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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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Adding, removing and changing elements

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⏱ 6 min read🧩 6 question types🎯 12 practice Q
The idea in one minute

In an addition series, the number of elements changes from frame to frame. Strokes grow, dots are added, or shapes disappear.

Count the elements in every frame and write the row of numbers. Then find the step.

01

What changes here

The figures do not slide or turn. They gain or lose something. The question is about how many, not about where.

Write the count of each element for every frame. The rule is usually plain arithmetic.

02

Counting the main element

Suppose the frames show 1, 3, 5 and 7 dots. The step is +2, so the next frame has 9.

The step can also be negative. 7, 6, 5, 4 falls by 1 each time. Taking away is as common as adding.

Rule: Count first. Only options with the right count are worth looking at.

03

Dots added round the frame

Some series keep the old dots and add one new dot each frame. The new dot goes to the next place on the border.

Note the starting place and the direction. Note the gap too: the new dot may land one, two or three places on.

frame 1     frame 2     frame 3
o . .       o . o       o . o
.   .       .   .       .   .
. . .       . . .       . . o

Here the new dot moves clockwise by two places each time.

04

When elements disappear

Some series remove one element each frame. A row of 6, 5, 4 strokes loses one stroke at a time.

Ask which element goes first. It is often the one drawn last, or the one at the top-left.

05

Steps that change

The step may alternate: +3, -1, +3, -1. It may also grow: +1, +2, +3, +4.

Write the differences as a list. Then look at the list as a small series of its own.

Tip: If the differences are not equal, take differences of the differences.

06

A second rule beside the count

Often the count grows while a small detail alternates. A shape may switch between a circle and a square. An arrow may flip up and down.

The two rules are independent. Find each one alone. The answer must satisfy both.

Watch: Wrong options often get one rule right. Check both before you choose.

07

Reading the options quickly

Wrong options are built from common slips. One has the wrong count. One has the right count but the wrong detail. One adds the new dot in the wrong place.

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

| ○  >  || □  >  ||| ○  >  |||| □

Options

(A)         (B)         (C)        (D)
||||| □     ||||| ○     |||| ○     |||||| ○

Solution

  • Strokes: 1, 2, 3, 4. Add 1 each time, so the next frame has 5 strokes.
  • Shape: circle, square, circle, square. It alternates, so the next is a circle.
  • Only the option with 5 strokes and a circle satisfies both rules.

Answer: (B).

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

Growing count

How to spot it:

Frames show 1, 3, 5 and so on of the same mark. The options differ mainly in how many.

an=an−1+sa_{n} = a_{n-1} + s
Method

  1. Count the marks in every frame.

  2. Find the constant step.

  3. Add it once more.

  4. Pick the option with that count.

Why it works:

A constant step turns the row of counts into a simple sum.

Try this

Find the figure that comes next in the series.

Series

o  >  ooo  >  ooo oo  >  ooo ooo o

Options

(A)            (B)               (C)             (D)
ooo ooo oo     ooo ooo ooo o     ooo ooo ooo     ooo ooo ooo oo
Show solution

  1. Count the dots: 1, 3, 5, 7.

  2. The step is +2 each time.

  3. Next: 7 + 2 = 9 dots.

Answer

(C)

Type 2common2 practice Q

Dots added round the frame

How to spot it:

Each frame keeps the old dots and adds one new dot at the next border place.

Method

  1. Number the border places 0 to 7.

  2. Find the place of each new dot.

  3. Find the gap between new dots.

  4. Add the next dot and keep all the old ones.

Why it works:

The new dot follows its own step, and the old dots never move.

Try this

Find the figure that comes next in the series.

Series

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

Options

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

  1. Frame 1 has a dot at 0. Frame 2 adds a dot at 2. Frame 3 adds a dot at 4.

  2. Each frame keeps the old dots and adds one new dot two places further clockwise.

  3. Frame 4 adds a dot at 6. The dots stay at 0, 2, 4, 6.

Answer

(D)

Type 3common2 practice Q

Count plus alternating detail

How to spot it:

The count grows while a shape or arrow switches between two forms.

Method

  1. Write the count row and find its step.

  2. Write the detail row.

  3. Find the cycle, often 2 frames long.

  4. Combine the next count with the next detail.

Why it works:

The two rules are independent, so the answer must agree with both.

Try this

Find the figure that comes next in the series.

