Possible Perfect Square Check

By divisibility tests, it's divisible by 3 but not by 9. So it's not a square.

Let's break this problem down into two subproblems. The number, let's say $x$ , is equal to $222...222 \times 10^{number of zeros}$ .

In order to be a square, the number of zeros must be an even number. So, let's ignore this first and deal with the $222...222$ and if it is a square then we can simply say that we do not have enough information.

So let's write $222...222$ as $2y$ and $y=111...111$ . $2y$ cannot be a square since $y$ is odd and $2y$ only has one $2$ as its factor.

Therefore, this is not a square number.

Consider 2 cases:

case 1: there are even number of "0"s

it suffices to show whether 222 is a perfect square number.

since 222=2 3 37, 22200....000 is not a perfect square

case 2: there are odd number of "0"s

it suffices to show whether 2220 is a perfect square number.

since 222=2 2 3 5 37, 22200....000 is not a perfect square

Hence the given number is not a perfect square number.