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Figure Series

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low importance~0-1 Q in Tier 11 formulas⚡ 6 shortcuts2 subtopics
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Movement and rotation

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⏱ 7 min read🧩 7 question types🎯 16 practice Q
The idea in one minute

In a movement series, nothing is added or removed. A mark slides round a square, an arrow turns, or shading hops from quarter to quarter.

Follow one element at a time. Write its step, then apply that step once more.

01

What the series asks

Four figures follow one another by a fixed rule. You pick the fifth figure from the options.

Here the elements only slide or turn. Their number never changes. The trick is to track one element at a time.

02

Number the border places

Many frames put a dot on the border of a square. There are 8 places. We number them clockwise from the top-left corner.

0 1 2
7   3
6 5 4

Now movement is just arithmetic. The next place is the last place plus the step, taken mod 8.

Rule: One place clockwise is +1. One place anticlockwise is -1. If the answer passes 7, take away 8.

03

Turning arrows

An arrow has 8 usual directions, 45 degrees apart. We count them clockwise from up.

Number01234567
Arrow↑↗→↘↓↙←↖

A 90-degree turn is 2 steps. A 135-degree turn is 3 steps. Anticlockwise turns are negative.

04

Shaded quarters

A square has 4 quarters: top-left, top-right, bottom-right, bottom-left. This is the clockwise order.

Shading moves the same way as a dot, but with 4 places instead of 8. Two shaded quarters move together as a pair.

Tip: Write the shaded quarters as numbers 0 to 3. Then you can spot the step at once.

05

Steps that are not constant

Some steps grow: 1, 2, 3, then 4. Some steps alternate: 45 degrees, then 90 degrees, then 45 degrees again.

If the plain step fails, write the jumps as a list. The pattern is usually in that list.

06

Several elements together

A frame may carry shading, an arrow and a dot. Each element has its own rule. The rules never mix.

Work through each element alone. Then pick the only option that agrees with every rule.

Watch: Two options often satisfy one rule each. Check every element before you choose.

07

Reading the options quickly

Wrong options are built from common slips. One goes the wrong way round. One moves by the wrong step. One forgets to take away 8.

Work out the answer first. Then find it among the options. Do not let the options steer you.

08

A worked example

Find the figure that comes next in the series.

Series

. o .  >  . . .  >  . . .  >  . . o
.   .  >  .   .  >  o   .  >  .   .
. . .  >  . . o  >  . . .  >  . . .

Options

(A)       (B)       (C)       (D)
. . .     . . .     . . .     o . .
.   .     .   .     .   o     .   .
. o .     o . .     . . .     . . .

Solution

  • Places: 1, 4, 7, 2. Jumps: +3, +3, +3 (7 + 3 = 10, which is 2).
  • Next: 2 + 3 = 5, the bottom-middle place.

Answer: (A).

09

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Rotating arrow or line

How to spot it:

Each frame shows an arrow or a line at a different angle.

θnext=θlast+s\theta_{next} = \theta_{last} + s
Method

  1. Write each direction as a number of 45-degree steps from up.

  2. Find the constant turn between frames.

  3. Add the same turn once more.

  4. Check the direction: clockwise is +, anticlockwise is -.

Why it works:

The turn is constant, so one more turn gives the next direction.

Try this

Find the figure that comes next in the series.

Series

↑  >  ↘  >  ←  >  ↗

Options

(A)     (B)     (C)     (D)
←       ↓       ↘       →
Show solution

  1. Count each direction in steps of 45 degrees clockwise from up: 0, 3, 6, 9 (which is 1 after taking away 8).

  2. The step is +3, which is 135 degrees clockwise.

  3. Next: 1 + 3 = 4, which is the arrow pointing down (↓).

Answer

(B)

Type 2very common2 practice Q

Element moving round the border

How to spot it:

A dot, ring or square sits on a different border place in every frame.

pnext=(plast+s) mod 8p_{next} = (p_{last} + s) \bmod 8
Method

  1. Number the 8 border places clockwise from the top-left corner.

  2. Write the place in each frame.

  3. Find the step, and take away 8 when the count passes 7.

  4. Add the step to the last place.

Why it works:

A constant jump on 8 places is plain addition mod 8.

