Solution to Problem 14

Congratulations to this week's winners

Mike Fitzpatrick, Ray Kremer

who will share this week's prize.

Correct solutions were also received from Jim Otermat and Eric Grennan. Many incorrect submissions were received.

Yes, the ant does reach the end of the strip of rubber, in about 10 minutes and 47 seconds. Many of you failed to notice that the point at which the ant stands also would be moved forward during each stretch.

Think of it this way. During the first minute, and before the first stretch, the strip is 3 feet long and the ant travels 1 foot, so it travels 1/3'rd of the way down the strip. During the second minute, after the first stretch but before the second one, the strip is 6 feet long and the ant travels 1 foot, so it travels 1/6'th of the way down the strip. In general, during the n'th minute, the strip is 3n feet long and the ant travels 1 foot, so it travels 1/3n'th of the way down the strip. The first question, whether it ever reaches the end of the strip, is answered by observing that the sum

1/3 + 1/6 + 1/9 + 1/12 + ...
will eventually exceed 1; in fact, the last time it's less than 1 is at 1/3 + 1/6 + 1/9 + ... + 1/27 + 1/30 which is about 0.9763. So, after 10 minutes the ant has traveled about 97.63% of the way down the strip, the strip being 33 feet long at that time (after the stretch).

When exactly does it get there? It took the ant 10 minutes to travel first 97.63% of the way. The ant had about 33(1 - .9763) feet to go and, since it travels 1 foot per minute, it reaches the end in about another 46.88 seconds. An exact calculation yield a time of 10 minutes seconds.

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Page last updated 21 September 1997.