How do you get better at number series?

Write the differences between neighbouring numbers. If they're constant, it's adding; if they grow by a fixed amount, look at the second differences; if the ratio is constant, it's multiplying; if each term is the sum of the two before, it's a Fibonacci-type series.

What's Your IQ

What is a pattern recognition test?

A pattern recognition test asks you to find the rule behind a sequence or a set of items, then apply it: complete a number, letter or shape series, or spot the one item that breaks the rule. Psychologists call this inductive reasoning.

Series and odd-one-out items have been part of intelligence testing since Louis Thurstone listed induction as a primary mental ability in 1938. They're still used in aptitude tests for jobs in finance, engineering and the armed forces, because they don't depend on specialist knowledge.

PATTERN RECOGNITION VS MATRIX REASONING

FormatA row of items, or a set to compareA 3×3 grid with one gap

ContentNumbers, letters and shapesShapes only

Main skillFinding one rule along a lineCombining rules across rows and columns

Take the matrix reasoning test instead →

SEQUENCE LAB

The trick for number series: look at the gaps

Pick a rule and a starting number. Underneath the series you'll see the differences between neighbours, and the differences of those. Each rule leaves a recognisable fingerprint in the gaps.

SERIES

2581114172023

1ST DIFFERENCE

3333333

2ND DIFFERENCE

000000

Starting number2

FINGERPRINTConstant first difference (3): an arithmetic series.

First differences are all the same. In a test you'd see the first four or five terms; working out the gaps on paper (or in your head) usually reveals the rule in seconds.

THE TEST

20 questions, 15 minutes

Before you start

1Twenty questions: number series, letter series, shape sequences and odd-one-out, mixed and getting harder.

2For series, choose what comes next. For odd one out, choose the item that breaks the rule.

3You have 15 minutes. Skip any question and come back using the numbered squares.

4At the end you'll see your score for each pattern type and the rule behind every answer.

Answers stay on this device.

Number seriesWhat comes next?

YOUR RESULT0 / 20Just startingThis is a practice test, not a normed IQ test, so it gives a raw score rather than an IQ. The breakdown shows which kinds of pattern come easily and which are worth practising.

Number series

0 / 5

Letter series

0 / 5

Shape sequences

0 / 5

Odd one out

0 / 5

Time 29854926:01Strongest even across all four typesTo practise none in particular

Every answer, explained

Q1 · SkippedNumber seriesAnswer C: 19Add 4 each time: 15 + 4 = 19.

Q2 · SkippedLetter seriesAnswer C: ISkip one letter each time: A, C, E, G, I.

Q3 · SkippedShape sequencesAnswer C: circle ×5, solidOne more solid circle each step: five solid circles.

Q4 · SkippedOdd one outAnswer C: circle, solidFour are squares; C is a circle.

Q5 · SkippedNumber seriesAnswer B: 162Multiply by 3 each time: 54 × 3 = 162.

Q6 · SkippedLetter seriesAnswer B: RGo backwards two letters at a time: T, then R.

Q7 · SkippedShape sequencesAnswer D: triangle, outlineA repeating cycle of circle, square, triangle. After the square comes the triangle.

Q8 · SkippedOdd one outAnswer E: triangle, outline, 180°Four triangles point up; E points down.

Q9 · SkippedNumber seriesAnswer C: 36Square numbers: 1², 2², 3², 4², 5², so 6² = 36.

Q10 · SkippedLetter seriesAnswer D: PThe gap grows by one: +1, +2, +3, +4, then +5. K + 5 = P.

Q11 · SkippedShape sequencesAnswer A: triangle, solidThe triangle turns 90° clockwise each step; after 270° it's back to pointing up.

Q12 · SkippedOdd one outAnswer D: 4816, 25, 36 and 64 are square numbers (4², 5², 6², 8²); 48 isn't.

Q13 · SkippedNumber seriesAnswer B: 0Subtract 1, 2, 3, then 4: 4 − 4 = 0.

Q14 · SkippedLetter seriesAnswer C: EVFirst letter steps forward through the alphabet, second steps backward from Z: EV.

Q15 · SkippedShape sequencesAnswer C: square, outlineTwo rules: one fewer square each step, and the fill alternates solid, outline. One outline square.

Q16 · SkippedOdd one outAnswer D: triangle, outlineIn the others the inner shape matches the outer one. D has a circle inside a triangle.

Q17 · SkippedNumber seriesAnswer C: 65Double and subtract 1 (the gaps 2, 4, 8, 16 double too): 33 × 2 − 1 = 65.

Q18 · SkippedLetter seriesAnswer D: ITwo alternating steps: forward 3, back 1. F + 3 = I.

Q19 · SkippedShape sequencesAnswer C: triangle, outlineShapes cycle every three, fills every two, so they drift out of step. Next: triangle, and the fill alternates to outline.

Q20 · SkippedOdd one outAnswer C: DIIn each pair the second letter is four after the first (A→E, B→F, C→G, E→I). D→I is five.

WORKED EXAMPLE

Solving the hardest number series, one step at a time

Question 17 is the hardest number item in this test. Step through how to crack it, then use the same moves on the other pattern types.

Number series:What's the gap?Write the differences. Constant means adding; growing evenly means squares; a constant ratio means multiplying.

Letter series:What number is each letter?Turn letters into positions (A=1, Z=26) and it becomes a number series. Watch for two series interleaved.

Shape sequences:What's changing, and how often?List each property separately (shape, count, fill, angle). Different properties can repeat on different cycles.

Odd one out:What do four of them share?Find the rule that fits most items first; the odd one is whichever breaks it. Check there isn't a second rule that singles out a different item.

3591733?

1Write the gaps…

2Look at the gaps themselves…

3Test a rule on the terms…

4Check it on every term…

5Apply it…

A century of studying patterns

From perception to puzzles to machines

1923Gestalt laws.Max Wertheimer describes how we group what we see by proximity, similarity and continuation.

1938Primary mental abilities.Louis Thurstone lists induction, finding rules from examples, as one of seven core abilities.

1963Letter series modelled.Herbert Simon and Kenneth Kotovsky build a computer model of how people solve letter-series problems.

1973Chess chunks.Chase and Simon show chess masters recall positions by recognising familiar patterns, not by better raw memory.

2012Machines learn to see.Deep neural networks dramatically improve automatic image recognition, reviving interest in how pattern recognition works.

TESTMatrix reasoning test15 3×3 matrices, from one rule to four.Open

TRAININGPattern recognition gamesPractise spotting rules every day.Open

GAMESequence CompletionEndless series, adapting to your level.Open

TESTInductive reasoning testFind the rule from examples.Open

2, 3, 5, 8, 13… you already know what comes next.

What is a pattern recognition test?

The trick for number series: look at the gaps

20 questions, 15 minutes

Solving the hardest number series, one step at a time

A century of studying patterns

Patterns are one piece of IQ. Measure the rest.