Meet Arthur, a young man about to fall in love for the first time. Arthur has promised his
mother that he will not marry the first girl who bewitches him, but being a romantic fellow
does intend to marry and live happily ever after with the first enamorata with whom he
experiences a quality of love greater than that in his first affair. How many affairs
should Arthur expect to have before he finds his intended spouse?
Shanghai Entrance Exam
The examination described below is taken by Shanghai high school graduates, all of whom have
studied mathematics, Chinese language, and English. Examinations for university entrance are given
in all three of these subjects; admittance to universities is largely based on total
scores attained on the three examinations. This version of the exam is only for students
from the city of Shanghai; similar extrance examinations are given elsewhere in China.
Some New Results on Nonattacking Chess Tasks
"Chess tasks" is a term for combinatorial problems that involve placing chessmen on square
or rectangular boards of side n^2 or m x n so as to meet certain provisos. They do not include
the most common type of problem, that of determining from a given position how to mate
in a specified number of moves. Indeed, chess tasks have almost nothing in common with chess
except for the use of chess pieces and how they move.
A Bonanza of Birthday Bewilderments
We are fascinated by coincidences. A pair of airplane hijackings to Cuba take place within 24 hours
of each other, the same machine in a plant breaks down twice within two days while other machines
continue to work smoothly, or in your advanced Calculus class of 24 there are two students
with the same birthday. The culprit here is the birthday problem.
March Mathness
The National Collegiate Athletic Association holds tournaments and competitions to determine national
championships in college and university sports. Probably the most popular of these is the
Division I men's basketball tournament, known to sports fans as "March Madness." In this
highly publicized competition, 64 schools are selected to compete in a single elimination
tournament. The schools are assigned a seed from 1 to 16, and placed into one of four
regions with the lower-numbered seeds representing the teams thought to have the best chance
of winning. Each region produces one winner who proceeds to the Final Four where a national
champion is crowned. A glance at the structure of the starting field for the 2000 tournament
suggests that the design and outcome of such
tournaments are rich in mathematical and statistical content.
What to Do on Your Summer Vacation
Summer vacation brings back fond memories; playing frisbee in the park, bike-riding till dusk,
sipping lemonade on the porch with Grandpa, collecting shells at the beach, solving that
difficult math problem you've been working on for the past several weeks. Wait a minute, you
say? Many mathematics majors don't realize that numerous summer opportunities exist (many
of them paid, so you don't need to get that job bagging groceries, too).
The Mathematical Web
In 1969 Stanford Research Institute, UCLA, UC Santa Barbara, and the University of Utah hosted
the first network of time-sharing computers (known as ARPANET). This was the birth of the
Internet. From this humble beginning, the Net has grown at an astronomical rate: there
were 1,000 hosts in 1984; 10,000 hosts in 1987; 100,000 hosts in 1989; 1,000,000 hosts in 1992;
10,000,000 hosts in 1996; and over 70,000,000 in January 2000. The first graphical
browser, Mosaic, was available in 1993, making the World Wide Web as we know it a reality.
But how can you, as a mathematician, effectively use the cyber-resources available to you?
Mathematical Life at the National Security Agency
Over the last several years, the visibility of the National Security Agency has increased
greatly in both the popular media and the mathematical community. While NSA mathematicians
have always maintained working relationships with academicians, NSA has only recently become a
major contender for U.S. mathematical talent. The Agency now recruits mathematicians at
all degree levels in unprecedented numbers, by visiting schools and attending national
meetings in search of potential applicants.
Problem Section
S-48.
Proposed by RGee Watkins, Hemet, CA. Four interlocking squares enclose ten regions, as shown in the diagram.
Each is to be filled with one of the integers 1 to 10. Each integer is used exactly once,
and the sum of the integers inside each of the four squares is the same. Which integer
is in the region marked by the question mark?
S-49.
Proposed by E. M. Kaye, Vancouver, BC. Let A_{n+1} = x^{A_n} for n = 0, 1, 2,... and
A_0 = 25 log 50 (base 5). Determine all real x satisfying the equation A_3 = 5.
S-50.
Proposed by E. M. Kaye, Vancouver, BC. Given that a_k a_{k+1} = b_k^p, k=1, 2, ..., 2n + 1; p
is a positive integer; and a_{2n + 2} = a_1. Prove that if p = 2m then a_k is a perfect m power
and if p = 2m + 1 then a_k is a perfect 2m + 1 power.
S-51.
Proposed by Mircea Ghita, Stuyvesant High School. Solve the equation
(sin x)^{1/n} + (cos x)^{1/n} = 1
for all x in [0, 2 pi] where n is an integer >=2.
The Final Exam: Sing! In the Name of Math
It happens at the most inconvenient times. You're tooling down the road with a friend,
coming back from a day at the beach or in the mountains, somewhere peaceful and relaxing.
You've left all your daily concerns way behind you and you feel wonderful. You've got the
car radio cranked, and you've been singing along with all the good traveling songs. Yes,
the Eagles are definitely your favorite band, at least right now: "So put me on a highway
and show me a sign, and take it to the limit one more tiii-iiime." Aaaaaack! Limits!