Try this

Find the figure that comes next in the series.

Series

o . .  >  . . .  >  . . o  >  . . .
.   .  >  .   .  >  .   .  >  o   .
. . .  >  . o .  >  . . .  >  . . .

Options

(A)       (B)       (C)       (D)
. . o     . . .     . . .     o . .
.   .     .   .     .   .     .   .
. . .     o . .     . . o     . . .
Show solution

  1. Number the places 0 to 7 clockwise from the top-left corner. The dot sits at 0, 5, 2, 7.

  2. From 0 to 5 is 3 places anticlockwise (0 - 3 = -3, which is 5). The step is -3 each time.

  3. Next: 7 - 3 = 4, the bottom-right corner.

Answer

(C)

Type 3common2 practice Q

Shaded part moving

How to spot it:

A shaded quarter, or a pair of quarters, jumps from frame to frame.

Method

  1. Number the quarters 0 to 3 clockwise from top-left.

  2. Write the shaded numbers for each frame.

  3. Find the step and its direction.

  4. Apply it once more, keeping a pair together.

Why it works:

Shading obeys the same step rule as a dot, but with 4 places.

Try this

Find the figure that comes next in the series.

Series

##  >  #.  >  ..  >  .#
..  >  #.  >  ##  >  .#

Options

(A)     (B)     (C)     (D)
.#      ..      #.      ##
.#      ##      #.      ..
Show solution

  1. The shaded pair is TL+TR, then BL+TL, then BR+BL, then TR+BR.

  2. Each frame the pair slides one quarter anticlockwise.

  3. One more slide takes TR+BR to TL+TR, which is the first frame again.

Answer

(D)

Type 4common2 practice Q

Two or three rules together

How to spot it:

Each frame carries two or three elements, such as shading and an arrow, and each one changes.

Method

  1. Take one element and follow it through all the frames.

  2. Write its rule.

  3. Repeat for the other elements.

  4. Choose the option that satisfies every rule.

Why it works:

The rules are independent, so the answer must match each of them.

Try this

Find the figure that comes next in the series.

Series

↗ ..  >  ↖ #.  >  ↙ .#  >  ↘ ..
  #.  >    ..  >    ..  >    .#

Options

(A)      (B)      (C)      (D)
↗ ..     ↙ ..     ↗ #.     ↗ ..
  #.       #.       ..       .#
Show solution

  1. Shading: BL, TL, TR, BR. It moves one quarter clockwise, so the next is BL.

  2. Arrow: up-right, up-left, down-left, down-right. It turns 90 degrees anticlockwise each time.

  3. Next arrow: down-right turned 90 degrees anticlockwise is up-right.

  4. Only the option with BL shaded and an up-right arrow satisfies both rules.

Answer

(A)

Type 5occasional

Step that grows

How to spot it:

The jumps between frames are not equal. They look like 1, 2, 3.

Method

  1. Write the place in each frame.

  2. Write the jumps in a list.

  3. Find how the jumps change, often +1 each time.

  4. Take the next jump and add it, mod 8.

Why it works:

A jump that grows by a fixed amount is itself a simple series.

Try this

Find the figure that comes next in the series.

Series

o . .  >  . o .  >  . . .  >  . . .
.   .  >  .   .  >  .   o  >  .   .
. . .  >  . . .  >  . . .  >  o . .

Options

(A)       (B)       (C)       (D)
. . .     . . o     . o .     o . .
o   .     .   .     .   .     .   .
. . .     . . .     . . .     . . .
Show solution

  1. The dot sits at 0, 1, 3, 6.

  2. The jumps are 1, 2, 3, so the jump grows by 1 each time.

  3. Next jump is 4: 6 + 4 = 10, which is 2 after taking away 8. The dot lands on the top-right corner.

Answer

(B)

Type 6occasional

Two marks moving in opposite ways

How to spot it:

Two dots move in the same figure, one clockwise and the other anticlockwise.

Method

  1. Label the dots and track each one on its own.

  2. Write each step, with its sign.

  3. Move each dot once more.

  4. Watch for frames where the dots overlap.

Why it works:

Overlap frames hide a dot, but each dot still follows its own step.

