Aisha says 1962 could be the square of some whole number, if the right number is found, since it…

Why

Aisha's guess is fair on its own — big numbers can certainly be perfect squares, so there's no obvious reason to rule 1962 out at a glance. But perfect squares carry a hidden pattern in their last digit that's worth knowing: check the squares of 0 through 9 (0,1,4,9,16,25,36,49,64,81) and their last digits are only ever 0, 1, 4, 5, 6, or 9 — never 2, 3, 7, or 8, for any whole number whatsoever, however large. This isn't a coincidence limited to small numbers; it holds for every whole number because a number's last digit alone decides its square's last digit. 1962 ends in 2, which is one of the four forbidden digits, so it is IMPOSSIBLE for 1962 to be any whole number's square — no amount of searching would find one. Test: try squaring several two-digit numbers ending in different digits and check their last digits — you'll never once land on a 2, 3, 7, or 8. Kunal is right: a number ending in 2, 3, 7, or 8 can never be a perfect square, and that can be known instantly, without testing a single square root.