Number sequence puzzles look as if they need a flash of insight, but almost all of them use a small set of rules. If you test those rules in the right order, you will find the answer to most puzzles in under a minute, and you will know when a puzzle is genuinely unusual. Below is the checklist, with a worked example for every step and practice puzzles at the end. Use it on the progressively harder rules in our free Pattern Sequence game.
The order matters. Steps 1 to 3 solve the large majority of puzzles in tests, apps and games, and they take seconds. Leave the exotic checks for last.
| Step | Check | Example | Next |
|---|---|---|---|
| 1 | Constant difference | 3, 7, 11, 15 | 19 (add 4) |
| 2 | Second differences | 2, 5, 10, 17, 26 | 37 (gaps 3, 5, 7, 9, 11) |
| 3 | Constant ratio | 3, 6, 12, 24 | 48 (times 2) |
| 4 | Squares, cubes, primes | 1, 4, 9, 16, 25 | 36 (square numbers) |
| 5 | Sum of previous terms | 2, 3, 5, 8, 13 | 21 (add the last two) |
| 6 | Alternating operations | 4, 8, 5, 10, 7 | 14 (×2, −3 in turn) |
| 7 | Two interleaved sequences | 1, 20, 2, 18, 3, 16 | 4 (odd places +1, even places −2) |
| 8 | Position-based formula | 2, 6, 12, 20, 30 | 42 (n × (n + 1)) |
| 9 | Digit tricks | 12, 21, 34, 43, 56 | 65 (reverse the digits, then add 13) |
Write the gaps between neighbouring terms underneath the sequence. If the gaps are all the same, you are done. If not, look at the gaps themselves: they often form their own simple sequence.
5, 8, 11, 14, ? Gaps: 3, 3, 3. Answer 17.
1, 3, 7, 13, 21, ? Gaps: 2, 4, 6, 8, so the next gap is 10. Answer 31.
4, 5, 8, 13, 20, ? Gaps: 1, 3, 5, 7. Next gap 9. Answer 29.
When the second differences are constant, the sequence follows a quadratic rule (a formula with n²). You do not need the formula; extending the gaps is enough.
If the terms grow quickly or shrink towards zero, divide each term by the one before. Doubling, tripling and halving are the usual suspects.
2, 6, 18, 54, ? Each term × 3. Answer 162.
80, 40, 20, 10, ? Each term × ½. Answer 5.
1, 2, 6, 24, 120, ? Ratios 2, 3, 4, 5. Answer 720 (these are factorials).
Learn to recognise a handful of families on sight. Knowing the first ten square numbers is especially useful; our square numbers page lists them.
Puzzles often disguise these families. 2, 5, 10, 17, 26 is the squares plus one. 0, 3, 8, 15, 24 is the squares minus one. 3, 4, 7, 11, 18 is Fibonacci-style with different starting numbers.
| Family | Starts | Rule |
|---|---|---|
| Square numbers | 1, 4, 9, 16, 25, 36 | n × n |
| Cube numbers | 1, 8, 27, 64, 125 | n × n × n |
| Triangular numbers | 1, 3, 6, 10, 15, 21 | add 2, 3, 4, 5… |
| Prime numbers | 2, 3, 5, 7, 11, 13 | only divisible by 1 and itself |
| Fibonacci | 1, 1, 2, 3, 5, 8, 13 | each term is the sum of the previous two |
| Powers of 2 | 1, 2, 4, 8, 16, 32 | double each time |
If the sequence zigzags up and down, it usually does one of two things: applies two operations in turn, or hides two separate sequences in the odd and even positions.
Alternating operations: 3, 6, 4, 8, 6, 12, ? The rule is ×2 then −2, in turn. Answer 10.
Interleaved: 10, 1, 9, 2, 8, 3, ? Odd places count down 10, 9, 8; even places count up 1, 2, 3. Answer 7.
A quick test: write out every other term. If the odd-position terms make sense on their own, you have found an interleaved sequence.
When nothing above works, number the positions 1, 2, 3, 4 and see whether each term is a simple formula of its position. 2, 6, 12, 20, 30 is 1×2, 2×3, 3×4, 4×5, 5×6, so the next term is 6 × 7 = 42.
Digit tricks are rare in serious tests but common in puzzle books: reversing digits, adding the digits of the previous term, or counting letters in number words. Treat them as the last resort.
Strictly, no. Any finite list of numbers can be continued in infinitely many ways, because you can always invent a more complicated rule that fits. Puzzle setters follow a convention: the intended answer is the simplest rule that fits every term. That is why the checklist starts with the simplest rules.
The Online Encyclopedia of Integer Sequences (OEIS) catalogues hundreds of thousands of sequences, and it is a good place to look up a pattern you cannot crack.
Then put the checklist under time pressure in Pattern Sequence. If you want a memory twist, Number Sequence shows digits one at a time and asks you to recall them in order, and Odd One Out trains the same rule-spotting with shapes. For logic beyond numbers, see how to solve logic puzzles and our sudoku tips for beginners.
| Puzzle | Answer | Rule |
|---|---|---|
| 7, 12, 17, 22, ? | 27 | +5 |
| 1, 2, 4, 7, 11, ? | 16 | gaps 1, 2, 3, 4, 5 |
| 5, 10, 20, 40, ? | 80 | ×2 |
| 1, 8, 27, 64, ? | 125 | cubes |
| 2, 2, 4, 6, 10, ? | 16 | add the previous two |
| 5, 15, 12, 36, 33, ? | 99 | ×3 then −3 |
| 100, 1, 90, 2, 80, ? | 3 | two interleaved sequences |
| 0, 2, 6, 12, 20, ? | 30 | n × (n − 1) |
Check the differences between terms first, then the differences of those differences, then ratios. Those three checks solve most puzzles. After that, look for famous families such as squares and Fibonacci, and for interleaved sequences.
Arithmetic sequences, where the same number is added each time, followed by sequences whose gaps grow steadily and geometric sequences, where each term is multiplied by the same number.
Mathematically, yes. By convention the correct answer is the simplest rule that fits all the terms given.
Sequence items appear on reasoning tests because they involve spotting rules, but a single puzzle or game score is not an intelligence assessment. Most of the skill is knowing the checklist.
Learn the checklist, memorise the common number families, and practise regularly with timed puzzles so the first three checks become automatic.
Differences, second differences, ratios, families, then the unusual rules: test them in that order and most sequences fall in seconds. Keep the checklist open and play Pattern Sequence to see how far up the difficulty ladder you can climb.