Made to Order - Futility Closet

Arrange the digits 0-9 into a 10-digit number such that the leftmost n digits comprise a number divisible by n. For example, if the number is ABCDEFGHIJ, the number ABC must be divisible by 3, ABCDE must be divisible by 5, and so on.

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3816547290. (By Nob Yoshigahara.) 09/14/2021 UPDATE: I came across this puzzle in Yoshigahara’s 2004 book Puzzles 101: A Puzzlemaster’s Challenge. In an article in Quanta last year, Pradeep Mutalik credits it to John Horton Conway. I don’t know who came up with it first; possible the two devised it independently. (Thanks, Dharmesh.) 09/17/2021 UPDATE: Reader Ben Hamula has worked out programmatically that this solution is unique. 09/17/2021 UPDATE: This is getting interesting. Reader Hans Havermann finds that neither Yoshigahara nor Conway originated the problem. The earliest known mention is in the December 1984 issue of the New Zealand Mathematical Society newsletter (Problem 15 on page 17). David B. Gauld, mentioned in that problem, is sometimes cited as its author; Gauld contributed OEIS sequence A181736 in 2010. Immediately after the problem appeared in the New Zealand newsletter, Kishor N. Gordhandas posed it in the January 1985 issue of Games (page 2) and it generally escaped into the wild. Hans contacted Gauld, who recalls that he encountered 3816547290 in François Le Lionnais and Jean Brette’s 1983 book Les nombres remarquables and passed it on to Teddy Zulauf, sub-editor of the NZMS newsletter. Many thanks to Hans for all this yeoman sleuthing; I’ll post a further update if Le Lionnais turns out to cite a source. 09/30/2021 UPDATE: Le Lionnais cites no source but calls the fact “Trés remarquable.” See Hans’ blog for more on the search. 10/04/2021 UPDATE: It appears Hans has identified the originator of the puzzle: Lea Gorodisky, in the Spanish puzzle magazine The Snark. See the link immediately above. (Thanks again, Hans.)

Select Click for Answer

3816547290.

(By Nob Yoshigahara.)

09/14/2021 UPDATE: I came across this puzzle in Yoshigahara’s 2004 book Puzzles 101: A Puzzlemaster’s Challenge. In an article in Quanta last year, Pradeep Mutalik credits it to John Horton Conway. I don’t know who came up with it first; possible the two devised it independently. (Thanks, Dharmesh.)

09/17/2021 UPDATE: Reader Ben Hamula has worked out programmatically that this solution is unique.

09/17/2021 UPDATE: This is getting interesting. Reader Hans Havermann finds that neither Yoshigahara nor Conway originated the problem. The earliest known mention is in the December 1984 issue of the New Zealand Mathematical Society newsletter (Problem 15 on page 17). David B. Gauld, mentioned in that problem, is sometimes cited as its author; Gauld contributed OEIS sequence A181736 in 2010. Immediately after the problem appeared in the New Zealand newsletter, Kishor N. Gordhandas posed it in the January 1985 issue of Games (page 2) and it generally escaped into the wild.

Hans contacted Gauld, who recalls that he encountered 3816547290 in François Le Lionnais and Jean Brette’s 1983 book Les nombres remarquables and passed it on to Teddy Zulauf, sub-editor of the NZMS newsletter. Many thanks to Hans for all this yeoman sleuthing; I’ll post a further update if Le Lionnais turns out to cite a source.

09/30/2021 UPDATE: Le Lionnais cites no source but calls the fact “Trés remarquable.” See Hans’ blog for more on the search.

10/04/2021 UPDATE: It appears Hans has identified the originator of the puzzle: Lea Gorodisky, in the Spanish puzzle magazine The Snark. See the link immediately above. (Thanks again, Hans.)