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 A Boy, a Girl, and a Dog (Posted on 2004-01-24)
A boy, a girl and a dog are standing together on a long, straight road. Simultaneously, they all start walking in the same direction: The boy at 4 mph, the girl at 3 mph, and the dog trots back and forth between them at 10 mph. Assume all reversals of direction instantaneous.

In one hour, where is the dog and in which direction is he facing?

 Submitted by DJ Rating: 4.0000 (8 votes) Solution: (Hide) Cannot be determined. The dog's position and direction are indeterminate, other than that the dog must be between the boy and girl (endpoints included). To see this, simply time reverse the problem. No matter where the dog starts out, the three of them wind up together in one hour. This argument is not quite adequate. It is possible to construct problems where the orientation changes an infinite number of times initially, but for which there can be a definite result. This would be the case if the positions at time t are uniformly continuous in the positions at time s, s small. But suppose that at time a the dog is with the girl. Then the boy is at 4a, and the time it takes the dog to reach the boy is a/6, because the relative speed is 6 mph. So the time b at which the dog reaches the boy is proportional to a. A similar argument shows that the time the dog next reaches the girl is b + b/13, and is hence proportional to b. This makes the position of the dog at time (t > a) a periodic function of the logarithm of a, and thus does not approach a limit as a -> 0.

 Subject Author Date I beg to differ with everybody jduval 2013-09-24 02:12:27 the solution is posted, but... sean 2005-08-23 02:08:12 Here Lassie ! Wendy 2004-02-18 17:19:47 re: Sit !! Stay !! Richard 2004-01-27 14:36:27 tee hee Cindy 2004-01-27 14:03:51 Any which way Jils 2004-01-26 08:55:19 Barking mad solution retiarius 2004-01-26 05:53:46 re: Sit !! Stay !! Ady TZIDON 2004-01-25 07:15:47 Sit !! Stay !! Penny 2004-01-24 22:47:48 re: Question: "Abandon All Common Sense..." TomM 2004-01-24 20:51:54 re: Question Richard 2004-01-24 20:43:45 Question Penny 2004-01-24 20:02:59 thoughts Charlie 2004-01-24 14:34:48 re: Official solution (spoiler) Richard 2004-01-24 12:25:45 Official solution (spoiler) Penny 2004-01-24 10:24:18

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