Enigma 476: A curious question
26 November 2018
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From New Scientist #1627, 25th August 1988 [link]
Kugelbaum wandered into a history lecture by mistake and, almost as quickly, but not by mistake, wandered out again. “1210 may well have been a dull year,” he said to himself, “but it’s an interesting number. The first digit gives the number of 0s in it, the next the number of 1s in it, the next the number 2s in it and so on, quite consistently, right up to the very last digit. And there are other such numbers too, such as 2020 and 3211000! Curiously, I can’t find one with six digits, though.”
As he sought vainly the room where he was to give his lecture on number theory he amused himself by calculating all the numbers having this property. “I wonder,” he remarked as he looked into broom cupboard, “if one were to take all the numbers having this property and add them together, what the result would be?”
What is the sum of all the numbers having the property that their first digit gives the number of 0s in the number, the next the number of 1s in the number, the next the number of 2s in the number and so on, consistently right through to and including the last digit of the number?
(Since 10, 11, 12 and so on are not digits, any such number containing more than 10 digits must have zeros in its 11th place and any other places after this).
Note: The original puzzle statement gave 211000 as an example, not 3211000.