A number series question shows a row of numbers with the next one missing. Behind the row sits a rule, and the rules come from a surprisingly short list. Learn the list and most questions stop looking like puzzles and start looking like recognition.
Why the list is short
Whoever writes a series has to be able to defend it. The rule must fit every number shown, it must be findable by hand, and it must lead to one answer. That constraint rules out anything exotic and leaves a handful of families.
The study of how people solve these tasks is old. Cognitive psychologists have examined how people infer rules from sequences since at least the 1960s (Simon and Kotovsky, 1963), and number series were analysed as a task of numerical rule induction in the early 1980s (Holzman, Pellegrino and Glaser, 1982). Series completion sits among the reasoning abilities mapped in the 1993 survey of factor-analytic studies by Carroll.
The seven below cover most of what you will meet on the JobCannon Number Series test and on employer numerical papers. For each one, here is how it looks and one series to try before you read the answer.
1. Add or subtract the same number
The steady staircase. Every step is the same size, so the gap between neighbours never changes.
6, 11, 16, 21, 26, ? The gap is always 5, so the next term is 31. Another: 17, 23, 29, 35, 41, ? adds 6 each time and gives 47.
2. Multiply or divide by the same number
The numbers grow (or shrink) quickly, and each term divided by the one before it is always the same.
2, 6, 18, 54, 162, ? multiplies by 3 each time, so the next term is 486. 3, 12, 48, 192, ? multiplies by 4 and gives 768.
3. Gaps that follow a pattern
Here the gaps are not equal, but the gaps themselves are regular. Write them down and look at that second row.
1, 2, 4, 7, 11, 16, ? has gaps 1, 2, 3, 4, 5, so the next gap is 6 and the term is 22. 2, 5, 11, 20, 32, ? has gaps 3, 6, 9, 12, so the next gap is 15 and the answer is 47. The gaps can fall as well: 100, 94, 85, 73, 58, ? has gaps of 6, 9, 12 and 15 taken away, so the next step takes away 18 and the term is 40.
A related version has square numbers as the gaps. 3, 4, 8, 17, 33, 58, ? has gaps 1, 4, 9, 16, 25, so the next gap is 36 and the term is 94.
4. Squares and cubes
These look odd at first and become easy once you know the lists. 4, 9, 16, 25, 36, ? is 2², 3², 4², 5², 6², so the next term is 7², which is 49. 1, 8, 27, 64, 125, ? is the cubes of 1 to 5, so the next is 6³, which is 216.
Squares and cubes are the one family where memory beats method. The tricks article lists the ones worth knowing by heart.
5. Each term built from the ones before it
The best-known case is the Fibonacci pattern, where each term is the sum of the two before. 2, 3, 5, 8, 13, 21, ? gives 34. A close cousin adds the previous three: 1, 2, 4, 7, 13, 24, ? gives 7 + 13 + 24 = 44.
These get a full article of their own, because they are the pattern people most often miss. The clue is that the numbers grow faster than a staircase but the gaps do not follow a simple rule.
6. Two series woven together
Every second number belongs to a different series. The row zigzags or looks irregular, because you are reading two rules at once.
5, 100, 8, 90, 11, 80, 14, 70, ? Take the odd positions: 5, 8, 11, 14 go up by 3. The even positions: 100, 90, 80, 70 go down by 10. The ninth term is in the odd group, so it is 17.
7. A pair of operations repeated
One operation is not enough, so the writer chains two. 3, 7, 15, 31, 63, ? doubles and then adds 1 every time, which gives 127. 5, 10, 8, 16, 14, 28, ? doubles, then takes away 2, then doubles again, so the next term is 26.
Chained rules fool the gap method, because the gaps (4, 8, 16, 32 in the first series) are themselves a rule. They are findable, but only if you keep looking after the gaps fail to be constant.
How the JobCannon test groups them
The JobCannon Number Series test groups its questions by these shapes: steady steps (rules 1 and 2), growing steps (rules 3 and 4), two series woven together (rule 6) and chained rules (rules 5 and 7). Seeing which group a question falls into is itself the first step of solving it.
If you want the broader numeric skill rather than series alone, that is what the JobCannon Numerical Reasoning test covers: percentages, ratios and reading tables. The two tests measure different things, and the series test is the narrower, rule-hunting one. Try the number series practice page when you want to drill the seven rules above.