Q:  What whole number can be added to 36, 300 and 596 so they all become perfect squares?

A: 925

36 + 925 = 961 (31^2)
300 + 925 = 12225 (35^2)
596 + 925 = 1521 (39^2)

Can try with a list of perfect squares subtract 36 and see if it is a perfect square.  If it is, now subtract 300 and check that answer.  If that one works, then try subtracting 596.
The lowest perfect square we can start with is 625 (25^2) as it is the first one larger than 596.
625 – 596 = 29 which is not a perfect square so we move to the next perfect square, 676 (26^2).

This was a modified problem from a 1989 contest:
https://www.artofproblemsolving.com/wiki/index.php?title=1989_AIME_Problems/Problem_7