If you observe carefully, there are actually 5 squares in total. 1 square is made up of 4 identical smaller squares and there are 4 smaller identical smaller squares. There are actually 4 one by one squares and 1 four by four square. Can you see a pattern emerging here?

So the total for a 4×4 is 16 + 9 + 4 + 1 = 30. The total for any nxn square is (n squared + n-1 squared + n-2 sqaured down to 1 when n=1.

Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.

The process of multiplying a number times itself is called squaring. Numbers whose square roots are whole numbers, (or more accurately positive integers) are called perfect square numbers….List of Perfect Squares.

A number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. Thus, the square root of 18 is not an integer, and therefore 18 is not a square number.

Perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.

List of perfect numbers