## Frank Morgan's Math Chat |

**ANSWER**. Neil Bitzenhofer, Jean-Pierre
Carmichael, Joe Conrad, Joel Foisy, G.
A. Michael, Jon Slutzman, and Walter Wright find a way to do it in one
trip. Turn on the first switch, wait ten minutes, and then turn off the
first switch and turn on the second switch. Run upstairs. If the light
is off but hot, it's the first switch. If the light is on, it's the
second switch. If the light is off and cool, it's the third switch.

Dean Thomas rightly advises waiting to make sure that all the switches have been off for a while before you start.

Erik Randolph creatively argues that, "There is no need to run upstairs at all: [just] watch for the blue spark, or even the buzzing sound [from] very, very slowly turning it on and off." His other methods involve a probe or compass. Eric Zeitler would "unscrew the cover plate and check the wiring to see which swich is connected."

"*THE PENGUIN DICTIONARY OF CURIOUS AND INTERESTING NUMBERS*" by David
Wells includes lots of fascinating PRIME numbers (not divisible by any
smaller number except 1). One example is 82 81 80 79 78 . . . 1 (with
all the numbers from 82 down to 1 in descending sequence). A second,
111. . .1 consists of 1031 repeating units 1; discovered by Williams
and Dubner in 1986, it is the largest known repunit (repeating-unit)
prime. A third prime has 1104 digits and all of them are prime. A
fourth, 1999. . .9 consists of a one followed by 3020 nines. The
largest prime listed has already been surpassed by 2^3021377 - 1,
discovered on January 27, 1998, by Roland Clarkson, a student at
California State University Dominguez Hills, from Norwalk, California.
It has almost a million digits.

**NEW 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?

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.

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