Number Sequence IQ Test Questions: Six Checks, Always in Order

A number series item hands you five or six numbers and asks for the next one. It feels like a test of inspiration, where the pattern either jumps out or it does not, and that feeling is why people do badly on them. Nothing jumps out. What happens inside someone reliably good at these is more boring: a short search over a finite list of checks, run in a fixed order, stopping at the first that fits every term. The list takes a minute to memorise. The order does the work.

The checks, and the order to run them

There are six, and between them they cover most of the number-series items you are likely to meet: on an aptitude battery — a set of timed subtests sat in one session — on an entrance exam, or online. The order is not a matter of taste: the cheapest operation goes first, and each check that fails narrows the next. A failure here is information, not a wasted step.

• First differences. Subtract each term from the one that follows it. If the row you get is constant, the rule is add the same amount every time, and you are finished.
• Second differences. Take the differences of that row. A constant second row means a squared rule — the position number multiplied by itself, with whatever else has been added on top.
• Ratios. Divide each term by the one before it. A constant ratio means multiplication: doubling, tripling, halving. If the ratios instead form a tidy series of their own, that series is the rule.
• Alternation. Read the odd positions as one sequence and the even positions as another. Two interleaved series look chaotic as one row and trivial as two.
• Index-linked terms. Compare each term with its own position number rather than its neighbours: n squared, n cubed, n times the next number up, factorials, the primes, and each of those shifted by a constant.
• Digit operations. The last resort, where the rule acts on how a number is written rather than what it is worth — its digit sum, its digits reversed, its digits treated as separate numbers.

Two things stop this being a fishing expedition. The first is a standard you can apply to your own answer: a well-formed item has exactly one rule, and it is simple enough to say out loud in a single sentence. Add four. Double it. The primes in order. If what you are holding needs a subordinate clause and an exception, go back a check. That is the standard we applied to matrix items, and it holds for the same reason: the wrong answers are written by someone who knows which half-solutions tempt.

The second sounds like a technicality and is not. Strictly, these items have no unique answer: a curve can be fitted through any finite set of points, so some rule always reproduces your five terms and then delivers whatever sixth you like. Items work anyway because everyone accepts a convention — the intended answer is the one the simplest fitting rule generates. That is why "sayable in one sentence" is a usable test and not a slogan.

Why first differences come first

Partly because subtraction is the cheapest thing you can do to a row of numbers, and partly because the family it detects — terms rising or falling by a steady step — is a very common one. The better reason is structural: a difference table is self-terminating, and it names the shape of the answer before you have the answer. Write the sequence along a line, put the differences underneath, then the differences of those, and keep going. If the terms come from a polynomial rule — one built by multiplying the position number by itself some fixed number of times — the table reaches a constant row and stops. Constant on the first row means the rule is linear; on the second, squared; on the third, cubed. The depth of the table gives you the degree of the rule, while you do nothing but subtract small numbers.

The failures are equally informative, and that is what makes the ordering pay. Difference a doubling sequence and you get the doubling sequence back, so a table that keeps reproducing itself is telling you to go to the ratio check. Differences that alternate in sign — up seven, down five, up nine, down seven — say the row is two sequences interleaved, before you have looked at a single value. Gaps with no structure at all say to stop looking at neighbours and start on the catalogue.

Six sequences, worked

Each is settled by one of the six checks, in the order the checks run. Try them first — the arithmetic is small enough to do in your head, which is itself a mark of a well-built item.

3, 7, 11, 15, 19, ? First differences: 4, 4, 4, 4. Constant on the first row, so check one has finished. The rule is add four to the previous term; the answer is 23. Four subtractions settled it, which is the whole argument for running this check before any other.

2, 6, 12, 20, 30, ? First differences: 4, 6, 8, 10 — not constant, but tidy. Second differences: 2, 2, 2. Constant on the second row, and you can finish without naming the rule: the next second difference is 2, so the next first difference is 12, so the next term is 30 plus 12. Check five arrives by another road: 2 is 1 times 2, 6 is 2 times 3, 12 is 3 times 4. The rule is each term is its position multiplied by the next number up; the answer is 42. When two checks agree, you have the intended rule and not a coincidence.

3, 6, 12, 24, 48, ? First differences: 3, 6, 12, 24 — the sequence you started with, one term shorter. Second differences: 3, 6, 12. This table will never settle, because differencing a doubling sequence returns a doubling sequence, and that non-termination is what sends you to check three. Ratios: 2, 2, 2, 2. The rule is double the previous term; the answer is 96.

