More chess masters reside in New York City than in the rest of the United States combined. We’re planning a chess tournament that all American masters are expected to attend, and we want to minimize the total intercity traveling done by the players. The New York players argue that, by this criterion, the tournament should be held in their city. The West Coast players argue that a city should be chosen near the center of the gravity of the players. Where should we hold the tournament?
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In New York. Label the New York players A1, A2, …, An, and label all the others B1, B2, …, Bm. Because most masters live in New York, n > m. Now let the straight-line distance between two given players, Ai and Bj, be denoted AiBj. If we hold the tournament in New York, the total distance traveled, T, will be A1B1 + A2B2 + … + AmBm. But if we hold the tournament elsewhere, then the total distance traveled by each pair Ai and Bi (i = 1, 2, … m) will be at least AiBi. Hence the total distance traveled by all the participants will be at least T plus the sum of the distances traveled by the unpaired New York masters Am+1, Am+2, …, An.
(Proposed and solved by Chester McMaster in Pi Mu Epsilon Journal 1:8 [April 1953], 328.)
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