Number systems

Why does our mathematics work so well in explaining the universe? Might alien cultures have different mathematics from ours? These intriguing questions appeared in an article, "Useful invention or absolute truth: What is math?" in The New York Times, Feb. 10. An accompanying graphic gave three highly debatable reasons why an alien culture might not even understand our number ("pi," or the ratio of the circumference of a circle to its diameter, about 3.1416).

1. The aliens might have a different number system. If they have four hands and twenty-four fingers, they might use a number system based on twenty-four rather than ten, with twenty-three different symbols such as 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C,.... before using the symbol 10 for twenty-four. In the two systems, would be written out in completely different ways. But our mathematicians and theirs could still recognize in each other's different symbols the same underlying number value .

2. Rather than working on flat desktops, the aliens might work on curvy surfaces, where the ratio of the circumference of a circle to its diameter (measured along the surface) is variable and perhaps less than . Nevertheless, for very small such circles the ratio approaches , which is still an important number even in such curvy geometries.

3. The aliens might, for example, measure distances with some crazy scale in which two inches are more than twice as long as one inch, and four inches are more than twice as long as two inches, and so on.

Actually, we ourselves use "logarithmic" and other scales sometimes, but the usual scale is the natural one in which 1 + 1 = 2 and 2 + 2 = 4, and so on. Their mathematicians would be sure to know and love this scale and our favorite constant .

Old hotel paradox

Three guys go into a hotel, each with $10 in his pocket. They book one room at $30 a night. A short while later a fax from headquarters directs the hotel to charge $25 a night. So the receptionist gives the bellhop $5 to take to the three guys sharing the room. Since the bellhop never got a tip from them and because he can't split $5 three ways, he decides to pocket $2 and give them each $1 back. So each of the three guys has now spent $9 and the bellhop has $2, for a total of $29. Where's the extra dollar?

Answer

The bellboy's $2 was included in the $27 the guests paid, so it should not be added but subtracted, yielding the remaining $25 the hotel got. James Turner notes that "adding $2 to $27 is meaningless, and only seems reasonable because the 'answer' of $29 is so close to the original number, $30."

Joe Shipman concludes that "this is a good puzzle because the faulty step is so brazenly stupid it can be hard to notice, sort of like Poe's purloined letter which was hidden in plain sight on a table." Robin Konicek suggests that "Tatum O'Neal and her father Ryan could well have used this in Paper Moon to try to swindle some more folks."Readers found ways to tell us gently that this is an old puzzle. Aubry Dunne wrote, "My father had posed this problem to me prior to World War II."

(Other winners for the best written of many correct answers were Eric Brahinsky, Ron Douglass, Steve Jabloner, Rajan Krishnaswami, Erik Randolph, and Mary WillAllen.)

New spring challenge (Ilan Vardi)

Why is the first day of spring in London sometimes on March 20 and sometimes on March 21?

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