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Property 6

Observe the following:

2
2
2
2 = 4 = (3 × 1) + 1, 3 = 9 = 3 × 3, 4 = 16 = 3 × 5 + 1,
2
2
6 = 36 = 3 × 12, 8 = 64 = 3 × 21 + 1, 11 = 121 = (3 × 40) + 1
2
Thus, we can say that,
The square of a natural number (other than 1) is either a multiple of 3 or exceeds a multiple of
3 by 1.

Interesting Patterns of Square Numbers

The square numbers can be arranged as a square while the numbers that are not square numbers
form other designs, such as a line, a triangle or a rectangle. Observe the following:

1 4 9 16 25
Square Numbers

or
2 3 5 6 7 10
Non-Square Numbers
There are many interesting patterns of square numbers. Let us learn about some of these.

Pattern 1. Adding triangular numbers
We know that numbers: 1, 3, 6, 10, 15, …
are triangular numbers. If we combine two Remember
consecutive triangular patterns, we get a dot Numbers whose dot patterns can be arranged as
pattern representing a square number as shown triangles are known as triangular numbers.
below.

1 3 6 10 15

1 + 3 = 4 = 2 2 3 + 6 = 9 = 3 2 6 + 10 = 16 = 4 2
Thus, we can say that, the sum of two consecutive triangular numbers is a square number.

Pattern 2. Adding consecutive odd numbers
Observe the following:

Sum of first one odd number = 1 = 1 2

Sum of first two odd numbers = 1 + 3 = 4 = 2 2

129 Squares and Square Roots