can we find a 4-digit number of the form aabb which is a square.
$\displaystyle aabb = 11(100a+b)$. If that is a square then 100a+b must be divisible by 11, which implies that a+b = 11. Also, a square has to end in 0,1,4,5,6 or 9. That doesn't leave many possibilities for you to check ... .