Figure Series & Counting
Crack the rule behind turning, moving and growing figures — and learn the counting system that never misses a triangle or a square.
1Turning shapes: 45°, 90° and 180°
Many figure-series questions show a shape turning step by step. Before you can find the next figure, you need to be sure about two things: which way it turns and how much.
| Turn | Fraction of a full turn | What it looks like |
|---|---|---|
| 45° | ⅛ turn | up → up-right (to the next slanting position) |
| 90° | ¼ turn (quarter turn) | up → right (a right angle) |
| 135° | ⅜ turn | up → down-right |
| 180° | ½ turn (half turn) | up → down (upside down) |
| 360° | full turn | back where it started |
- Clockwise = the direction clock hands move: up → right → down → left.
- Anticlockwise (counter-clockwise) = the opposite: up → left → down → right.
- Turning 90° anticlockwise ends in the same place as turning 270° clockwise.
Track one clear part of the shape — an arrow head, a dot, a black corner — and watch only that. It is much easier than looking at the whole shape.
An arrow points up. At every step it turns 90° clockwise. Where does it point after 3 steps?
2Figure series: finding the rule
A figure series is a row of figures that change by a fixed rule. Ask these questions, in this order:
- Turning? Is something rotating? Which way? By 45°, 90°, 180°?
- Moving? Is a dot or small shape moving from corner to corner, or along the sides?
- Adding or removing? Do lines, dots or sides increase or decrease by a fixed number?
- Alternating? Do two things take turns — filled/empty, big/small, circle/square?
- Two rules together? Hard questions often change two things at once. Check each part separately.
The arrow points up, up-left, left, down-left. What comes next?
- Track the arrow head: up → up-left is a 45° turn anticlockwise.
- Check the next steps: up-left → left (45°), left → down-left (45°). The rule holds.
- One more 45° anticlockwise turn from down-left gives down.
Answer: An arrow pointing straight down.
A dot sits in a square. It is filled at top-left, empty at top-right, filled at bottom-right, empty at bottom-left. What is figure 5?
- Rule 1 (position): top-left → top-right → bottom-right → bottom-left. The dot moves one corner clockwise. Next: back to top-left.
- Rule 2 (shading): filled, empty, filled, empty … Next is filled.
- Put both rules together.
Answer: A filled dot at the top-left corner.
An arrow turns anticlockwise by 45°, then 90°, then 135°. By how much will it turn next?
- The turns grow by 45° each time: 45°, 90°, 135°, …
- Next turn = 135° + 45° = 180° (half a turn).
Answer: 180° — so the arrow will point the opposite way.
Do not stop after finding one rule! If the option you picked matches the turning but not the shading (or the number of lines), it is wrong. Check every part of the figure against every rule.
Clockwise or anticlockwise? Many options show the right amount of turn in the wrong direction. Say the direction out loud: “up to up-left is anticlockwise”.
A shape turns 45° anticlockwise at every step. It starts pointing right. Which way does it point after 2 steps?
3Figure matrix, analogy and adding/subtracting figures
In a figure matrix the figures sit in a grid (usually 3 × 3) and one is missing. The rule works along rows and/or columns.
Find the figure that goes in place of “?” in the grid above.
- Row 3 has a triangle and a circle, so the missing shape is a square (every row has one of each).
- Row 3 has half-shaded and fully shaded, so the missing shading is white.
- Check column 3: triangle (full), circle (half), square (white) — one of each shape and shading. ✔
Answer: A white (unshaded) square.
Figure analogy
“Figure 1 is to figure 2 as figure 3 is to ?” Find what changed from 1 to 2 — size swapped, shading changed, turned 90°, one line added — and do exactly the same to figure 3. For example, if a big triangle with a small circle inside becomes a big circle with a small triangle inside (inside and outside swap), then a big square with a small star inside becomes a big star with a small square inside.
Adding and subtracting figures
- Subtraction: remove every line of the second figure from the first. What is left is the answer.
- Addition: put one figure on top of the other (overlap them). Lines that are in either figure appear in the answer.
4Missing numbers in figures
Numbers placed in triangles, wheels, squares or grids follow a hidden rule. Try the simple rules first: add, subtract, multiply, then combinations like “multiply and add 1”.
Find the missing number in the third triangle.
- Try adding the corners of the first triangle: 3 + 4 + 5 = 12 ✔ (the middle number).
- Check with the second: 2 + 6 + 1 = 9 ✔.
- Third: 4 + 7 + 2 = 13.
Answer: 13
Find the number in place of “?” in the wheel.
- Pair up numbers that are directly opposite: 3 ↔ 17, 8 ↔ 12, 5 ↔ 15.
- Each pair adds to 20: 3 + 17 = 20, 8 + 12 = 20, 5 + 15 = 20.
- “?” is opposite 11, so ? = 20 − 11 = 9.
Answer: 9
Find the missing number in the grid.
- Row 1: 2, 3, 6 → 2 × 3 = 6.
- Row 2: 4, 5, 20 → 4 × 5 = 20 ✔ (the same rule).
- Row 3: 3 × 7 = 21.
