Counting systems and the relationship

between numbers and the real world

by Martin Gardner

Ah why, ye gods! should two and two make four?

The Dunciad, Book 2

Anthropologists have yet to find a primitive society whose members are unable to count. For some time they assumed that if an aboriginal tribe had no words for numbers except “one,” “two” and “many,” its members could not count beyond two, and they were mystified by the uncanny ability of such people to look over a herd of sheep, for example, and say one was missing. Some anthropologists believed these tribesmen had a phenomenal memory, retaining in their heads a Gestalt of the entire herd, or perhaps knew each sheep personally and remembered its face. Later investigators discovered that the use of the same word for all numbers above two no more meant that a tribesman was unaware of the difference between five and six pebbles than the use of the same word for blue and green meant that be was unaware of the difference in color between green grass and blue sky. Tribes with limited number vocabularies had elaborate ways of counting on their fingers, toes and other parts of their anatomy in a specified order and entirely in their heads. Instead of remembering a word for 15 a man simply recalled that he had stopped his mental count on, say his left big toe.

Most primitive counting systems were based on five, 10 or 20, and one of few things on which cultural anthropologists are in total agreement (an in agreement with Aristotle) is that the reason for this is that the human animal has five fingers on one hand, 10 on both and 20 fingers and toes. There have been many exceptions. Certain aboriginal cultures in Africa, Australia and South America used a binary system. A few developed a ternary system; one tribe in Brazil is said to have counted on the three joints of each finger. The quaternary, or 4‑base, system is even rarer, confined mostly to some South American tribes and the Yuki Indians of California who counted on the spaces between their fingers.

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A 54‑move solution to C. L. Baker’s order‑2 solitaire game, with the starting pattern shown in June, was published in July. Hundreds of readers sent shorter solutions. By July 10, 106 readers had sent 50‑move solutions and 35 had sent 49‑move versions. Forty-nine is probably the minimum. Although the 49‑move solutions differed considerably, all had in common the placing of the 13 hearts on a P cell in 32 moves, with the remaining 17 moves being used to put the remaining spades on the other P cells.

The readers who sent 49‑movers by July 10 were B. B. Abercrombie, Philip S. Abrams, Frank Anger, Gene C. Barton, Harold A. Beatty, P. H. Browning, William J. Butler, Jr., Donald B. Charnley, Roger D. Coleman, Charles W. Disbrow, Jr., Ralph Dumain, F. J. Dyson, Terry C. Gleason, John W. Gosling, A. W. Greig, Gordon G. Heath, Neale S. Hyatt, Peter Johansson (with Douglas Jacobs), Larry B. Klaasen, S. Kogan, R. C. Koleszar, William V. Lavin, Jr., Sanford Libman, Charles L. McClenon, John Mallinckrodt, Warren H. Olrich, John L. Sampson, Joan and Susanne Schwartz, Walter C. Siff, Theodore and Judith Simon, Hereford A. Stuerke, A. Ungar, Samuel L. Ward, Charles B. Weinberg and Jack Whitney.

Dyson also sent a 54‑move solution using only one T cell, and although he believes the pattern has no solution without a T cell, he is not yet sure. Ohlrich’s 49‑move solution, the first received, is given on the preceding page.

In August, problem 28 was incorrectly stated, and many readers pointed that the ribbon’s length is actually minimized when AB = 0. For a correct statement of the puzzle see problem 66 in the book cited in the answer. Many readers also saw that the answer to problem 31 involves a logical contradiction. My parenthetical “Thanks to Epimenides the Cretan” was a hint that this is a variant on the old liar paradox and was intended as a joke.


SOURCE: Gardner, Martin. “Mathematical Games: Counting Systems and the Relationship between Numbers and the Real World,” Scientific American, September 1968 (vol. 219, no. 3), pp. 218-230. [Boldface in text added by RD]

Note: Gardner's column discussing Baker's Solitaire appeared in the June 1968 issue of Scientific American, and was collected in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American (New York : Alfred A Knopf / Random House, 1977), Chapter 7. Readers' responses reported in successive columns did not always make it unabridged into the books, such as the list of acknowledged persons above, including R. Dumain. Feedback on the Baker's Solitaire column was reported in the July, August, and September (above)1968 issues.

Homage to Martin Gardner (October 21, 1914 – May 22, 2010)
Mathematical Games Web Links

Algorithmic Puzzles: Selected Bibliography

Vladimir Nabokov: Science Fiction, Artificial Languages, Ars Combinatoria, Narrative Structure, Martin Gardner, Play:
An Arbitrary Bibliography


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