What are the patterns and properties in square numbers?

Progressive increase in difference The difference between consecutive square numbers increases progressively. For instance, the difference between 1 (1x1) and 4 (2x2) is 3, between 4 and 9 (3x3) is 5, and so on. Each time, this difference increases by 2. Sum of odd numbers Each square number is the sum of the first 'n' odd numbers, where 'n' is the root of the square. For example, 1+3 = 4 (2x2), 1+3+5 = 9 (3x3). Geometric representation Square numbers can be represented geometrically as squares with a certain number of points on each side, illustrating their name. Even and odd properties If the square root is even, the square number is even, and if the square root is odd, the square number is odd. Perfect squares and square roots Square numbers are the only integers that have an integer square root, known as perfect squares. Relationship with triangular numbers A square number is the sum of two consecutive triangular numbers. For example, 16 (4x4) is the sum of the 3rd (6) and 4th (10) triangular numbers.