Seemingly simple games can serve as thought-provoking exercises in mathematical logic. They can provide deep insights into subtle issues that confront logicians who are interested in the foundations of mathematics.So-called Ehrenfeucht games have proved particularly useful for tackling certain aspects of mathematical logic. They were developed in the 1960s by Andrzej Ehrenfeucht, who is now a computer science professor at the University of Colorado in Boulder.Ehrenfeucht games can also be studied for their own sake as interesting and often surprisingly subtle games, an approach adopted by Caro...

The curiously looping movements of the planets relative to the stars have presented all sorts of puzzles to keen, patient observers of the night sky.In 1601, Johannes Kepler (1571-1630) undertook the challenge of deciphering the orbit of Mars and developing a mathematical theory of its motion to fit observations of the planet's changing position in the sky. In assuming that Earth itself traveled around the sun, Kepler's immediate hurdle was to find a way to disentangle Mars' motion from that of Earth. He then faced the daunting task of choosing an appropriate geometry for the two planetary orb...

The lure of easy money brings gullible bettors back again and again to the game of video poker--an immensely popular pastime in casinos and other gambling venues throughout the United States.Most players are bound to lose money, says Todd D. Mateer, a recent graduate of Clemson University, who has studied video poker machines in South Carolina. Moreover, imposing limits on how much a gambler may win per machine increases potential losses, even when the gambler plays a long time and makes the best possible choices in each game.In video poker, a player receives five cards, displayed on a video m...

The ancient Greeks, especially the Pythagoreans, were fascinated by whole numbers. They defined as "perfect" numbers those equal to the sum of their parts (or proper divisors, including 1). For example, 6 is the smallest perfect number-the sum of its three proper divisors: 1, 2, and 3. The next perfect number is 28, which is the sum of 1, 2, 4, 7, and 14.The Pythagoreans were also interested in what we now call amicable numbers--pairs in which each number is the sum of the proper divisors of the other. The smallest such pair is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10,...

In a book completed in the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on?The total number of pairs, month by month, forms the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Each new term is the sum of the previous two terms. This set of numbers is now called the Fibonacci sequence.The Fibonacci numbers, F[x] (starting with 0), display a variety of patterns, inc...

Anyone trying to refold an opened road map is wrestling with the same sort of challenges confronted by origami designers and sheet metal benders.The problem of returning a creased sheet to its neatly folded state gets tougher when you're not sure if the sheet can be folded into a flat packet and when you're not permitted to change the crease directions. Such conundrums can arise, for example, when designers determine how to bend sheet metal to produce, say, car doors, airplane parts, or heating ducts.Now Erik D. Demaine of the computer science department at the University of Waterloo in Ontari...