## Frank Morgan's Math Chat |

July 1, 1999

**SHOR AND WEEKS WIN MACARTHUR AWARDS.** Winners of the thirty-two MacArthur
Foundation "genius awards" announced last week include computer scientist
Peter Shor and mathematician Jeffrey R. Weeks. Each receives about $300,000
over five years to pursue his own interests. Shor, at AT&T Research Labs,
has shown how the effect of quantum mechanics at small scales could
theoretically enable a computer to quickly solve problems that would
otherwise take longer than the age of the universe. Weeks, unaffiliated
with any university or business, is perhaps the most notable free-lance
mathematician working today. His computer program SNAPPEA will be used on
data from the 2001 Microwave Anisotropy Probe (MAP) satellite to help
cosmologists figure out the shape of the universe.

**LEIBNIZ AND FOLDING TRIANGLES.** John Sullivan observes that even Leibniz,
inventor of the calculus, originally made the mistake addressed in the June
4 Math Chat on WHY IT'S HARD TO FOLD A TRIANGLE IN HALF. Leibniz wrongly
thought that any fold though the center of a triangle would divide the area
in half and wrote of a conversation with Huygens:

. . . I thought that a straight line drawn through the center of gravity always cut a figure into two equal parts; since that clearly happened in the case of a square, or a circle, an ellipse, and other figures that have a center of magnitude, I imagined that it was the same for all other figures. Huygens laughed when he heard this, and told me that nothing was further from the truth.

(C. H. Edwards, Jr., The Historical Development of the Calculus, Springer, 1979, p 239.)

**OLD CHALLENGE.** Neville Barnard likes the equation 2 + 2 = 2 x 2, and asks
for other similar equations, involving two or more numbers on each side of
the equation.

**WINNING ANSWER** (Joseph DeVincentis):

3 + 1.5 = 3 x 1.5

p + p/(p - 1) = p x p/(p - 1)

1 + 2 + 3 = 1 x 2 x 3

Eric Brahinsky, in a very thorough analysis, also provides some using exponentials:

2 + 2 = 2^{2}

.346323362 + .346323362 = .346323362^{.346323362 } (approximately)

(the only two numbers that work like this?)

**NEW CHALLENGE** (Joe Shipman). Abel writes down two distinct positive
integers from 1 to 4. Beta sees one of them at random and tries to guess
whether it is the larger of the two. What are Beta's chances? (What if Abel
can choose from more numbers?)

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. Math Chat
appears on the first and third Thursdays of each month. Prof. Morgan's
homepage is at www.williams.edu/Mathematics/fmorgan. Prof. Morgan was just
elected as Second Vice-President of the Mathematical Association of
America.

Copyright 1999, Frank Morgan.