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Number series: how to find the rule in under a minute

A number series looks like a memory test and is not one. There is no list to learn: practically every series on the paper runs on one of four rules, and there is a single first move that tells you which.

Take the differences between consecutive terms and write them underneath. If the differences are all the same, you are done. If the differences themselves form a pattern, that is the answer. If they grow by multiplying rather than adding, it is a ratio.

Start by taking the differences

This is the whole method, and it takes about five seconds once it is a habit. Below the series, write the gap between each pair of terms.

4710131619?+3+3+3+3+3+3every gap is the same, so the rule is add 3
Differences written underneath. When they are all equal, the series is arithmetic and the next term is one more gap along.

Worked example · constant difference

What number completes the series?

4, 7, 10, 13, 16, 19, ___
  1. A16
  2. B19
  3. C22
  4. D25

The series follows one rule throughout: each term is 3 more than the one before it. Applying that rule to the last printed term gives 22. Check a rule against every printed term before continuing a series; a rule that explains only the first pair is usually the wrong one.

  • AThis moves backwards through the series instead of forwards.
  • BThis repeats the last printed term instead of continuing the series.
  • DThis continues the series one term too far.

When the differences are not equal, look at them again

Most people stop when the first row of differences comes out uneven, and that is exactly the moment to keep going. Take the differences of the differences. Very often they are constant, and the series is what a reviewer would call a second-difference series - the gaps are growing by a fixed amount.

151824334560?+3+6+9+12+15+18+3+3+3+3+3the second row is constant, so the gaps grow by a fixed amount
A second row of differences. The gaps grow by 3 each time, so the next gap is 18 and the answer is 60 + 18.

Worked example · growing difference

What number completes the series?

15, 18, 24, 33, 45, 60, ___
  1. A60
  2. B81
  3. C78
  4. D57

The series follows one rule throughout: the differences between terms start at 3 and grow by 3 each time. Applying that rule to the last printed term gives 78. Check a rule against every printed term before continuing a series; a rule that explains only the first pair is usually the wrong one.

  • AThis repeats the last printed term instead of continuing the series.
  • BThis continues the series one term too far.
  • DThis moves backwards through the series instead of forwards.

When the numbers grow too fast to be adding

If the terms roughly double or triple, stop taking differences and try dividing instead. A geometric series multiplies by a fixed ratio, and the giveaway is scale: adding gets you from 31 to 62, but it will not get you from 248 to 496 with the same step.

Worked example · constant ratio

What number completes the series?

31, 62, 124, 248, ___
  1. A248
  2. B246
  3. C498
  4. D496

The series follows one rule throughout: each term is 2 times the one before it. Applying that rule to the last printed term gives 496. Check a rule against every printed term before continuing a series; a rule that explains only the first pair is usually the wrong one.

  • AThis repeats the last printed term instead of continuing the series.
  • BThis moves backwards through the series instead of forwards.
  • CThis continues the series one term too far.

The distractor to watch there is the one just below the right answer. Somebody who spots the doubling but subtracts a little on the last step - or who doubles the wrong term - lands two away, which is exactly where the wrong option sits.

Letters are numbers wearing a hat

A letter series is the same problem with the alphabet as the number line. Give A the value 1 and count. Once you write the positions underneath, the rule is as visible as it was for the digits.

Y25U21Q17M13I9E5?−4−4−4−4−4−4A = 1, so counting back four from E lands on A
Positions written underneath. Y is the 25th letter, and each step goes back four, so the series ends at A.

Worked example · alphabet series

What comes next in the series?

Y, U, Q, M, I, E, ___
  1. AZ
  2. BA
  3. CC
  4. DB

The series follows one rule throughout: each letter is 4 places before the previous one in the alphabet. Applying that rule to the last printed term gives A. Check a rule against every printed term before continuing a series; a rule that explains only the first pair is usually the wrong one.

  • AThis is 1 letter away from the letter the pattern requires.
  • CThis is 2 letters away from the letter the pattern requires.
  • DThis is 1 letter away from the letter the pattern requires.

The two mistakes that cost the most

Reading the printed order as the answer. A series ending 45, 60 makes 60 feel like a candidate, and 60 is almost always one of the options. It is the last term, not the next one.

Giving up on the first row. Uneven differences are not a dead end, they are the second half of the method. If the first row is 3, 6, 9, 12 you have already solved it and stopped one line too early.

Practising it

The trouble with practising series from a list is that you learn the list. By the third pass you are recalling which answer went with which row rather than taking any differences, and the format stops measuring anything.

Every example above came from our own bank at a fixed seed, and the number-series practice generates them fresh rather than drawing from a fixed set - so the series is new each time and the only way through it is the method. Each item carries the reason it is decided the way it is, not just a letter.