2, 9, 4, 13, 6, 17, 8, ? First differences: up 7, down 5, up 9, down 7, up 11, down 9. The alternating sign is a flag about structure, not values. Split by position: the first, third, fifth and seventh terms read 2, 4, 6, 8; the second, fourth and sixth read 9, 13, 17. One counts up by two, the other by four, and the missing eighth term belongs to the second. The rule is two interleaved sequences, one adding two and one adding four; the answer is 21.

1, 8, 27, 64, 125, ? If you recognise the cubes on sight, check five finishes in one step: each term is its position number cubed — position times position times position — so the answer is 216. If you do not recognise them, the table still gets there. First differences: 7, 19, 37, 61. Second: 12, 18, 24. Third: 6, 6. Constant on the third row, which says cubic before a single cube has been identified. Run it back down: next third difference 6, next second 30, next first 91, next term 125 plus 91.

16, 7, 49, 13, 169, ? Every value-based check fails: the differences swing wildly, the ratios are nothing, and splitting by position gives 16, 49, 169 against 7, 13 — which helps only once you notice that 49 is 7 times 7 and 169 is 13 times 13. The rule is acting on the digits. 1 plus 6 is 7, 7 squared is 49, 4 plus 9 is 13, 13 squared is 169, and 1 plus 6 plus 9 is 16. Said in a sentence — alternately take the digit sum and square the result — it passes the standard. Note what it quietly depends on: base-ten notation.

Number series are not culture-fair, and it matters

This is where the parallel with matrix items stops. A matrix puzzle can be built with no numbers, no words and no arithmetic in it: shapes rotate, counts go up, a feature in two rows is absent from the third. A number series cannot. Attempting one needs fluent arithmetic, multiplication tables solid enough that 6 times 7 is recognised rather than computed, enough exposure to spot a square going past, and, for digit rules, base-ten notation. All of that is taught.

So performance here mixes two things that are not the same: reasoning about a novel problem, which the item means to measure, and acquired numeracy, which makes that reasoning possible inside the time allowed. Someone who has not looked at a multiplication table in twenty years may well be slower on the second sequence above for reasons unrelated to finding rules. Batteries disagree about how to handle that — some group a number-series test with fluid reasoning, meaning problem-solving that does not lean on stored knowledge, others put series items in a quantitative section — and the disagreement is really about how much of the item is schooling.

It is also why assessments built to travel between languages and school systems reach for figural items instead — a choice that sits close to the centre of how an IQ test is built. Guidance on adapting tests across cultures exists because content that looks neutral often is not, and arithmetic notation is a clear case. A culture-fair format attempts to reduce that dependence rather than remove it: no format is free of culture, and the label is a claim about degree. It is a different instrument from a number-series test, not a better one.

Using the checks when the clock is running

• Write the sequence and its differences down before you think. The thinking is cheap; the remembering is what fails.
• Do not extrapolate before you have a rule. Guessing the next term from the drift of the numbers is what the distractors — the wrong options — are built to catch, exactly as on matrix items.
• Test your rule against every term you were given, the first included. A rule that fits terms three to six and misses term one is the wrong rule, and skipping that check is how a solved item turns into a wrong answer.
• When two rules both fit, take the shorter one — that is the convention the item was written under.
• Set a time budget per item and keep it. If four checks have failed and you are over it, mark your best candidate and move on.
• Expect practice to buy recall more than reasoning: squares, cubes and times tables back within reach, which shortens the search rather than deepening it.

Be clear about what a good score on sequences alone would mean, which is not much. Saying anything about reasoning in general needs many items across several formats, graded in difficulty and scored against a defined reference sample. That is what a full IQ test is for. If you already hold a result, our percentile calculator turns it into a percentile — the share of that sample scoring at or below you. An IQ figure quoted without its mean and standard deviation, meaning the centre of the scale and the typical distance scores sit from it, is not yet a number you can reason about.

The reason to learn the six checks is not the marks they are worth. It is that they turn a feeling into a procedure, and a procedure can be run calmly while a feeling cannot. You stop waiting for the pattern to announce itself and start asking six questions with cheap answers. And when the procedure runs out, the likeliest conclusion is that you have misread a term or slipped a subtraction, not that the item wants something baroque. A well-formed item, by the standard set out above, does not.

Common questions