How many of the numbers from 10 through 96 have the sum of their digits equal to a perfect square?

8+9=17, 1+0=1. So, your biggest perfect square is 4, and the smallest one is 1.

So, one number is: 1+0=1=1^2

1+3=3+1=4+0=2+2=2^2. So, 4 more numbers: 13, 31, 40 and 22. (5 so far).

9+0=1+8=2+7=3+6=4+5=3^2
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There four numbers are "symmetric", so you have also 81, 27, 36, and 45, to add up to 9. So, those that add up to 9, there are 9 of them. (14 so far)

7+9=8+8=16=4^2, so two more. (Note: 97 is not in there ...)

Overall you have 16 numbers.