Odd numbers make cubes

The idea

When the odd numbers 1, 3, 5, 7, … are taken in successive groups of one, two, three, … numbers, the sum of the nth group is n³.

Cut the odd numbers

Write the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, … Cut them into groups of 1, 2, 3, 4, … numbers:

1 | 3 5 | 7 9 11 | 13 15 17 19 | …

Add up each group. What will the sums look like?

• Always odd numbers
• Squares: 1, 4, 9, 16, …
• Something else

Add up the rows

Each row is one group. Tap a row to add its numbers. What do the sums remind you of?

Sum of the nth group

When the odd numbers 1, 3, 5, 7, … are taken in successive groups of one, two, three, … numbers, the sum of the nth group is n³.

Why? The middle of group n is n². There are n numbers, balanced on both sides of the middle, so their sum is n × n² = n³.

Notes

When the odd numbers 1, 3, 5, 7, … are taken in successive groups of one, two, three, … numbers, the sum of the nth group is n³.

Check yourself

Which of these odd-number sums is equal to 4³ = 64?

The 7th group is 43 + 45 + 47 + 49 + 51 + 53 + 55. Without adding, find its sum.

Answer: 343

The 7th group adds up to 7³ = 7 × 7 × 7 = 343.

The 5th group is 21 + 23 + 25 + 27 + 29. What is its middle number?

Answer: 25

The middle of group 5 is 5² = 25, and 5 × 25 = 125 = 5³.

The 8th group has 8 odd numbers and is centred on 8² = 64. What is its smallest number?

Answer: 57

4 odd numbers below 64 are 63, 61, 59, 57 and 4 above are 65, 67, 69, 71. The group 57 + … + 71 adds up to 512 = 8³.

• 7 + 9 + 11. 7 + 9 + 11 = 27 = 3³. That is the 3rd group.
• 13 + 15 + 17 + 19 — correct. Yes! The 4th group: 13 + 15 + 17 + 19 = 64 = 4³.
• 9 + 11 + 13 + 15. Four numbers, but the group must start right after the 3rd group (7, 9, 11). 9 + 11 + 13 + 15 = 48.
• 21 + 23 + 25 + 27 + 29. That is the 5th group: 125 = 5³.