Clearly Out of Ideas...

Perfect Square - Number obatined when exponent number is 2

Since 1^2 = 1, 2^2 = 4, 3^2 = 9 it follows that 1, 4 and 9 are   Perfect Squares.

Properties and Pattersn of some Square NUmbers:

• Numbers with any of the digits, 2, 3, 7, 8 in the unit value are perfect squares
• In other words, no square number ends in 2, 3, 7, 8. (WTF? >.>)
• The number of 0’s at the end of a perfect square is always even.  In other words, a number ending in an odd number of 0’s is never perfect.
• Squares of even numbers are always even and vice versa for odd. (Too lazy to write it. XD)
• The square of a natural number other than 1 is either a multiple of 3 or exceeds a multiple of 3 by 1. E.G. 8^2 = 64 = 3 x 2 (divided!) 1
• Perfect Square leaves a remainder of 0 or 1 when divided by 4 E.g 5^2 = 25 = 6 x 4 + 1, 4^2 = 16 = 4 x 4 + 0
• Square of Natural Numbers having all digits as 1 follow the pattern:

1. 11^2 = 121
2. 111^2 = 12321
3. 1111^2 = 123421

End Notes. :D