Question

How many two-digit numbers have digits whose sum is a perfect square?

$13$

$16$

$17$

$18$

$19$

There is $1$ number whose digit sum is $1$ : $10.$

There are $4$ numbers whose digit sum is $4$ : $13, 22, 31,$ and $40.$

There are $9$ numbers whose digit sum is $9$ : $18, 27, 36, 45, 54,$ $63, 72, 81,$ and $90.$

There are $3$ numbers whose digit sum is $16$ : $79, 88,$ and $97.$

Therefore, there are $17$ numbers that satisfy the problem statement.

Thus, C is the correct answer.