Try this

Find the figure that comes next. A and B are two dots. A star means both are on the same place.

Series

A . .  >  . A .  >  . . *  >  . B .
.   .  >  .   B  >  .   .  >  .   A
. . B  >  . . .  >  . . .  >  . . .

Options

(A)       (B)       (C)       (D)
. B .     A . .     B . .     . B .
.   .     .   .     .   .     .   .
. . A     . . B     . . A     . A .
Show solution

  1. A moves 0, 1, 2, 3: one place clockwise each time. Next is 4.

  2. B moves 4, 3, 2, 1: one place anticlockwise each time. Next is 0.

  3. In frame 3 both were on place 2. That is why the star appeared. It does not change either rule.

Answer

(C)

Type 7occasional

Turn that alternates

How to spot it:

An arrow turns by different amounts, such as 45 degrees then 90 degrees.

Method

  1. Write the directions as numbers.

  2. Write the turns in a list.

  3. The list repeats in a short cycle, for example 1, 2, 1, 2.

  4. Take the next turn from the cycle.

Why it works:

A repeating cycle of turns is the next simplest rule after a constant turn.

Try this

Find the figure that comes next in the series.

Series

↑  >  ↗  >  ↘  >  ↓

Options

(A)     (B)     (C)     (D)
↙       ↖       →       ←
Show solution

  1. Directions in 45-degree steps: 0, 1, 3, 4.

  2. The jumps are 1, 2, 1. They repeat as 1, 2, 1, 2.

  3. Next jump is 2: 4 + 2 = 6, which is the arrow pointing left.

Answer

(D)

10

Formula sheet

Position after k steps
pk=(p0+k s) mod 8p_{k} = (p_{0} + k\,s) \bmod 8

s = step (positive = clockwise) on the 8 border places.

11

Shortcuts that save time

⚡ One element, one line

Write one short row for each element: the places in every frame, then the step, then the next place. Any option that fails a row is out.

Example

A dot sits at places 2, 5, 0, 3. Where next?

Show solution

  1. Jumps: 2 to 5 is +3. 5 to 0 is +3 (5 + 3 = 8). 0 to 3 is +3.

  2. Next: 3 + 3 = 6.

Answer

Place 6, the bottom-left corner.

⚡ Arrows as clock hours

Think of the arrow as the hand of a clock. A 90-degree clockwise turn goes 12, 3, 6, 9.

Example

An arrow points up, right, down. What comes next?

Show solution

  1. Up to right is 90 degrees clockwise.

  2. Right to down is again 90 degrees clockwise.

  3. Down turned 90 degrees clockwise points left.

Answer

Left.

⚡ Cross out by one element

Test the easiest element first, such as the arrow. Cross out the options that fail it before you look at anything else.

12

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Counting the places from a different starting point in each frame.

Number the places once, clockwise from the top-left corner, and never change the numbering.

Mistake 02

Mixing up clockwise and anticlockwise.

Clockwise is +. Anticlockwise is -. Check the direction with the first two frames.

Mistake 03

Reading the step from one pair of frames only.

Check the step on every pair. A growing or alternating step shows up only when you compare all of them.

Mistake 04

Forgetting to take away 8 when the count passes 7.

After place 7 comes place 0. Take away 8 from any number of 8 or more.

Mistake 05

Choosing an option that fits one element but not the others.

Every element has its own rule. The answer must satisfy all of them.

Mistake 06

Treating a turn of 90 degrees as one step.

One step is 45 degrees. A 90-degree turn is 2 steps.

13

Quick revision

Read this the night before the exam.

  • Follow one element at a time. Write its step, then apply the step once more.

  • Border places: 0 to 7 clockwise from the top-left corner. Next = last + step, mod 8.

  • Arrows: 8 directions, 45 degrees apart. A 90-degree turn is 2 steps.

  • Shaded quarters: 4 places, moving in the clockwise order TL, TR, BR, BL.

  • If the step is not constant, write the jumps as a list. They may grow or alternate.

  • With several elements, the answer must satisfy every rule.

14

Practice: 16 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.

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