Knowing the squares from 1 to 20 by heart saves time in almost every topic that comes after them: square roots, area, Pythagoras, algebra and many exam questions. Start with the chart below, then use five patterns that turn a list of 20 numbers into something you can rebuild and check in seconds.
A square is a number multiplied by itself. The answer is called a perfect square. For example, 7 × 7 = 49, so 49 is the square of 7, written 7².
| n | n × n | n² | n | n × n | n² |
|---|---|---|---|---|---|
| 1 | 1 × 1 | 1 | 11 | 11 × 11 | 121 |
| 2 | 2 × 2 | 4 | 12 | 12 × 12 | 144 |
| 3 | 3 × 3 | 9 | 13 | 13 × 13 | 169 |
| 4 | 4 × 4 | 16 | 14 | 14 × 14 | 196 |
| 5 | 5 × 5 | 25 | 15 | 15 × 15 | 225 |
| 6 | 6 × 6 | 36 | 16 | 16 × 16 | 256 |
| 7 | 7 × 7 | 49 | 17 | 17 × 17 | 289 |
| 8 | 8 × 8 | 64 | 18 | 18 × 18 | 324 |
| 9 | 9 × 9 | 81 | 19 | 19 × 19 | 361 |
| 10 | 10 × 10 | 100 | 20 | 20 × 20 | 400 |
The first ten sit on the diagonal of the times tables chart, so if you know your tables, half the job is done already. The patterns below help with the other half.
Look at the gap from one square to the next:
The steps are the odd numbers 3, 5, 7, 9, 11 and so on. The pattern keeps going all the way up the chart. For example, 12² is 144, and the next odd step is 25, so 13² = 144 + 25 = 169.
Going from n² to the next square adds one more row of n, one more column of n, and one corner square. In algebra that is (n + 1)² = n² + 2n + 1, so the step from n² is 2n + 1, which is always odd. It also means the squares are running totals of the odd numbers: 1 + 3 + 5 + 7 = 16, the fourth square (see square numbers on Wikipedia).
If you forget a square, step from a neighbour you know. Say you remember that 16² is 256 but not 17². The step from 16² is two 16s plus 1, which is 33, so 17² = 256 + 33 = 289. You can step down too: 20² is 400 and the step below it is 39, so 19² = 400 − 39 = 361.
A square can only end in 0, 1, 4, 5, 6 or 9. It can never end in 2, 3, 7 or 8 (Wikipedia lists this property). So if you write 17² as 287, you know at once it is wrong: no square ends in 7.
The last digit of a square depends only on the last digit of the number, and the digits come in pairs:
Numbers ending in 5 always give a square ending in 25: 5² is 25 and 15² is 225. So 17² must end in 9 (like 7²), 18² must end in 4 (like 8²) and 14² must end in 6 (like 4²). That one fact catches most slips.
To square a number ending in 5, take the digits in front of the 5, multiply them by the next number up, then write 25 on the end.
Every number from 11 to 19 is 10 plus a small number. Square it in three parts: 100, then twenty times the small number, then the small number squared. In algebra that is (10 + b)² = 100 + 20b + b².
Two of the teens are easy to picture. 11² is 121, which reads the same forwards and backwards. 12² is 144, which starts with 1 and then a double 4. Hang the others on these anchors: 13² is just 25 more than 144.
For 17, 18 and 19, start from 20² and come down. The rule is (20 − b)² = 400 − 40b + b²:
Check these against the chart: 361, 324 and 289 all match. Notice that 18² came out the same, 324, in both Pattern 4 and Pattern 5. When two methods agree, you can trust the answer.
Twenty facts at once is a lot. Split them into four sets and learn one set at a time:
Practise for a few minutes a day rather than in one long session. Short, spaced practice is the same approach that works for learning times tables fast, and squares are part of the same tables.
Cover the chart. Write down five numbers from 1 to 20 at random, write each square, then uncover the chart and check. Many learners mix up the middle teens, so put 13, 14, 16, 17 and 18 in your quiz most often. Use the last-digit check from Pattern 2 before you uncover the chart.
Once the chart feels familiar, drill it:
When you are ready to go beyond 20, read Squaring in Seconds for larger numbers and Vedic Maths square tricks for more shortcuts. Knowing the chart also makes square roots quick, as Roots in a Flash shows.
They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361 and 400.
Learn them in four sets of five, a few minutes a day. Use the patterns to rebuild any you forget: odd-number steps, 10 plus something for the teens, and 400 minus something for 17 to 19.
To grow a square of side n into a square of side n + 1, you add a row of n, a column of n and one corner. That is two lots of n plus 1, which is always an odd number.
From 1 to 20 there are four: 1, 4, 9 and 16. From 1 to 400 there are twenty, one for each number from 1 to 20, ending with 400 itself.
Only 0, 1, 4, 5, 6 or 9. A number ending in 2, 3, 7 or 8 is never a perfect square.
Multiply the digits in front of the 5 by the next number up, then write 25 after the result. For 15², 1 × 2 = 2, so the answer is 225.
Start from 400, which is 20². For 19², take away 40 and add 1 to get 361. For 18², take away 80 and add 4 to get 324. For 17², take away 120 and add 9 to get 289.
It varies, but with a few minutes of practice a day, learning one set of five at a time, most students know them well within a couple of weeks. Keep reviewing the harder teens until they come back instantly.