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1. Of numbers from 10 to 19, the sums of their digits are the integers from 1 to 10.
Of them, there are 3 numbers with the sums of digits 4 and 9. (3)
2. Of numbers from 20 to 29, the sums of their digits are the integers from 2 to 11.
Of them, there are 2 numbers with the sums of digits 4 and 9. (2)
3. Of numbers from 30 to 39, the sums of their digits are the integers from 3 to 12.
Of them, there are 2 numbers with the sums of digits 4 and 9. (2)
4. Of numbers from 40 to 49, the sums of their digits are the integers from 4 to 13.
Of them, there are 2 numbers with the sums of digits 4 and 9. (2)
5. Of numbers from 50 to 59, the sums of their digits are the integers from 5 to 13.
Of them, there is 1 number with the sum of digits 9. (1)
6. Of numbers from 60 to 69, the sums of their digits are the integers from 6 to 14.
Of them, there is 1 number with the sum of digits 9. (1)
7. Of numbers from 70 to 79, the sums of their digits are the integers from 7 to 14.
Of them, there is 1 number with the sum of digits 9. (1)
8. Of numbers from 80 to 83, the sums of their digits are the integers from 8 to 11.
Of them, there is 1 number with the sum of digits 9. (1)
9. Now compute the sum of the numbers in parentheses
3 + 3*2 + 4*1 = 3 + 6 + 4 = 13
to get the answer to the problem's question.
Answer. Among the numbers from 10 through 83, there are 13 numbers that have the sum of their digits to a perfect square.
