The number that describes its own digits

Question

Find a ten-digit number with the following property: its first digit is the number of zeros in the number, its second digit is the number of ones, its third digit is the number of twos, and so on, up to its tenth digit, which is the number of nines. Find such a number and argue that it is the only one.

Accepted Answer

6,210,001,000. Check it. The number contains six 0s, two 1s, one 2, and one 6. Its digits say: position 1 (zeros) = 6, position 2 (ones) = 2, position 3 (twos) = 1, position 4 (threes) = 0, position 5 (fours) = 0, position 6 (fives) = 0, position 7 (sixes) = 1, position 8 (sevens) = 0, position 9 (eights) = 0, position 10 (nines) = 0. Every claim is true. Why the digits sum to 10. Each of the ten digit positions counts how many digits of a certain kind there are, and every digit is of exactly one kind, so the counts add up to the total number of digits, 10. Narrowing down. Let z be the first …

The number that describes its own digits

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