Mathematical Brainteasers with Surprising Solutions by Owen O'Shea & Owen O'Shea

Author:Owen O'Shea & Owen O'Shea [O'Shea, Owen]
Language: eng
Format: azw3
ISBN: 9781633885851
Publisher: Prometheus
Published: 2020-04-13T16:00:00+00:00

SOURCE

Martin Gardner, Mathematical Circus (London: Penguin Books, 1981), pp. 188–90.

73

AGAINST THE WIND

You are told that a cyclist on a 1-mile journey rode his bicycle at the rate of 1 mile in 3 minutes with the wind to his back, but on the return trip with the wind blowing against him, he managed to cycle at the rate of only 1 mile in 4 minutes. If he always applied the same force to his pedals, how long would it take him to ride 1 mile if there were no wind?

SOLUTION

On being asked this puzzle, most people, it appears, answer it as follows. The cyclist went on a journey of 1 mile. He rode the outward part of the journey at 3 miles per minute and rode the return part of the trip in 4 minutes. Therefore, if there were no wind, his speed must have been one-half of 3 plus 4, or 31⁄2 miles per minute.

This answer appears plausible, but it is incorrect!

The surprising answer is that if there were no wind, the cyclist would ride 1 mile in 3 and 3⁄7 minutes.

The incorrect answer is arrived at by assuming that the effects of the wind on the cyclist are the same for the outward and homeward trips. But this is not the case. On the outward journey, the wind has helped the cyclist for only 3 minutes. On the homeward trip, the wind has hindered the cyclist for 4 minutes.

One method of obtaining the correct solution is as follows.

On the outward journey, with the wind blowing behind him, the cyclist travels 1 mile in 3 minutes. On the homeward trip, with the wind blowing against him, the cyclist travels 1 mile in 4 minutes, which is equivalent to three-quarters of a mile in 3 minutes. It is this 3-minute time period in both directions that we are interested in. Since the wind is blowing in the cyclist’s favor for 3 minutes on the outward trip and the wind is blowing against him for 3 minute on the return trip, the effects of the wind on the cyclist cancel.

We know that in this 6-minute period, with the effects of the wind cancelled, the cyclist travels 13⁄4 miles. This is equivalent to the cyclist—in 24 minutes—traveling 7 miles, which is equivalent to cycling 1 mile in 33⁄7 minutes.

Thus, with no wind assisting him or hindering him, the cyclist will cycle 1 mile in 33⁄7 minutes.

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