Number Series Puzzles
A row of numbers, one rule running underneath, and a gap at the end. Work out what turns each term into the next, then choose the number that continues it — and see the rule named, with a reason for every wrong option.
Also known as number sequences, find the next number and series completion.
Related Practice
Finding The Rule Behind A Run Of Numbers
A number series puzzle shows a row of numbers built by a single rule, with the last one missing. Your job is to work out what turns each term into the next and apply it once more. The rule might be as plain as adding four every time, or it might be that the gaps themselves grow, that each term is the sum of the two before it, or that two separate series have been woven together and you are looking at alternate terms of each.
What makes it a puzzle rather than arithmetic is that the rule is not stated. Several rules will usually fit the first two or three terms, and only one fits all of them — which is why the habit that matters is testing a rule against every term you were given before committing to it.
Start With The Gaps Between The Terms
There is a reliable order, and following it beats intuition on everything except the very easiest series. The point of a fixed order is that you stop rediscovering where to look.
- Write the differences underneath. Subtract each term from the next. If every gap is the same, the rule is a constant addition and you are finished in one step. This alone settles the largest single family.
- Take differences again. If the gaps are not equal, subtract the gaps from each other. A constant second difference means the gap grows by a fixed amount — the family people most often miss, because the terms look irregular until you write the gaps down.
- Divide if it grows fast. When a series outruns any sensible addition, try the ratio instead. A constant ratio means multiplication. Growth faster still usually means squares, cubes, or a two-step rule such as multiply then add.
- Check the known families. Squares, cubes and primes account for a large share of series, and they are recognised rather than calculated. Learning them by sight is the single biggest speed gain available.
- Split the positions. If nothing fits the whole row, read every other term. Two simple series interleaved look like one complicated series, and separating them makes both trivial.
The Ten Rule Families, And What Gives Each One Away
Every puzzle here comes from one of ten families. Knowing the tell for each is faster than testing rules one at a time.
| Family | Example | What gives it away |
|---|---|---|
| Steady gaps | 4, 9, 14, 19 | The differences are all equal. |
| Growing gaps | 3, 5, 9, 15, 23 | The differences are not equal, but the differences between them are. |
| Multiplying | 3, 6, 12, 24 | Dividing each term by the last gives the same number every time. |
| Two-step rules | 2, 7, 22, 67 | Grows faster than a ratio explains; try multiply-then-add. |
| Squares | 9, 16, 25, 36 | Recognised on sight; the gaps rise by a steady 2. |
| Cubes | 8, 27, 64, 125 | Grows far faster than squares and outruns most ratios. |
| Looking back two | 2, 5, 7, 12, 19 | Neither differences nor ratios work, but each term is the last two added. |
| Two series woven | 4, 40, 7, 36, 10, 32 | The row lurches up and down; alternate terms are each well behaved. |
| Primes | 5, 7, 11, 13, 17 | Irregular gaps that never settle, and every term is prime. |
| Alternating operations | 4, 12, 17, 51, 56 | The step changes character every other term. |
Why Every Wrong Option Here Is A Named Mistake
The three wrong answers are not numbers picked near the right one. Each is generated by making a specific, common error with the actual rule — applying it one step too many times, continuing the last gap when the series was multiplying, using the operation that has just been used rather than the one due next.
That is what lets the explanation say why an option was tempting rather than only that it was wrong. If you picked one of them, the reason given is very likely the reasoning you actually used, which is more useful than being told the answer was something else.
Number Series Puzzles Frequently Asked Questions
How do you find the rule in a number series?
Work through a fixed order rather than staring at it. Write the differences between consecutive terms underneath; if they are equal, the rule is a constant addition. If not, take differences again — a constant second difference means the gap grows steadily. If the series grows faster than that, divide instead of subtracting to test for a ratio. Only then check squares, cubes and primes, and last of all try reading alternate terms as two woven series.
What are the most common number series patterns?
Six cover most of what you will meet: a constant difference, a difference that grows by a fixed amount, a constant ratio, a two-step rule such as multiply then add, the well-known families of squares and cubes, and each term being the sum of the two before it. Two more turn up often enough to be worth knowing: prime numbers, and two separate series interleaved.
Why does my answer fit but still get marked wrong?
Usually because the rule was tested on too few terms. Almost any two terms can be joined by several rules, and three by more than one, so a rule that fits the start and fails later is not the rule. The other common cause is finding a rule that works but is not the simplest one available — series are built on the simplest consistent rule, and that is the convention every setter follows.
Are the puzzles here the same every time?
No. Each of the 60 patterns names a rule family, and the numbers are generated fresh from the seed, so the same pattern gives you a different series every visit. The one you played keeps a permanent link if you want to send it to someone or come back to it.
What is the difference between a number series and a wrong number puzzle?
A number series shows a run that obeys its rule all the way through and asks you to continue it. A wrong number puzzle shows a run in which exactly one term has been broken and asks you to find it. The skill is the same — identify the rule — but the second is harder, because you cannot trust any single term until you have tested it.
How can I get faster at number series?
Work through the same checks in the same order every time rather than hunting for inspiration. Look at the gaps between terms first; if the gaps are not constant, look at the gaps between the gaps; then try ratios; then check whether each term is built from the two before it. Most series fall to one of those four.
What is a number series?
A row of numbers built by a rule, with one of them missing — usually the next one. The rule might add a fixed amount, multiply, square, alternate between two operations, or weave two separate sequences together. You are not asked to recognise the numbers, only to work out what turns each one into the next.