Answer: 21
Always test your rule on at least two figures before using it. “Add them” may work for the first figure by luck, but fail for the second — then try another rule (like “multiply and add 1”: 3 × 4 + 1 = 13).
Find the missing number. Row 1: 3, 4, 12 · Row 2: 5, 2, 10 · Row 3: 6, ?, 42
5Counting triangles
“How many triangles?” questions are easy to get wrong because we miss triangles or count one twice. The cure is to be systematic: label the points with letters and count in an order.
How many triangles are there in triangle ABC with lines AD and AE?
- Every triangle here has corner A at the top and its base on BC.
- Bases made of 1 part: BD, DE, EC → 3 triangles (ABD, ADE, AEC).
- Bases made of 2 parts: BE, DC → 2 triangles (ABE, ADC).
- Base made of 3 parts: BC → 1 triangle (ABC).
- Total = 3 + 2 + 1 = 6.
Answer: 6 triangles
How many triangles are there in the figure above (one standing line from the top, two flat lines across)?
- The standing line cuts the base into 2 parts → 1 + 2 = 3 triangles sit on the base.
- Each flat line acts like another base: it also has 3 triangles sitting on it (with the same top corner).
- There are 3 “bases” (level 1, level 2 and the real base): 3 × 3 = 9.
- The pieces between two flat lines are not triangles (they have 4 sides), so we don't count them.
Answer: 9 triangles
How many triangles are in a square with both diagonals and both middle lines?
- Small triangles (1 piece): count the pieces → 8.
- Medium triangles (2 pieces): each has one whole side of the square as its base, with its top at the centre → 4.
- Large triangles (4 pieces): each diagonal cuts the square into 2 halves, and there are 2 diagonals → 4.
- Total = 8 + 4 + 4 = 16.
Answer: 16 triangles
Square + both diagonals
4 small + 4 half-squares = 8 triangles.
Triangle with midpoints joined
Join the middle points of the 3 sides: 4 small + 1 big = 5 triangles.
Six-pointed star
6 small point triangles + 2 big ones = 8 triangles.
The two classic misses: forgetting the big outer triangle, and forgetting triangles made of two or more pieces put together. Count by size — smallest first, biggest last.
Three lines are drawn from the top corner of a triangle to its base, cutting the base into 4 parts. How many triangles are there?
6Counting squares, rectangles and lines
Squares in a grid
How many squares are there in a 3 × 3 grid (a big square cut into 9 small squares)?
- Small (1 × 1) squares: 3 × 3 = 9.
- 2 × 2 squares: they can start in 2 places across and 2 down → 2 × 2 = 4.
- 3 × 3 square: just the big one → 1.
- Total = 9 + 4 + 1 = 14.
Answer: 14
How many squares are there in a grid of 2 rows and 3 columns?
- 1 × 1 squares: 2 × 3 = 6.
- 2 × 2 squares: they fit in 1 place down and 2 places across → 2.
- 3 × 3 squares cannot fit (only 2 rows).
Answer: 6 + 2 = 8 squares
Rectangles
Remember: every square is also a rectangle, so squares are included when a question asks for rectangles.
How many rectangles are there in a 2 × 2 grid (a square cut into 4)?
- Across: 2 boxes in a row → 1 + 2 = 3 ways to choose the width.
- Down: 2 boxes in a column → 1 + 2 = 3 ways to choose the height.
- Rectangles = 3 × 3 = 9 (4 small + 2 wide + 2 tall + 1 big).
Answer: 9
Counting straight lines
A straight line keeps going through crossing points — count it once, from one end to the other. Sort the lines into flat (horizontal), standing (vertical) and slanting, and add.
In line counting, a long line that passes through a crossing is still one line, not two. In rectangle counting, don't forget that squares count as rectangles. In square counting, look out for a tilted square (a diamond) — it still counts!
How many squares are there in a 2 × 2 grid?
A strip is divided into 3 equal boxes in a row. How many rectangles are there in it?
7Watch and explore
8Quick recap
- 45° = ⅛ turn, 90° = ¼ turn, 180° = ½ turn. Clockwise = the way clock hands move.
- In a series, check: turning → moving → adding/removing → alternating → two rules at once.
- Track one clear part (arrow head, dot, black corner) to see a turn.
- Figure matrix: each row/column often has one of each shape and one of each shading.
- Figure subtraction: remove the second figure's lines; addition: overlap both figures.
- Missing numbers: test your rule on at least two figures (add, multiply, opposite pairs, × then + 1…).
- Label points with letters and count triangles by size, smallest first.
- Lines from the top to a base in n parts: 1 + 2 + … + n triangles (3 parts → 6, 4 parts → 10).
- Each extra flat line across multiplies: bases × triangles on each base.
- Square + both diagonals = 8 triangles; add both middle lines = 16; six-pointed star = 8.
- Squares in n × n grid = 1² + 2² + … + n²: 2×2 → 5, 3×3 → 14, 4×4 → 30.
- Rectangles include squares. A strip of n boxes has 1 + 2 + … + n rectangles.
- A straight line through crossings is counted once.