Adding odd numbers

A beautiful pattern: start adding the odd numbers from 1.

• 1 = 1
• 1 + 3 = 4
• 1 + 3 + 5 = 9
• 1 + 3 + 5 + 7 = 16
• 1 + 3 + 5 + 7 + 9 = 25
• 1 + 3 + 5 + 7 + 9 + 11 = 36

Every answer is a square number!

Why? Draw a square grid of 6 rows of 6 dots. Start at one corner with 1 dot. Around it, an L-shaped layer of 3 dots makes a 2 by 2 square. The next L-shaped layer has 5 dots and makes a 3 by 3 square. The layers go on, 1, 3, 5, 7, 9, 11, and together they fill the 6 by 6 square of 36 dots.

Each L-shaped layer has 2 more dots than the one before, because it runs along two sides that are each one dot longer. To grow a 5 by 5 square into a 6 by 6 square you add 6 + 6 − 1 = 11 dots (the corner dot is shared, so it is counted once).

The picture works for a square of any size, so the pattern never stops. Adding the first n odd numbers gives n × n.

• First 10 odd numbers: 10 × 10 = 100.
• First 16 odd numbers: 16 × 16 = 256.
• First 100 odd numbers: 100 × 100 = 10,000.