, to be sixty. How fast must he drive on the way back?
Almost every hand went up with the same answer: ninety. Thirty and ninety, they said, average sixty.
The teacher asked them to check it with the clock instead of with their instinct. Suppose the towns are sixty kilometres apart. Going there at thirty takes two hours. The whole journey is a hundred and twenty kilometres, and to average sixty he must finish it in two hours.
The room went quiet. He has already used the two hours. The return trip must take no time at all.
"So the answer is not ninety," she said. "The answer is that there is no speed. Not a large one, not an enormous one. None."
One boy said that if the driver went fast enough he would surely get close. The teacher agreed, and said that was exactly the interesting part. At a hundred and fifty the average is fifty. At three hundred it is a little over fifty-four. At a thousand it is a little over fifty-eight. The average creeps towards sixty and never arrives.
"Some questions have an answer you cannot reach," she said, "and the arithmetic tells you so before you waste a single litre of fuel finding out."