SOLUTION: How many of the numbers from 10 through 85 have the sum of their digits equal to a perfect square?

Question 580761: How many of the numbers from 10 through 85 have the sum of their digits equal to a perfect square?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The sum of the digits would be at least 1 (as for 10, which I am assuming is included). It could be at most 18 for a 2 digit number (as in 99), but since the highest we can go is 85, we may not get to 18 with the sum of the digits.
The only perfect squares between 1 and 18 are 1, 4, 9, and 16.
The only number between 10 and 85 whose digits add up to 1 is .
The only numbers between 10 and 85 whose digits add up to 4 are:
, , , and (4 numbers).
The only numbers between 10 and 85 whose digits add up to 9 are:
, , , ... and (8 numbers).
The only number between 10 and 85 whose digits add up to 16 is (88 and 97 are too high).
So I am counting numbers whose digits add up to a perfect square.
My answer is .
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