### Instruction: How many two-digit numbers have digits whose sum is a perfect square?### Response: A perfect square is a number that can be written as the product of two equal factors, such as 1, 4, 9, 16, 25, and so on.
The sum of the digits of a two-digit number can range from 1 (for 10) to 18 (for 99).
The only perfect squares in this range are 1, 4, 9, and 16.
So I need to find how many two-digit numbers have digits that add up to one of these four values.
To do this, I can use a table to list all the possible combinations of digits that give these sums.
For example, for the sum of 1, the only possible two-digit number is 10.
For the sum of 4, there are three possible two-digit numbers: 13, 22, and 31.
I can fill in the rest of the table in a similar way, making sure not to repeat any numbers or use any leading zeros.
Here is the complete table:

Sum | Two-digit numbers | Count
1 | 10 | 1
4 | 13, 22, 31 | 3
9 | 18, 27, 36, 45, 54, 63, 72, 81 | 8
16 | 79, 88, 97 | 3
To find the total number of two-digit numbers with digits whose sum is a perfect square, I just need to add up the counts in the last column of the table.
This gives me 1 + 3 + 8 + 3 = 15.