PROBLEM:
Find all four-digit numbers which are a complete square and in which the first two digits are equal and the last two digits are also equal.
Working mod 100 we have that 44 is the only possibility for x^2 to have two digits. Which means this number AB (where A,B are digits) must end in 44. This means B=2 or B=8. Thus, 12,...,92 and 18,...,98. But since it is 4 digits the only possible choices are 32,...,92 and 38,...,98 and these are easy to check.