Some series zigzag. They climb, drop, climb and drop, or the gaps jump around with no pattern at all. That is usually a sign of two simple series taking turns, and the way to solve them is to pull the two threads apart.
The method
- Number the terms: 1, 2, 3 and so on under the row.
- Write the odd-numbered terms (1st, 3rd, 5th, …) as one list.
- Write the even-numbered terms (2nd, 4th, 6th, …) as another.
- Find the rule for each list with the usual tools: gaps, ratios, and so on.
- Work out whether the missing term is in the odd list or the even list, and extend that one.
A worked example
5, 100, 8, 90, 11, 80, 14, 70, ?
Odd positions: 5, 8, 11, 14, which add 3 each time. Even positions: 100, 90, 80, 70, which take away 10 each time. Eight numbers are shown, so the missing term is the ninth, which is odd. The odd list continues 14 + 3 = 17. The answer is 17, not 60. Sixty is the next term of the other list, and it is the tempting wrong answer.
Two more
3, 50, 6, 45, 9, 40, 12, 35, ? has the odd list 3, 6, 9, 12 (adds 3) and the even list 50, 45, 40, 35 (takes away 5). The ninth term is in the odd list, so the answer is 15.
6, 30, 12, 25, 18, 20, 24, 15, ? has the odd list 6, 12, 18, 24 (adds 6) and the even list 30, 25, 20, 15 (takes away 5). Again the ninth term is odd, so 24 + 6 gives 30.
When both lists use the same kind of rule
The two lists do not have to be different in kind. 2, 3, 4, 6, 8, 12, 16, ? has the odd list 2, 4, 8, 16, which doubles, and the even list 3, 6, 12, which also doubles. Seven numbers are shown, so the missing term is the eighth, which is even. The even list continues 12 × 2 = 24.
Because both lists double, you could mistake the whole row for one series that stutters. Splitting is what reveals the structure.
Not every zigzag is two series
A single series can alternate its operations instead. 5, 10, 8, 16, 14, 28, ? goes up and down, but it is one rule applied in turn: double, then take away 2. Doubling 5 gives 10, taking away 2 gives 8, doubling gives 16, taking away 2 gives 14, doubling gives 28, and taking away 2 gives 26.
The difference: in a true two-series row each list stands alone and the lists do not depend on one another. In an alternating-operation row, each term depends on the one just before it. Try both; one will give a clean rule and the other will not.
Why splitting works
Splitting a row into repeating lists mirrors what research on sequence learning has long described as detecting the period of a pattern: how many terms pass before the structure repeats (Simon and Kotovsky, 1963). Two interleaved series have a period of two, and once the period is found, each list is an ordinary series.
Common traps
- Counting the position wrongly, so you extend the wrong list. Number the terms.
- Reading the first rule that appears in the odd list as the rule for the whole row.
- Giving up when the gaps look random. Random-looking gaps are the signal to split, not to stop.
The JobCannon Number Series test counts these as their own group, two series woven together, so your result shows how you do on them separately. Drill them on the number series practice page; the JobCannon Numerical Reasoning test covers the table-and-percentage side of numeric work.