As the square of SUM of digits is 18 more than that of the number, so the square of the sum of digit must be greater than or equal to 28 (18+10 as 10 is the smallest 2-digit number) and should be less than or equal to 117 (18+99 as 99 is the largest two-digit number)

So the possible squares are:

36 and hence the possible number can be (36-18) =18 or (1+8)2 = 81 =! 36 and hence not possible.

49 and hence the possible number can be (49-18) =31 or (3+1)2 = 16 =! 49 and hence not possible.

64 and hence the possible number can be (64-18) =46 or (4+6)2 = 100 = 81 =! 64 and hence not possible.

81 and hence the possible number can be (81-18) =63 or (6+3)2 = 81 = 81 and hence possible.

100 and hence the possible number can be (100-18) =82 or (8+2)2 = 100 = 100 and hence not possible.

So only 2 possible values i.e. 63 and 82.

As the square of sum of digits is 18 more than that of the number, so the square of the sum of digit must be greater than or equal to 28 (18+10 as 10 is the smallest 2-digit number) and should be less than or equal to 117 (18+99 as 99 is the largest two-digit number)

So the possible squares are:

36 and hence the possible number can be (36-18) =18 or (1+8)2 = 81 =! 36 and hence not possible.

49 and hence the possible number can be (49-18) =31 or (3+1)2 = 16 =! 49 and hence not possible.

64 and hence the possible number can be (64-18) =46 or (4+6)2 = 100 = 81 =! 64 and hence not possible.

81 and hence the possible number can be (81-18) =63 or (6+3)2 = 81 = 81 and hence possible.

100 and hence the possible number can be (100-18) =82 or (8+2)2 = 100 = 100 and hence not possible.

So only 2 possible values i.e. 63 and 82.