## Frank Morgan's Math Chat |

November 19, 1998

**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.