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"Unsolved Math Mysteries," by John Allen Paulos. ABCNews, "Who's Counting" monthly column, 1 December 2002.
Paulos tackles explaining three unsolved mathematics problems (problems not on the Clay Mathematics Institute's list of Millennium Problems, which if solved will each garner a US$1 million award). The first is a problem discovered by Lothar Collatz in 1937: "Pick any whole number you wish and subject it to the following rule: If the number is even, cut it in half, but if it is odd, multiply it by 3 and add 1. Whatever number results from this, apply the rule to it as well. If you do this over and over again ... you get to 1, and then 4,2,1 will repeat itself indefinitely. Every number that has been tried does eventually return to 1, but it's never been mathematically proved that every number does." The second problem is traced to a paper by Frank Ramsey published in 1928: "It is still not known how many people a group must contain in order for there always to be at least 5 people in it who are mutually acquainted or or at least 5 who are mutual strangers. Despite supercomputers, almost nothing is known about this phenomenon for numbers larger than 5." The third problem involves a game called Chomp, invented by mathematician David Gale in 1974. Despite the game's simple rules, no one has yet found a winning strategy for it that always works. "The game requires you to consider a rectangular array of cookies (in rows and columns) in which the cookie in the lower left corner is poisoned. Players A and B must take turns picking any cookie they wish and then eating it as well as any cookies in the region to the right of it and above it. The player who is forced to eat the poisoned cookie loses." The article shows examples and contains more details.
--- Annette Emerson