Series

| ○  >  || □  >  ||| ○  >  |||| □

Options

(A)         (B)         (C)        (D)
||||| □     ||||| ○     |||| ○     |||||| ○
Show solution

  1. Strokes: 1, 2, 3, 4. Add 1 each time, so the next frame has 5 strokes.

  2. Shape: circle, square, circle, square. It alternates, so the next is a circle.

  3. Only the option with 5 strokes and a circle satisfies both rules.

Answer

(B)

Type 4occasional2 practice Q

Shrinking count with a moving marker

How to spot it:

The count falls each frame while a dot slides round the border.

an=an−1−sa_{n} = a_{n-1} - s
Method

  1. Write the count row and continue it downwards.

  2. Write the dot's place in each frame.

  3. Find its step, mod 8.

  4. Combine both results.

Why it works:

Count and movement are separate rules. Each one is easy alone.

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. Strokes: 7, 6, 5, 4. Take away 1 each time, so the next frame has 3 strokes.

  2. Dot: 1, 3, 5, 7. It moves two places clockwise. 7 + 2 = 9, which is 1.

  3. Only the option with 3 strokes and the dot at place 1 satisfies both rules.

Answer

(A)

Type 5occasional

Add then take away

How to spot it:

The count goes up, then down, then up again.

Method

  1. Write the count in each frame.

  2. Write the changes: for example +3, -1, +3.

  3. See that they repeat as a pair.

  4. Use the next change in the cycle.

Why it works:

Two changes that take turns form a repeating cycle.

Try this

Find the figure that comes next in the series.

Series

oo  >  ooo oo  >  ooo o  >  ooo ooo o

Options

(A)         (B)            (C)             (D)
ooo ooo     ooo ooo oo     ooo ooo ooo     ooo oo
Show solution

  1. Count the dots: 2, 5, 4, 7.

  2. The changes are +3, -1, +3. They alternate between +3 and -1.

  3. Next change is -1: 7 - 1 = 6 dots.

Answer

(A)

Type 6occasional

Step that grows

How to spot it:

The count increases, but the amount it increases by keeps getting bigger.

Method

  1. Write the counts.

  2. Write the differences: for example 1, 2, 3.

  3. Find how the differences change, often +1.

  4. Add the next difference to the last count.

Why it works:

The differences form their own simple series.

Try this

Find the figure that comes next in the series.

Series

o  >  oo  >  ooo o  >  ooo ooo o

Options

(A)               (B)             (C)                 (D)
ooo ooo ooo o     ooo ooo ooo     ooo ooo ooo ooo     ooo ooo ooo oo
Show solution

  1. Count the dots: 1, 2, 4, 7.

  2. The changes are +1, +2, +3. The change itself grows by 1.

  3. Next change is +4: 7 + 4 = 11 dots.

Answer

(D)

10

Shortcuts that save time

⚡ Count first

Count the elements in each frame. Only options with the right count survive. Then check positions or details.

Example

Strokes are 2, 4, 6, 8. Next?

Show solution

  1. The step is +2.

  2. 8 + 2 = 10.

Answer

10 strokes.

⚡ Differences of differences

If the differences are not equal, write them as a list. A list like 1, 2, 3 grows by 1 each time.

Example

Counts are 3, 4, 6, 9. Next?

Show solution

  1. The differences are 1, 2, 3.

  2. The next difference is 4.

  3. 9 + 4 = 13.

Answer
⚡ Cross out by the detail

Test the small detail first, such as a shape or an arrow. It removes options quickly.

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Guessing from the look of the figure without counting.

Count the elements in every frame and write the row of numbers first.

Mistake 02

Assuming the step is always plus.

The count can fall. Check the sign on the first two frames.

Mistake 03

Finding the step from the first two frames only.

Check every pair. An alternating or growing step shows up only later.

Mistake 04

Ignoring the second detail, such as a shape or an arrow.

Write a row for the detail too. It usually alternates.

Mistake 05

Forgetting that old dots stay when a new one is added.

In an adding series, each frame keeps the earlier elements and adds the new one.

12

Quick revision

Read this the night before the exam.

  • Count the main element in every frame and write the row of numbers.

  • Find the step. It may be plus or minus.

  • If the steps differ, look at them as a list. They may alternate or grow.

  • Added dots keep their old places. Note the start, direction and gap.

  • A second detail usually alternates. Find its rule separately.

  • The answer must satisfy the count rule and the detail rule together.

13

Practice: 12 questions

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

Topic test · 12 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.

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