Digital Root of Square

Theorem
Let $n^2$ be a square number.

Then the digital root of $n^2$ is $1$, $4$, $7$ or $9$.

Proof
Let $\map d n$ denote the digital root base $10$ of $n$.

From Digital Root is Congruent to Number Modulo Base minus 1, $\map d n \equiv n \pmod 9$.

So, let $n = 9 k + m$ where $1 \le m \le 9$.

Thus:

We enumerate the squares of the digits:

Hence the result.