Odd numbers build squares

The idea

Adding consecutive odd numbers starting from 1 gives consecutive squares: the sum of the first n odd numbers is n², and consecutive squares differ by consecutive odd numbers.

The jumps between squares

Look at the squares: 1, 4, 9, 16, 25. From 1 to 4 we add 3. From 4 to 9 we add 5. From 9 to 16 we add 7.

What do we add to go from 16 to 25?

• 8
• 9
• 10

Wrap a layer

Press + to wrap an L-shaped layer around the square. Watch how many tiles each layer has.

Odd numbers make squares

Adding consecutive odd numbers starting from 1 gives consecutive squares. The sum of the first n odd numbers is n². So consecutive squares differ by consecutive odd numbers.

• Odd numbers added | Sum | Square
• 1 | 1 | 1²
• 1 + 3 | 4 | 2²
• 1 + 3 + 5 | 9 | 3²
• 1 + 3 + 5 + 7 | 16 | 4²
• 1 + 3 + 5 + 7 + 9 | 25 | 5²
• 1 + 3 + 5 + 7 + 9 + 11 | 36 | 6²

Notes

The sum of the first n odd numbers is n²: consecutive squares differ by consecutive odd numbers 3, 5, 7, 9, …

Check yourself

Which sum is equal to 36?

What is 1 + 3 + 5 + 7 + 9 + 11 + 13?

Answer: 49

Seven odd numbers from 1, so the sum is 7² = 49.

What is the sum of the first 10 odd numbers?

Answer: 100

The first 10 odd numbers add to 10² = 100.

9² = 81. What must we add to 81 to get 10²?

Answer: 19

100 − 81 = 19, the 10th odd number.

• 1 + 3 + 5 + 7 + 9. That is only five odd numbers: 25 = 5².
• 1 + 3 + 5 + 7 + 9 + 11 — correct. Yes! Six odd numbers starting from 1 give 6² = 36.
• 3 + 5 + 7 + 9 + 11. This does not start from 1. It adds up to 35.
• 1 + 3 + 5 + 7 + 9 + 11 + 13. That is seven odd numbers: 49 = 7².