number of numbers

you may look the average row .. and so you can get the answer is 1

any number of the form aabb will always be divisible by 11, now perfect squares have property that they contain prime factors even number of times that is say for 16 [ 2exp4] or say 49 [7 exp 2]. so ideally for aabb checking with just 11 is not enough , aabb should also be divisible by 11exp2[ minimum] that is 121. so the list goes as follows 11^2x1^2=121 , 11^2x2^2=484, 11^2x3^2=1089, 11^2x4^2=1936 [ can be split into 11^2x2^4] , 11^2x5^2=3025 , 11^2x6^2=4356 [can be written as 11^2x 2^2x3^2] , 11^2x7^2=5929 , 11^2x8^2=7744[ only number of the form aabb], 11^2x9^2=9801, 11^2x10^2= >5 digits rest its useless to check as all are >4 digits...

so answer is just 1