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Series (Figures)

Reasoning · Non-Verbal Reasoning

8 Part2 worked examples6 practice questions

In a number series we look at 2, 4, 6, 8, … and say the next number. In a figure series the numbers are replaced by figures: five figures are given, a “?” stands in place of the sixth, and you choose the right figure from four options.

In every series one or two elements change by a fixed rule — an arrow turns, a dot moves along the border, the number of lines grows, or the shading changes in turn. You must catch the rule and run it one step further.

Every series here is built by computer from a rule, and while building it was checked that no other simple rule fits the five figures — so the answer is always unique. The options are made from real slips: repeating the 5th figure, going the wrong way, going one step too far.

How to use this chapter

  1. Read — first understand one kind of change, then the way to spot it.
  2. Study the solved example — every example shows the series and its rule, and the completed series below.
  3. Try it yourself — the solution is hidden; look at each element separately first, then choose.
  4. Know the level — Basic (one rule, a fixed step), Standard (a growing or alternating step), Advanced (two rules together).
NoteGood practice: before looking at the options, draw the sixth figure in your head. Then match it with the options. This way the options cannot confuse you.
1

Foundation — what can change in a series

Four kinds of change and the way to find the rule

The kinds of change in a series and a four-step method to catch them.

Kinds of change in a series

Turning · moving · counting · shading

Four basic changes and how to recognise each at a glance.

Almost every figure series is made from one or two of these four changes. Below are four small series, one in each row:

Top to bottom: a turning arrow, a dot moving along the border, growing lines, changing shading
ChangeWhat to look forForm of the rule
TurningThe direction of the figure changes; its shape staysThe same number of degrees clockwise / anticlockwise each time
MovingA part changes its place inside the boxThe same number of cells in one direction each time
CountingSides, lines, dots or shaded boxes grow or shrink+1, +2, −1 or a changing step
ShadingThe figure is filled or emptyUsually in turn

More than one element can change in a series — for example a triangle turns and is filled in turn. Then find the rule of each element on its own; the two rules are independent of each other.

RememberAn element that never changes is not part of the rule (for example the box is the same every time). Look for the rule only in the elements that really look different from figure to figure.
?ABCD
Example 1 · Basic
Which figure replaces the “?” in the series above?
Solution, step by step
  1. 1The arrow turns 90° clockwise every time; now it goes from down-right to down-left.
  2. 2Answer C. The rest: A = like the 4th figure (assumed a 2-step repeat); B = one step too far; D = off by just one place.
Answer Figure C
Only one element — the arrow — is changing here, so finding the rule is direct.
The completed series: the 6th figure is the answer found above
?ABCD
BasicTry it yourself first
Which figure replaces the “?” in the series above?
?ABCD
BasicTry it yourself first
Which figure replaces the “?” in the series above?
?ABCD
BasicTry it yourself first
Which figure replaces the “?” in the series above?

The four-step method

One element → the step → check → the next

A straight order to solve any series and the common traps.

Four steps for every series
1. Pick one element (arrow, dot, lines…)2. Measure its change between figure 1 and 2 (how many degrees / cells / numbers?)3. Test the same change on 2→3, 3→4, 4→54. When the rule holds, make the 6th figure, then match the options

Step 3 is the most important. If the change is not equal in every pair, the rule is not a fixed step; then look at these three forms:

What the changes look likeForm of the ruleThe next change
1, 2, 3, 4 (one more each time)Growing step5
2, 1, 2, 1 (two steps in turn)Alternating step2
Figures 1, 3, 5 and 2, 4 in separate seriesTwo interleaved seriesThe next term of the other series

To recognise turns, treat the arrow as the hand of a clock: a 90° turn equals 3 hours, a 45° turn equals one and a half hours and a 180° turn equals 6 hours. Up is 12 and right is 3.

CarefulThe commonest trap: deciding the rule from the first two figures only. The arrow turned 45°, then 90° — if you assumed “45° each time” from the first pair your answer will be wrong. Always check all four pairs.
?ABCD
Example 2 · Standard
Which figure replaces the “?” in the series above?
Solution, step by step
  1. 1The triangle turns 45°, then 90°, then 135°… — each turn is larger by 45°. The next turn is 135°, so it points up-left.
  2. 2Answer B. The rest: A = like the 4th figure (assumed a 2-step repeat); C = repeats the 5th figure; D = one step too far.
Answer Figure B
Here the step of the turn grows — you would slip if you stopped at the first pair.
The completed series: the 6th figure is the answer found above
?ABCD
BasicTry it yourself first
Which figure replaces the “?” in the series above?
?ABCD
BasicTry it yourself first
Which figure replaces the “?” in the series above?
?ABCD
BasicTry it yourself first
Which figure replaces the “?” in the series above?
2

Series of turning

How far did the arrow, triangle or pointer turn?

Clockwise and anticlockwise turns, steps of 45°/90°/180°, growing and alternating turns.

Plain turns: equal every time

Clockwise and anticlockwise, 45° / 90° / 135° / 180°

How to measure the angle of a turn and a trick to find its direction.

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Growing and alternating turns

45°, 90°, 135°, … or 45° and 135° in turn

When the angle is not equal each time: growing angle, alternating angles and two elements turning together.

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3

Series of moving

A dot or square moving along the border of a box

Counting the eight border cells, direction and equal, growing or alternating steps.

Equal steps along the border

Eight cells: clockwise or anticlockwise

Giving the border cells numbers and counting steps with them.

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Growing and alternating steps

1, 2, 3, … cells or 2 and 1 in turn

When the step is not equal: growing steps, alternating steps and two separate series.

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4

Series of counting and shading

Sides, lines, dots, shaded boxes and shaded parts

When it is not the position but the number or the shading that changes.

Series of numbers of things

Count, write down, solve like a number series

The rule of the number of sides, lines, dots and shaded boxes.

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Shading and shaded parts

Alternating shading and a turning shaded quarter

The turn of filled-empty, the shaded quarter of a circle and shading together with another rule.

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5

Alternating and mixed rules

Two interleaved series and two elements changing together

The structure of difficult series and a firm check of a unique answer.

Two interleaved series

Figures 1, 3, 5 and figures 2, 4, 6 separately

When an alternating series is built from two separate rules.

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Two elements, two rules

A separate rule for each element, the answer is their combination

The order of solving when two elements change together and a way of cutting options.

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6

Strategy guide

The order of checks and the slips to avoid

What to look at first in any figure series, common traps and time-savers.

The order of checks

Which element changes and how?

A straight order to follow for each kind of change.

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Common traps and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in figure series.

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7

Mixed practice

All types mixed

Series of different types in three levels.

Basic practice

One element, an equal step

Series with one rule and an equal step.

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Standard practice

Growing or alternating steps, with shading

Series with changing steps and two elements.

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Advanced practice

Two rules, interleaved series, long counts

The hardest kinds: two rules together and interleaved series.

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8

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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