(the question is formatted a bit weird, but I'm assuming the third statement is meant to be '1/3 of the number is a square number'. If not, let me know and I'll adjust my answer accordingly)
Answer:
The number is 27
Explanation:
Let the number be called x.
10% of x is greater than 1 but less than 7. Written mathematically, this is 1<0.1x<7. Multiply this by ten to get 10<x<70.
1/3 of x is a square number. The squares 4, 9, and 16 are the only squares that satisfy this statement, because the if the square was 1, x would be 1, which doesn't fit x>10, and if the square was 25 or higher, x would be greater than 70, which doesn't fit x<70.
x isn't a multiple of six. If the square is 4, x is 12, which is a multiple of six so this doesn't work. If the square is 16, x is 48, which is a multiple of six too, so this doesn't work. But if the square is 9, x is 27, which isn't a multiple of six and satisfies the statement.
Double check that x=27 is a valid answer.
Is 27 a whole number between 1 and 100? Yes
Is 10% of 27 greater than 1 and less than 7? Yes (10% of 27 is 2.7)
Is 1/3 of 27 a square number? Yes (1/3 of 27 is 9, which is the square of 3)
Is 27 not a multiple of 6? Yes (27 isn't divided evenly by 6)
Therefore, the number is 27.