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(b) Odd numbers and Square numbers

1 = 1 = 1 × 1

1 + 3 = 4 = 2 × 2
1 + 3 + 5 = 9 = 3 × 3 Square numbers can be represented by

1 + 3 + 5 + 7 = 16 = 4 × 4 counting the dots in a square grid. We can
1 + 3 + 5 + 7 + 9 = 25 = 5 × 5 partition these dots into consecutive odd
numbers, i.e., 1, 3, 5, 7, .... This can be shown
1 + 3 + 5 + 7 + 9 + 11 = 36 = 6 × 6 using reverse of L shape. Here, every L shape
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 7 × 7 shows odd numbers of dots while the sum of
............................................................ consecutive layers makes a square number.

Quick Check

Can you make another pattern using the dots by your own?
Look at the pattern below and try to show it using dots.
1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
..........................................................................
(c) Cubic numbers and Odd numbers: Here is a relation depicted to show how cubic numbers
are related to odd numbers.
1 × 1 × 1 = 1, the first odd number
2 × 2 × 2 = 8 = 3 + 5, the sum of the next two odd numbers

3 × 3 × 3 = 27 = 7 + 9 + 11, the sum of the next three odd numbers
4 × 4 × 4 = 64 = 13 + 15 + 17 + 19, the sum of the next four odd numbers
5 × 5 × 5 = 125 = 21 + 23 + 25 + 27 + 29, the sum of the next five odd numbers

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Can you justify why this happens? Do you think it will happen forever?
(d) Triangular numbers and Tetrahedral numbers: The tetrahedral numbers are the sums of
the consecutive triangular numbers beginning from 1.
1 = 1
1 + 3 = 4 Knowledge Desk
1 + 3 + 6 = 10 Johann Carl Friedrich Gauss

1 + 3 + 6 + 10 = 20 (April 30, 1777 – Feb. 23, 1855)
1 + 3 + 6 + 10 + 15 = 35 is known as the Prince of
1 + 3 + 6 + 10 + 15 + 21 = 56 Mathematics.
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A relation between two different numbers has been shown above. Now you can explore the
same for pentagonal and hexagonal numbers as well.

15 Patterns in Mathematics