Wednesday, September 29, 2021


I've been looking very hard for the earliest occurrence in print of the number 3816547290 as the solution of a famous divisibility challenge. For an explanation, see OEIS A181736. David Gauld, the author of that sequence, wrote about this number way back in December of 1984 (problem 15 on page 17, here; the following month it appeared in Games magazine, page 2). I recently asked David wherefrom he got that number. He replied that it was in the 1983 book "Les nombres remarquables" by Fran├žois Le Lionnais. Indeed:

There's more! A Google book search for 3816547290 comes up with a Russian hit, Nauka i zhizn' (Science and life) - page 148, supposedly 1980:
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Once I figured out the trailing numbers, this did look like a bona fide mention. The snippet view, alas, showed nothing. And Google's date assignments can't generally be trusted, so I longed for an inside-the-journal look. Playing with it (try searching for 381654), I eventually conjured up this snippet:
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Showing a '1980' at the top-right made me hopeful that this is an actual earlier posing of the divisibility curiosity than that of Le Lionnais. If that were so, it would not be unreasonable to suppose that this dissemination was responsible for Le Lionnais (being the well-connected individual that he was) coming to be acquainted with 3816547290.

A variant of the divisibility puzzle that excludes the digit 0 appeared just prior to Le Lionnais in an "Acorn User" competition in November 1982:
Acorn User: November 1982, page 71
Acorn User: February 1983, detail from page 55
The February 1983 answer to the puzzle is significant for noting that the problem appeared in the 'Brainteaser' column of the Sunday Times (in 1982), which is how David Wells credited 381654729 in "The Penguin Dictionary of Curious and Interesting Numbers" (1986). More relevant (to me) is the assertion that it was first posed in Scientific American "five years ago". Of course! That would make it Dr. Matrix (Martin Gardner) in Chautauqua (Mathematical Games, Scientific American, December 1978, page 23):

"Jaime Poniachik of Buenos Aires, who edits an excellent Spanish puzzle magazine titled The Snark, happened to be visiting the Chautauqua Institution at the time of the ASMOF demonstration. When it was his turn to ask a question, he said he had a friend in the U.S. with a curious social security number. Its nine digits include every digit from 1 through 9, and they form a number in which the first two digits (reading from left to right) make a number divisible by 2, the first three digits make a number divisible by 3, the first four digits make a number divisible by 4 and so on until the entire number is divisible by 9. What is the number?"

Gardner's solution (Mathematical Games, Scientific American, January 1979, page 24):

"The social security number is 381-65-4729. Adding 0 at the end gives the unique solution to the same problem with the ten digits from 0 through 9."

So 3816547290 was implicitly noted as of this date and the explicit Russian mention a year or so later is no longer a surprise. In "The Magic Numbers of Dr. Matrix" (1985), Gardner cleans up the ambiguity of his "ten digits" solution by adding "..., and the ten-digit number divisible by 10." There is considerably more in the book but I will quote only a couple of paragraphs:

"Michael R. Leuze worked with a computer to examine solutions to this problem in number systems other than base 10. He found that there are no solutions in any odd base or in base 12. In base 2 there is the trivial solution 1. In base 4 there are two solutions, 123 and 321; in base 6, 14325 and 54321; in base 8, 3254167, 5234761, and 5674321. In base 14, as in base 10, there is a unique solution: 9 12 3 10 5 4 7 6 11 8 1 2 13. Leuze conjectures that there are no solutions in higher bases."

"Leuze sent a thorough analysis of the problem in negative bases. For other generalizations, see 'Progressively Divisible Numbers', by Stewart Metchette, in Journal of Recreational Mathematics, Vol. 15, No. 2 (1982-83), pages 119-22. The problem surfaced again in 'What's In a Number?', Mathematical Gazette, Vol. 67 (December 1983), pages 281-282. The editor added in a footnote that when the problem appeared in London's Sunday Times in 1982, irate readers made the same careless mistake some of my readers did — they complained that the answer was not unique because they had worked on the problem with an eight-digit calculator!"

Finally, Gardner reveals the true author of the puzzle:

"Jaime Poniachik (the person who posed the problem in my column) told me the problem had been invented by his wife, Lea Gorodisky, who worked with him on the Snark."

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