For which digits d, is it possible to add d to every digit of a square and get another square?
For example, adding 3 to each digit of 16 gives 49.However, adding zero to each digit in this manner is NOT permissible.

For which digits d are there infinitely many such squares?

*** Digit sums greater than 9 are not allowed. For example, you could not add 8 to the digits of 81 to get 169.