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PUZZLE
EXPLORING CARTER’S CRATER
On Mercury there is a crater which, for this problem’s pur poses, we assume to have a perfectly circular rim. It is called Carter’s Crater after President Jimmy Carter’s greatgrandson, who became an astronaut and was the first to set boot on Mercury. Carter landed near the crater that now bears his name. At two random spots on its rim he estab lished supply stations. Although Carter returned to earth safely, an unfortunate fire on the spaceship destroyed his records of where the supply stations had been placed. Two years later, astronauts Smith and Jones were sent on a mission to explore Carter’s Crater. They landed at a random spot near the rim, picked a direction (clockwise or counterclockwise) by flipping a flat piece of Mercury rock, and started walking along the rim to the nearest supply station. “Assuming that the two stations, and the spot on the rim where we started walking, are all randomly and inde pendently selected points,” said Ms. Jones, “how far is the expected distance we have to walk before we reach the first station?” “You mean,” said Dr. Smith, “that if we repeated this event many, many times, what in the long run would be the average distance we would have to travel?” “Precisely,” said Ms. Jones. “Of course we have to in clude in each repetition the initial random selection of two points for the two supply stations. I figure it like this. The second station can be any distance from the first, from zero to the length of the crater’s circumference. So the average distance will be half the circumference. Now we had the same chance of landing in one of those semicircles as the other. In either case our average distance from the two sta tions would be half a semicircle. Therefore the expected distance to the nearest station must be one-fourth of the circumference.” “Sounds plausible,” said Dr. Smith, whose Ph.D. was
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