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Editor's note: This article originally appeared in the May 1961 issue of Scientific American. Sherman K. Stein is currently professor emeritus of mathematics at University of California, Davis. He is the author of several popular books on mathematics published after this article; his latest is "Survival Guide for Outsiders: How to Protect Yourself from Politicians, Experts and Other Insiders: (BookSurge Publishing, 2010).
The nature of mathematics is elucidated by one mathematician's account of how a memory word used by drummers in ancient India led him to the classic problem of the traveling salesman's route
Mathematics, like every branch of knowledge, is the product of the interplay between past and present, between accumulated knowledge and curiosity, between an autonomous structure and the tastes and needs of the time. What one age considers a pressing question, another may not ask at all. The pure mathematics of one era may be applied in another, perhaps centuries later.
A problem with which I recently tangled illustrates these aspects of the growth of knowledge, and its solution captures the adventurous flavor of mathematical research. Pushing into the unknown, the mathematician is an explorer who is likely to find what he did not seek and who cannot predict how others will use his discoveries.
This particular adventure began when the composer George Perle told me about an elaborate theory of rhythm that had been developed in India more than a thousand years ago. "While reading about this theory," he said, "I learned my one and only Sanskrit word: yamátárájahánsalagám." I asked him what it meant.
"It's just a nonsense word invented as a memory aid for Indian drummers."
"If a drummer can remember that, " I replied, "he can remember anything. "
"There is a lot in those ten syllables, " said Perle. "As you pronounce the word you sweep out all possible triplets of short and long beats. The first three syllables, ya má tá, have the rhythm short, long, long. The second through the fourth are má tá rá: long, long, long. Then you have tá rá já: long, long, short. "Next there are rá ja bhá: long, short, long. And so on." I wrote down the word and saw that what Perle said is true. Each successive triplet of syllables displays a different pattern, and the whole word displays all the possible patterns, giving each one once and only once. As a mathematician I was fascinated to find that such a sequence could exist.
That night I returned to the ancient word. To strip it of irrelevancies I replaced the syllables with digits, letting 0 stand for short beats and 1 for long. In this notation yamátárájahánsalagám became 0111010000l.
Staring at the simplified string for a while, I noticed a lovely thing. The first two digits are the same as the last two; so if I bent the string into a loop, it would look like a snake swallowing its own tail. That is, the last 01 could be placed over the first 01, so that the two pairs of digits would merge into a single pair. Instead of a line of 10 digits I now saw a circle of eight.
I could begin anywhere on this "memory wheel" and move around it in either direction, sweeping out triplets of 0's and l's. Starting at the top and reading clockwise, for example, gave 011, 111, 110, 101, 010, 100, 000, 00l.
The next thing that occurred to me, as it would automatically to any mathematician,was to generalize what I had found. Is there a "word" for listing all quadruplets of 0's and l's once and only once? For quintuplets? For groups of any size? And if so, does the snake always swallow its tail?
Before attacking the problem for quadruplets it seemed sensible to go back and look at couplets. Is there a word that lists each of the four couplets 00, 01, 10, 11 exactly once? Does it close up on itself? 'Writing down the sequence 0011, I saw that I already had the three couplets 00, 01, 11. Adding one 0 more, to form 00110, gave the final couplet: 10. I noted that the first and last digits are O's, so that the snake swallows its tail. The five-digit word could be bent into a four-digit wheel containing each couplet once and only once.