Number sequence puzzles: patterns, examples, and how to solve them
A number sequence puzzle gives you ordered terms and asks for the missing one. You find the rule that produces each term from the ones before it — a constant gap, a growing gap, multiplication, two interleaved lists, or a known family such as squares — then apply that rule one step further.
Published 2026-10-27
Start with the gaps, not a guess
Write the difference between neighbours before you invent a clever rule. A steady difference is an arithmetic sequence. A difference that itself grows by a constant is still a simple rule, just one level down. If the differences look random, check multiplication, then check whether odd and even positions are separate sequences.
Say the rule in one sentence before you commit to an answer. “Add 3 each time” is a rule. “It feels like it doubles” is a hunch. Hunches are useful, but the sentence is what you can test against every term, not only the last one.
Pattern types worth recognising
These families cover most short sequence puzzles. They are not a ranking of intelligence. They are a checklist so you do not stare at the last term and hope.
- Constant difference: 4, 7, 10, 13 adds 3.
- Second difference: the gaps themselves make a simple sequence, often the powers of two or the odd numbers.
- Squares and factorials: 1, 4, 9, 16, 25 or 1, 2, 6, 24, 120.
- Product of neighbours: 2, 6, 12, 20, 30 is n × (n + 1) if you start n at 2.
- Two sequences woven together: odd positions follow one rule, even positions another.
Worked examples
Try each one before you open the answer. Every sequence below was written for this page.
Easy
4, 7, 10, 13, 16, ?
Answer and reasoning
Answer. 19
The gap is +3 every time. 16 + 3 = 19.
Easy
1, 4, 9, 16, 25, ?
Answer and reasoning
Answer. 36
These are the squares 1², 2², 3², 4², 5². The next square is 6² = 36.
Medium
2, 6, 12, 20, 30, ?
Answer and reasoning
Answer. 42
Each term is a product of two consecutive whole numbers: 1×2, 2×3, 3×4, 4×5, 5×6. The next product is 6×7 = 42.
Medium
5, 6, 8, 12, 20, ?
Answer and reasoning
Answer. 36
The gaps are +1, +2, +4, +8. The next gap is +16. 20 + 16 = 36.
Medium
3, 8, 6, 11, 9, 14, 12, ?
Answer and reasoning
Answer. 17
Odd positions: 3, 6, 9, 12 (+3). Even positions: 8, 11, 14, and the next even position is 17 (+3).
Harder
1, 2, 6, 24, 120, ?
Answer and reasoning
Answer. 720
Multiply by 2, then 3, then 4, then 5, then 6. 120 × 6 = 720. These are the factorials 1!, 2!, 3!, 4!, 5!, 6!.
Harder
7, 8, 11, 16, 23, 32, ?
Answer and reasoning
Answer. 43
The gaps are the odd numbers +1, +3, +5, +7, +9. The next gap is +11. 32 + 11 = 43.
Mistakes that throw the next term off
The usual miss is fitting a rule to the last two terms and never checking the first. Doubling explains 8, 16 but not 5, 6, 8, 12, 20. Another miss is mixing two interleaved lists into one difference row, which looks noisy until you split odd and even positions.
A third miss is treating a puzzle score as a measure of general ability. Practising sequences makes you faster at sequences. A review of commercial brain-training programs found little convincing evidence that practice on one task lifts unrelated abilities.
Sources
On Two5
Enjoy number sequences? Two5’s Numbers category — Number Sprint, Sequence Hunt, and Target Number — is part of what we are building first. Rounds are not playable on this site yet. Join the waitlist and we will tell you when early access opens.