Number series questions: how to solve them fast

The seven patterns

1. Add or subtract the same amount. 4, 11, 18, 25, ... adds 7 each time, so 32 is next.

2. Multiply or divide by the same amount. 3, 6, 12, 24, ... doubles, so 48. And 400, 200, 100, ... halves, so 50.

3. A growing (or shrinking) difference. 2, 3, 5, 8, 12, ... adds 1, 2, 3, 4, so +5 gives 17.

4. Alternating rules. 5, 10, 8, 16, 14, ... alternates x2 and -2, so the next step is x2: 28.

5. Squares and cubes. 1, 4, 9, 16, 25 are 1 x 1 to 5 x 5. 1, 8, 27, 64 are cubes. Watch for "square plus one": 2, 5, 10, 17.

6. Combined steps. 2, 5, 11, 23, ... is "double, then add 1", so 47 is next.

7. Two series interleaved. 1, 20, 3, 18, 5, 16, ... has odd positions going up by 2 (1, 3, 5) and even positions going down by 2 (20, 18, 16), so 7 comes next.

A few tests also use the Fibonacci pattern, where each number is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13.

A fast method

1. Write the differences between neighboring numbers underneath. Most series reveal themselves here: a constant difference (pattern 1) or a pattern in the differences (pattern 3).
2. If the differences grow fast, try ratios instead: divide each number by the one before (patterns 2 and 6).
3. If the series zigzags, split it into odd and even positions (pattern 7) or look for alternating operations (pattern 4).
4. If nothing fits, check for squares, cubes or Fibonacci.
5. Check your rule against every number, not just the last two. A rule that fits most of the series is not the rule.

This takes seconds once it is a habit, and it stops you guessing from the answer options.

Worked examples

7, 10, 16, 25, 37, ... Differences: 3, 6, 9, 12. They grow by 3, so the next difference is 15 and the answer is 52.

3, 9, 27, 81, ... Each number is 3 times the one before, so 243.

64, 32, 34, 17, 19, ... Alternating: halve, add 2, halve, add 2. The next step is halve: 9.5.

4, 9, 19, 39, ... Double and add 1: 4 x 2 + 1 = 9, 9 x 2 + 1 = 19, 19 x 2 + 1 = 39, so 79.

2, 12, 4, 10, 6, 8, ... Two series: 2, 4, 6 (up by 2) and 12, 10, 8 (down by 2). The next number is in the first series: 8.

How number series questions look in job tests

The same patterns turn up in several question formats. Knowing the format in advance saves you from reading the instructions twice under the clock.

Next number. The most common format: four to six numbers and a choice of four or five answers.

Missing number in the middle. 4, 8, ?, 32, 64. Treat the gap as two steps. Here each number doubles, so the gap is 16. Check both sides: 8 x 2 = 16 and 16 x 2 = 32.

Odd one out. "Which number does not belong?" 3, 6, 12, 25, 48. The series doubles, so 24 should sit where 25 is. These items reward checking every term, which is the habit from step 5 of the fast method.

Letter series. A, C, F, J, ... uses the same logic with letters. Convert to positions in the alphabet (A = 1, C = 3, F = 6, J = 10). The gaps are 2, 3, 4, so the next gap is 5 and the answer is position 15, the letter O.

Pattern questions inside broader tests. Fast general cognitive tests mix series with word, math and logic items. Wonderlic, for example, lists "pattern identification" among the question types on its 50-question, 12-minute cognitive test (Wonderlic). Criteria describes a "math and logic" question type on the CCAT (Criteria).

Harder patterns you may meet

When the seven basic patterns do not fit, one of these usually does.

Differences that multiply. 2, 3, 5, 9, 17, ... The differences are 1, 2, 4, 8, which double each time. The next difference is 16, so the answer is 33. (You can also read it as "double, then subtract 1": 17 x 2 - 1 = 33. Two routes to the same answer is a good sign.)

Multiply by a growing number. 1, 2, 6, 24, 120, ... multiplies by 2, then 3, then 4, then 5. The next step is x6, giving 720.

Second-level differences. 1, 2, 6, 15, 31, ... The differences are 1, 4, 9, 16, which are squares. The next square is 25, so the answer is 31 + 25 = 56.

Prime numbers. 2, 3, 5, 7, 11, 13, ... are the primes (numbers divisible only by 1 and themselves). The next is 17. Keep the primes up to 30 in your head: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Alternating signs. 3, -6, 12, -24, ... multiplies by -2 each time, so 48 comes next.

Fractions. 1/2, 2/3, 3/4, 4/5, ... Both top and bottom go up by 1, so 5/6 is next. Treat the top and bottom as two separate series.

The traps test writers set

Wrong answer options are rarely random. Each one is usually what you get from a common mistake, so knowing the mistakes helps you spot the bait.

• Stopping at the first rule that fits. In 2, 4, 8, 14, ... doubling fits the first three numbers, so 28 looks tempting. But 14 breaks it. The differences are 2, 4, 6, so the next is +8: 22.
• Taking the wrong series. In an interleaved series, count which position the blank is in. In 3, 30, 6, 27, 9, 24, ... the next number is the 7th term, which belongs to the odd positions (3, 6, 9), so it is 12, not 21.
• Small arithmetic slips. When the differences are large, add them on paper or in your head twice. An option that is off by exactly 1 or 10 is often there for this reason.
• Mixing up the direction. In a decreasing series such as 96, 48, 24, ... it is easy to multiply instead of divide. The answer is 12, and 48 or 96 are likely options.
• Reading a missing-middle question as next-number. Check where the question mark is before you start.

Using the answer options

The options are information. Use them, but only after you have tried to find the rule yourself.

Estimate first. If a series grows by multiplying, the answer should be much larger than the last term. If two options are close to the last term and two are far above it, you can often rule out half the list straight away.

Check odd and even. If every term is odd and the rule adds an even number, the answer must be odd too.

Test an option backward. If you are stuck between two answers, put each one at the end of the series and look at the new differences. The right answer keeps the pattern tidy; the wrong one breaks it.

Do not reverse-engineer from options alone. Picking the option that "looks right" is how the trap answers work. Spend your first few seconds on the differences.

Pacing number series under time pressure

On fast cognitive tests, time per question is short. Wonderlic's cognitive test is 50 questions in 12 minutes (Wonderlic), about 14 seconds each. The CCAT is 50 questions in 15 minutes (Criteria), about 18 seconds each. Neither allows a calculator, according to their publishers (Wonderlic FAQ, Criteria).

At that pace, a simple add or multiply series should take you under 15 seconds. A two-step or interleaved series may take 25. If you have not found the rule in about 30 seconds, pick the most likely option and move on. Criteria says there is no penalty for wrong answers on the CCAT, so an informed guess costs nothing there. Wonderlic advises candidates not to spend too long on any one question and not to skip around hoping to come back.

In numerical or logical reasoning tests with fewer, longer questions, you have more time, so check every term before you commit.

How to practice number series

Short daily sets work better than one long session. Ten series a day for four or five days builds the pattern recognition you need.

Always write the differences. Even when you think you can see the answer, writing the difference row trains the habit you will rely on under pressure.

Time yourself. Start untimed to learn the patterns, then set a limit of 20 seconds per series, then 15.

Review every miss. Note which pattern you missed. Most people find one or two patterns (usually interleaved and second-level differences) cause most of their errors.

Keep your mental math sharp. Series questions sit next to percentages, ratios and rates on most cognitive tests. Our guides to percentage questions, ratio questions and speed, distance and time cover those, and syllogisms covers the logic items that often appear alongside.