Math: Fun and Games
If you have a 3-gallon jug and a 5-gallon jug, how can you measure exactly 4 gallons of water?
Probably everyone has encountered this problem or something like it in grade school, but most people are perhaps unaware that it is an opening wedge into a delightful and deceptively serious branch of knowledge called recreational mathematics. When we told our friends that we were going to a conference on this subject at the University of Calgary, in Canada, they looked puzzled. “We’ve heard of recreational drugs,” they said, “and we’ve heard of recreational sex. But recreational mathematics sounds like a contradiction in terms.”
Nearly 100 people spent last week in Calgary noodling on just this subject, and we can report that we have never attended a conference punctuated by so much laughter. The participants talked about games and puzzles and intellectual oddities of all kinds. They amused themselves, but they weren’t wasting time. Amusement, after all, is one of humanity’s greatest sources of progress, and recreational mathematics may be a paradigm for all knowledge.
“I do it for fun,” said Leon Bankoff, a 77-year-old Los Angeles dentist who, though an amateur, is one of the deans of the field. “To me, all of mathematics is recreational.”
But the participants could not even agree on what recreational mathematics is, or what distinguishes it from “real” mathematics.
“Recreational mathematics is what somebody does for his own amusement,” said David Singmaster of the Polytechnic of the South Bank in London. “But that includes almost all of mathematics--and all of science. I can’t define mathematics, but I can smell it. There’s no way to define physics, either. It’s what physicists do. Chemistry is what chemists do. And mathematics is what mathematicians do.”
Singmaster, a burly man with a white beard (many of the conference’s participants were bewhiskered), recalled an old problem in the field:A maiden of 16 is worth 32 dinars. How much is a maiden of 20 worth? “It’s done by proportion,” he said, “but I cannot remember whether the maiden of 20 is worth more or less than the maiden of 16.”
Unlike serious mathematics, the problems in recreational mathematics can usually be stated easily and do not require much background to attack. So many people can play, and there is evidence that many people do. Hundreds of puzzle books are published every year, and the public bought about 100 million Rubik’s cubes in three years, compared with fewer than 90 million Monopoly sets in 50 years.
But the Calgary meeting was no place for the fainthearted or the careless. Jack Eidswick of the University of Nebraska was giving a talk on a mechanical puzzle that he had analyzed, and concluded that the solution would require 4 million moves. “If I did one move per second,” he said, “it would take five years.”
Almost without missing a beat, Solomon R. Golomb of the University of Southern California piped up from the audience, “Four million seconds is only 46 days.” (It was an impressive calculation in his head. Forty-six days contain 3,974,400 seconds.)
“I stand corrected on my arithmetic,” an embarrassed Eidswick replied.
Perhaps the tone of the meeting can be summed up by the reaction of the speaker whose slides got out of order. “I don’t know whether I’m going backwards or forwards,” he said, adding, “I don’t think it matters.”
As a group, these people are the most delightful around, and it’s a joy to try to follow the curlicues of their minds. Society needs them, and is fortunate to be wealthy enough to support their playful work.
Oh, about the 3-gallon jug and the 5-gallon jug: There are two ways to measure exactly 4 gallons of water. Here is one of them:
(1) Fill the 3-gallon jug and pour it into the 5-gallon jug. (2) Refill the 3-gallon jug, and use it to top off the 5-gallon jug, leaving 1 gallon in the 3-gallon jug. (3) Empty the 5-gallon jug, and pour the 1 gallon from the 3-gallon jug into it. (4) Refill the 3-gallon jug, and pour it into the 5-gallon jug, which already contains 1 gallon. Thus, 3 gallons plus 1 gallon equals 4 gallons.
