November 19, 1998
OLD CHALLENGE. Walter Wright's neighbor Joan was entered to run a special 26.5-mile marathon, and she hoped to average under nine minutes per mile over the total distance. She had a number of friends measure her time over various mile segments of the course, and for each mile that was measured, IN FACT FOR EACH POSSIBLE MILE THAT COULD HAVE BEEN MEASURED (starting anywhere), her time was exactly nine minutes. Was she disappointed? NO--because she claims she met her goal of averaging UNDER nine minutes per mile! Is this possible?
ANSWER. Yes, it is possible, as best explained by Vladimir Annenkov, Jean-Pierre Carmichael, Timur Dogan, Joel Foisy, Art Pasternak, and John Snygg. For example, Joan could run the race in alternating four- and five-minute half-mile segments. Every mile would measure nine minutes. For example, if the mile starts 80% of the way through a four-minute segment, it would have 20% of that four-minute segment, the entire following five-minute segment, and 80% of the next four-minute segment, for a total of nine minutes. Nevertheless, since she starts and ends with a four-minute segment, there would be 27 four-minute segments and only 26 five-minute segments, and the average would be UNDER nine minutes per mile. Indeed, the total time would be 27x4 + 26x5 = 238 minutes, for an average of 238/26.5 = 8.98 minutes per mile. This can happen because the total distance is not an even number of miles.
It turns out that all that can be deduced mathematically is that she averaged over 8.83 minutes per mile (allowing unlimited speeds and neglecting any effects of Einstien's theory of special relativity).
It all goes to show how tricky averages and statistics can be. Walter Wright concludes, only somewhat facetiously, "In my work as a casualty actuary, the only difficult mathematical aspect is figuring out how to compute an average."
"HOW TO WIN MORE" by Norbert Henze and Hans Riedwyl (A K Peters) describes how to avoid sharing your lottery winnings by avoiding popular combinations of numbers. It does not however, emphasize the tiny probability of winning or how the odds favor the state over the participants. There are better places to place your treasures and hopes, such as in understanding math and life a little better.
NEW CHALLENGE. What are the largest and smallest objects anyone has ever seen with the naked eye? heard? felt? Answers should include estimates of sizes. (For the first part, the whole object must be seen, though not necessarily in full detail or from every side.)
Copyright 1998 Frank Morgan
Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat, which appears on the first and third Thursday of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.
NOTE TO READERS.Thanks to excellent reader response, Math Chat will continue here on the MAA web page, the first and third Thursdays of each month.