A number whose sum of digits is 1977.
Prove that the number will not be a perfect square.
Note: in this problem, a "number" means a positive integer.
Consider the digital root of a number.
Add the digits together.
If the sum has more than one digit, add its digits together.
Continue until we have a one-digit sum.
That is the digital root of the number.
A number can have only nine possible digital roots.
Examine the digital roots of their squares.
The digital root of a square must be:
Since the digital root of is , it cannot be a square.