Figure series, mirrors, embedded shapes and paper folding.
About Series
A figure series shows four figures that change by a fixed rule and asks which figure comes next. The rule is nearly always a count or a turn: the number of sides goes up by one, the number of dots goes up by two, or a line turns through the same angle each time. Finding that one number is the whole question.
What you need to understand
Most series change by a constant step, so the difference between consecutive figures is the same throughout.
A count series is read by counting the sides of each polygon, or the dots in each figure.
A turning series is read by measuring how far a line or a shape turns from one figure to the next, in the same direction each time.
Two rules can operate at once, for example a shape that both gains a side and turns.
Two figures are never enough to fix a rule. A rule is confirmed only when the third and fourth figures obey it too.
The answer is the figure whose value continues the sequence, not the figure that merely resembles the others.
A series can also repeat with a period, as in a pattern of three that starts again, which is a different rule from a constant step.
How to work through these questions
Decide what is changing: the number of sides, the number of dots, the position, or the angle of a line.
Write the four values down as numbers, for example 3, 4, 5, 6.
Subtract to find the step. If the step is constant, that is the rule.
State the rule in words: the number of sides goes up by one each time.
Apply it once to the last figure to get the value the answer must have.
Look for the option with that value, counting rather than estimating.
Check that no other option has the same value, which would make the question ambiguous.
Mistakes that cost marks
Fixing the rule from the first two figures. A constant step needs at least three figures to be confirmed, since any two numbers can be joined by many rules.
Counting the sides of a figure with rounded corners as though the corners were extra sides.
Ignoring the direction of a turn, and accepting an option that turns the wrong way by the right amount.
Choosing the option that looks closest to correct instead of counting it exactly.
Assuming the series increases when it may decrease, or alternates between two rules.
Overlooking a second rule that runs alongside the first, such as a shape that grows and also moves.
Worked example
A series shows a triangle, then a four-sided figure, then a five-sided figure, then a six-sided figure. Which figure comes next?
Count the sides of each figure: 3, 4, 5, 6.
Subtract consecutive values: 4 - 3 = 1, 5 - 4 = 1, 6 - 5 = 1. The step is 1 and is constant, so the rule is confirmed by four figures.
Apply the rule once more: 6 + 1 = 7.
The next figure must have 7 sides.
Count the sides of each option rather than judging by size, and take the one with exactly 7.
Answer: The figure with 7 sides.
Practice questions with answers
A few Series questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Study the series, then decide which figure comes next.
Which of the four figures should replace the question mark?
A
B
C
D
Answer: Option C — with explanation
In this series the number of sides goes up by one each time. The four figures given are 5, 6, 7, 8, so the next must be 9.
Only one of the four figures continues the pattern.
Common mistakes
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
Question 2
Study the series, then decide which figure comes next.
Which of the four figures should replace the question mark?
A
B
C
D
Answer: Option B — with explanation
In this series the number of dots goes up by 1 each time. The four figures given are 1, 2, 3, 4, so the next must be 5.
Only one of the four figures continues the pattern.
Common mistakes
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
Question 3
Study the series, then decide which figure comes next.
Which of the four figures should replace the question mark?
A
B
C
D
Answer: Option A — with explanation
In this series the line turns through 45 degrees clockwise each time. The four figures given are 270, 315, 360, 405, so the next must be 450.
Only one of the four figures continues the pattern.
Common mistakes
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
Question 4
Study the series, then decide which figure comes next.
Which of the four figures should replace the question mark?
A
B
C
D
Answer: Option C — with explanation
In this series the number of sides goes up by one each time. The four figures given are 4, 5, 6, 7, so the next must be 8.
Only one of the four figures continues the pattern.
Common mistakes
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
Question 5
Study the series, then decide which figure comes next.
Which of the four figures should replace the question mark?
A
B
C
D
Answer: Option C — with explanation
In this series the number of dots goes up by 2 each time. The four figures given are 1, 3, 5, 7, so the next must be 9.
Only one of the four figures continues the pattern.
Common mistakes
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
- carried the pattern on by the wrong amount, or repeated one of the figures already given
Frequently asked questions
How many figures do I need before I trust the rule?
At least three, and preferably all four given. Two figures can be joined by a straight line, a curve or a step rule, so a rule confirmed only by two figures is a guess.
What if the number of sides goes up by one but the figures are also turning?
Then two rules are running together. Find the step for the count and the angle for the turn separately, and check the answer satisfies both.
What if the sequence repeats instead of growing?
That is a periodic series. Find the length of the repeating block, then work out where the answer falls inside it.
Should I count or just look?
Count. Two figures that differ by one side can look very similar, and options in these questions are usually drawn to differ by exactly one, so an estimate will not separate them.
What if the difference is not constant?
Then look for a rule in the differences themselves, such as 1, 2, 3, or for a different property such as the number of enclosed regions or the position of a mark.
Take the Series test
Two timed papers on the same syllabus — sit the foundation paper first,
then the advanced one. Both use the real exam paper format with a full
step-by-step review of every question once you submit.
