Cube numbers from 1 to 20
A cube number is a number multiplied by itself three times: 4³ = 4 × 4 × 4 = 64. Here are the first twenty, and the pattern that lets you take the cube root of any perfect cube up to a million in your head.
The table
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1,000 |
| 11 | 1,331 |
| 12 | 1,728 |
| 13 | 2,197 |
| 14 | 2,744 |
| 15 | 3,375 |
| 16 | 4,096 |
| 17 | 4,913 |
| 18 | 5,832 |
| 19 | 6,859 |
| 20 | 8,000 |
The last-digit pattern
Look at the last digit of each cube from 1³ to 10³: 1, 8, 7, 4, 5, 6, 3, 2, 9, 0. Every digit appears exactly once. So the last digit of a cube tells you the last digit of its root:
ends in 1 → root ends in 1 ends in 8 → root ends in 2 ends in 7 → root ends in 3 ends in 4 → root ends in 4 ends in 5 → root ends in 5 ends in 6 → root ends in 6 ends in 3 → root ends in 7 ends in 2 → root ends in 8 ends in 9 → root ends in 9 ends in 0 → root ends in 01, 4, 5, 6, 9 and 0 map to themselves. The other four swap in pairs: 2 ↔ 8 and 3 ↔ 7.
Cube roots in two seconds
Someone gives you a perfect cube below 1,000,000, say 185,193. Drop the last three digits: 185. Find the largest cube that is not more than 185: 5³ = 125 (6³ = 216 is too big), so the first digit is 5. The last digit of 185,193 is 3, so the root ends in 7. The answer is 57.
It only works when the number really is a perfect cube, which is why it makes a good party trick: have someone cube a two-digit number on a calculator and read you the result.
Cubing a number in your head
Square it, then multiply by the number again. For 13³: 13² = 169, then 169 × 13 = 1,690 + 507 = 2,197. Fast squaring does most of the work, which is covered in how to square numbers in your head.
Drill squares