Guide
How to solve number series questions
A number series gives you a short list of numbers and asks for the next one. The arithmetic is almost always easy. The work is in finding the rule, and a short, fixed routine finds most rules in under a minute.
Want your own number? The test is 25 visual puzzles, about 12 minutes, no sign-up.
What a number series asks
You see five or six numbers followed by a question mark, and you choose the next number from a set of options. On IQ Octopus these items make up the Quantitative Reasoning section, 3 items ordered from easier to harder. They test the same thing as the visual puzzles: reasoning with new material, which psychologists call fluid intelligence (Wikipedia: Fluid and crystallized intelligence). The numbers just make the rule easier to write down.
The rule types, from easy to hard
Nearly every series you will meet uses one of a small set of rules. These are the ones the IQ Octopus engine uses, with a made-up example of each.
| Rule | Example | Next | Why |
|---|---|---|---|
| Steady gaps | 4, 9, 14, 19, 24 | 29 | Every gap is 5. |
| Steady shrink | 50, 43, 36, 29, 22 | 15 | Every gap is -7. |
| Doubling | 3, 6, 12, 24, 48 | 96 | Each number is twice the one before. |
| Tripling | 2, 6, 18, 54 | 162 | Each number is three times the one before. |
| Growing gaps | 9, 12, 16, 21, 27 | 34 | Gaps 3, 4, 5, 6 grow by 1. |
| Two series | 2, 20, 5, 18, 8, 16 | 11 | 2, 5, 8 go up by 3; 20, 18, 16 go down by 2. |
| Squares | 1, 4, 9, 16, 25 | 36 | 1x1, 2x2, 3x3 and so on. |
| Alternating steps | 3, 6, 10, 20, 24 | 48 | Double, add 4, double, add 4. |
| Double plus a constant | 2, 5, 11, 23, 47 | 95 | Each number is double the last, plus 1. |
| Sum of the two before | 2, 3, 5, 8, 13 | 21 | Each number is the two before it added together. |
| Gaps that double | 1, 3, 7, 15, 31 | 63 | Gaps 2, 4, 8, 16 double each time. |
Easy items use one steady step: add, subtract or double. Medium items use growing gaps, two series woven together, square numbers or tripling. Hard items use two steps that take turns, a multiply-then-add rule, sums of the two previous numbers, or gaps that themselves double.
Try one · Quantitative Reasoning
Which number comes next?
3, 8, 15, 24, 35, ?
Tap the option you think is right, or show the answer.
Show the answer
The next number is 48 (option 5).
How to solve it
- The gaps (5, 7, 9, 11) grow by 2 each time, which is the pattern of square numbers.
- Each number is a square number minus 1: 2 × 2 - 1, 3 × 3 - 1, 4 × 4 - 1, 5 × 5 - 1, 6 × 6 - 1.
- So the next number is 7 × 7 - 1 = 48.
That was 1 puzzle. The full test has 25.
A routine that finds the rule
1. Write the gaps
Subtract each number from the next and write the results under the series. This one step solves most easy and medium items. If every gap is the same, add it once more. If the gaps change by a steady amount, continue the gaps, then add. If the gaps double, double the last gap.
2. If the gaps jump around, split the series
Look at the 1st, 3rd and 5th numbers, then the 2nd, 4th and 6th. Two separate series taking turns is a common trick, and each half is usually simple. The missing number continues whichever half comes next.
3. If the numbers grow fast, try multiplying
When gaps keep getting larger, compare each number with the one before it. Is it two or three times as big? If it is close to double but not exact, check for double plus or minus a constant. If the multiplication fits every other step, look for two steps that alternate.
4. Try the two-before rule
If nothing above fits, add the two previous numbers and see if you get the next one. This rule is easy to miss because the gaps look irregular.
5. Check against every number shown
A rule that fits the last three numbers but not the first is the wrong rule. Wrong options are built from exactly these near misses: the answer you get if you assume steady gaps when they actually grow, or if you continue the wrong half of a split series. Checking the whole series before you choose catches almost all of them.
A worked example
Series: 9, 12, 16, 21, 27, ?
- Gaps: 3, 4, 5, 6. Not steady, so this is not plain adding.
- The gaps grow by 1 each time. That is a steady change in the gaps.
- The next gap is 7, so the next number is 27 + 7 = 34.
- Check: 9 + 3 = 12, 12 + 4 = 16, 16 + 5 = 21, 21 + 6 = 27. The rule fits every step.
A tempting wrong option here is 33, which you get by adding the last gap again. Another is 35, a near miss on the arithmetic. The routine avoids both.
Pacing
Give each item a fixed budget. If two passes through the routine find nothing, choose the most likely option and move on; a later item may be easier for you. On a timed test, the gaps step alone is worth doing on every item, because it is quick and solves the most items.
Does practice help?
A little, mostly by making the format familiar. Retesting raises scores on cognitive tests by about a quarter of a standard deviation on average (Hausknecht et al., 2007). For number series, the most useful practice is simply knowing the rule types above so you check them in order instead of guessing.
Try the number series puzzle library, where every item comes with its rule explained step by step, or take the timed number sequence practice test. The Quantitative Reasoning practice page shows one worked example from the test itself, and the pattern recognition guide covers the visual version of the same skill.
Sources
- Fluid and crystallized intelligence. Wikipedia. en.wikipedia.org/wiki/Fluid_and_crystallized_intelligence
- Hausknecht, J. P., Halpert, J. A., Di Paolo, N. T., & Moriarty Gerrard, M. O. (2007). Journal of Applied Psychology, 92(2), 373-385. doi.org/10.1037/0021-9010.92.2.373