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May 20, 1999

OLD CHALLENGE (John Snygg). A mathematician, lost in the desert, hears a single toot from a train due west. He knows that the train goes at constant speed in a straight line, but he forgets in which direction. What direction should he walk?

ANSWER (Joe Shipman). The most interesting question turns out to be: Which direction provides the best chance of reaching the tracks in time to intercept the train? The answer is: due north or due south. Of course if the train is heading westward, you have no chance, unless you can outrun a train. Your only hope is that the train is heading a relatively small angle q north of east [or south of east, you have to guess which]. In this case, due north gives you the best chance. You will reach the track before the train if the ratio of your speed to the train's speed is greater than the sine of q. This is true for anyone, no matter how far east of the toot. If you take some other direction to the track, you'll travel farther than someone else heading due north to the same spot, and thus need a greater velocity.

Unfortunately, even if the mathematician runs due north at 1/4 the speed of the train, he has only about a 4% chance that the train is heading a small enough amount north of east for him to intercept it.

Shipman concludes, "Personally, I would . . . go due west and flag the next train, judging that it is more likely there'll be another train before I die of thirst." Many readers agree. In particular, Joe Conrad agrees that this mathematician should not expect to be lucky enough to intercept the first train: "True, he could get lucky, but he's already lost in the desert so luck is not part of this mathematician's 'track' record."

John Snygg reports that at a problem-solving conference, he was the only one to solve this problem. His more complicated solution found the circle of points where he and the train might arrive simultaneously, drew the most distant track to meet that circle, and showed that it met the circle due north. The most elegant solution he encountered was one by a Russian student, Michael Chetchelnitsky, who used the law of sines.

NEW CHALLENGE (John, RiverGlen School). For a circular piece of paper, the lines along which you can fold it "in half" (with half the area on each side) are precisely the lines through the center. What other shapes work the same way?

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Copyright 1999, Frank Morgan.