I found this puzzle and variants of it in many places; a search yields about 5k hits.
A four-digit number which is a perfect square has the form aabb, with two digits repeated. What is this number?
Short and sweet! See the Comments for a solution and explanation.
Comments
Solution and explanation
The number has the form aabb, so it equals 1000a+100a+10b+b = 1100a+11b = 11 x (100a+b). We know it is a perfect square and since we see it is a multiple of 11 we can simply square the multiples of 11 up to 99 until we find the only answer, 882 = 7744.