Coal Miner's Daughter

I grew up in a very small town. My father worked in a coal mine. He was an engineer in a coal mine, and of course while I was growing up I didn't notice, but in retrospect coal-mining towns are very special kinds of small towns. I mean, there is just one big employer and he controls the whole life, even the social life, of the town. My mother, well, for her generation in Belgium it was not common to have university education, but she did. She had expected to have a career, but after marrying my father she didn't. Partly because there was no opportunity, but also because in this very small paternalistic town, two generations behind the wide world, it just was not done for wives of engineers to work. There was one wife who worked, she was a nurse, and everybody knew that that was why her husband never got a promotion. It wasn't said that way, of course, and even then they couldn't write things down that way. So my mother didn't work.

Beware of Geeks Bearing Grifts

What is a grift? It is circus slang for a swindle, a game rigged so the customer is at a disadvantage. The guys operating the games in the midway are known as grifters. As for geeks, I don't know how to define them, but I recognize them when I see them. If you are wondering what this has to do with mathematics, read on.

The Wild Numbers

That first night, I set out in high spirits, following the path Riedel had cut into the flanks of the Wild Number Problem. The point that he had reached in 1912, his proof that there were an infinite number of tame numbers, served me as a base camp. From there, equipped with my specialized mountain gear, that is, with Dimitri's new concepts, I could continue my ascent. Every step I took required my fullest concentration; now and then I had to stop to catch my breath, giving me a moment's rest to marvel at the wondrous mathematical landscape all around me.

GIMPS Finds Another Prime!

YOU TOO CAN BE A WINNER!! No, this pitch is not coming from Ed McMahon or the Publishers' Clearing House. If you own a 200MHz (or better) Pentium computer, you too can participate in the Great Internet Prime Search (GIMPS). The Electronic Frontier Foundation is offering $100,000 to the individual or team which finds the first ten-million-digit prime number; and up to $250,000 for larger primes. In 1996 George Woltman, using networking software by Scott Kurowski, created the database GIMPS, which coordinates the efforts of more than 8,000 computer users internationaly in a practice known as "distributed internet data processing." Woltman's powerful "Prime Net" server distributes and collects work from participating individuals, effectively operating as a single, gigantic parallel processor. Its ambition is to find whether certain large numbers are prime.

Three Estimation Challenges

Do you pick up pennies? Most people, it seems, don't bother anymore-and with good reason. It's hard to see how picking up a penny or two will have any material effect on your life, unless you're just short of the cash you need for that doughnut you'd like to buy. One way to decide if it's worth your time to pick up pennies is to perform the following thought experiment: imagine that there are an unlimited number of pennies scattered all over the ground, far enough apart that you could only pick up one at a time. Estimate how long it would take you to bend down for each coin, then figure how much money you'd collect in an hour: Would you be willing to do such menial labor for the hourly wage you've computed?

The Duplicity of Two

As we round out the second millennium, and begin a second year of the WordWise column (two reasons!), we thought it would be fitting to take a glance or two at the word "two" itself. One, two, buckle my shoe . two is the first number that a child learns. And well it should be: the child can point to pairs of eyes, ears, feet, and shoes. In contrast, three, the next largest whole number, is not so biological, while zero and one are actually rather subtle. As the German scholar Karl Menninger has put it, "The number two is a frontier in counting" (Menninger; 1969, p. 15).

Problem Section

S-39.

Proposed by Gregory Galperin, Eastern Illinois University. Prove, for positive real numbers x1, x2, ..., xn, that
(x1 + 1/x1)...(xn + 1/xn) >= (x1 + 1/x2)...(xn + 1/x1) >= 2^n.

S-40.

Proposed by Titu Andreescu, American Mathematics Competitions. Find all solutions to the system of equations
6(x - y^{-1}) = 3(y - z^{-1}) = 2(z - x^{-1}) = xyz - (xyz)^{-1}
in nonzero real numbers x, y, z.

S-41.

Proposed by Gerald Heuer, Concordia College. If a and b are rational and a is nonzero, the linear function ax+b assumes rational values when x is rational and irrational values when x is irrational. Are there polynomials of degree greater than 1 with this property?

S-42.

Proposed by Titue Andreescu, American Mathematics Competitions. Let a1, ..., an be real numbers such that a1 + ... + an >= n^2 and a1^2 + ... + an^2 <= n^3 + 1. Prove that n - 1 <= ak <= n + 1 for all k.

The Final Exam: Do You Haiku?

(To read the winning entry to September's Final Exam, click here.)

Augarithms is the biweekly newsletter of the Augsburg College Mathematics Department. Each week we run a brainteaser, "The puzzle of the week," whose solution generally requires cleverness, insight, or sideways thinking rather than prowess in technical mathematics. Collected below are a baker's dozen of puzzles featured recently in Augarithms. This month's Math Horizons contest is to solve these puzzles. Send your solutions in by April 1. Several randomly selected respondents from among the most successful solvers will receive Math Horizons t-shirts.