Rotations, reflections, folded paper, cube nets, pattern matrices, and careful counting of the parts of a figure. Everything is described in words as well as drawn, so nothing here depends on colour or on picking out fine detail.
What this category trains
Mental rotation and reflection
Paper folding and net visualisation
Systematic counting of visual parts
Viewpoint and orientation changes
25 puzzles · about 55 minutes end to end, as a guide only · untimed
The ladder
The puzzles below run in difficulty order, easiest first. Each one checks on its own and shows its explanation straight away, so you get the method for one rung before you attempt the next.
Work the ladder in order: the early puzzles fix one idea at a time and the later ones combine them. Nothing here is timed and nothing is scored. Sketch freely — most spatial questions get easier the moment something is on paper, and turning a real scrap of paper is a legitimate way to check a folding puzzle. Read the explanation even when you were right: it names the method, and the method is what carries over to the next puzzle.
1. Turning the corner piece
Difficulty 1 of 5 · Warm-up
The figure below is turned 90 degrees clockwise, staying flat on the page. Which description matches the result?
The starting figure, drawn on a three-by-three block grid. A hash is a filled block.
#..
#..
###
Describe this figure in words
A three-by-three grid of blocks. The whole left-hand column is filled, top to bottom, and the whole bottom row is filled, left to right. Together they form an upright L: a tall arm rising from the bottom-left corner and a foot running right along the bottom.
Show a hint
Decide where the tall arm ends up first, then attach the foot to the corner they share.
2. The mirrored floor plan
Difficulty 1 of 5 · Warm-up
A room is drawn from above with north at the top. A desk stands against the north wall, towards the east corner; a door sits in the middle of the west wall. The plan is reflected in a mirror line running north to south down its middle. Where are the desk and the door in the reflected plan?
Show a hint
Ask which compass directions the mirror line separates, and leave the other pair alone.
3. Edges of the pointed solid
Difficulty 1 of 5 · Warm-up
A solid has a square base, and four triangular sides that rise from the base and meet at a single point above it. How many edges does the solid have in total?
Show a hint
Split the count into the edges you would trace around the base and the edges that climb.
4. Finishing the small square
Difficulty 1 of 5 · Warm-up
In the two-by-two pattern below, the same change happens from left to right in each row. What belongs in the empty cell?
A two-by-two pattern; the bottom-right cell is blank.
( ) (o)
[ ] ?
Describe this figure in words
A two-by-two arrangement of small figures. The top row shows an empty circle on the left and a circle containing a single dot on the right. The bottom row shows an empty square on the left and a blank cell on the right that has to be filled.
Show a hint
Name in words the one thing that happens between the two figures in the top row.
5. Squares inside the strip
Difficulty 1 of 5 · Warm-up
How many squares of any size can be found in the figure below? Count the small cells and any larger squares that the cells form together.
A grid two cells tall and three cells wide.
+--+--+--+
| | | |
+--+--+--+
| | | |
+--+--+--+
Describe this figure in words
A rectangle divided into six equal cells arranged in two rows of three, so the whole figure is two cells tall and three cells wide. All cells are the same size and none is subdivided.
Show a hint
Work out how many positions a two-cell-wide square can occupy before you count any.
6. The card turned right round
Difficulty 2 of 5 · Straightforward
A rectangular card has a small notch cut from its top-left corner and a hole punched near its bottom edge, slightly right of centre. The card is turned 180 degrees in its own plane; it is not flipped over. Where are the notch and the hole now?
Show a hint
A half turn is two quarter turns; check that both features move, not just the obvious one.
7. Which flat pattern makes a cube
Difficulty 2 of 5 · Straightforward
Each description below is a flat arrangement of six identical squares joined edge to edge. Which one can be folded up into a closed cube?
Show a hint
Wrap the longest row into a band in your head, then ask what is left to close the two ends.
8. The stacked cubes
Difficulty 2 of 5 · Straightforward
Identical cubes are stacked in a corner. The bottom layer is a solid three-by-three square of cubes, a solid two-by-two square of cubes sits on top of that, and one single cube sits on top of those. No cube is missing from any layer. How many cubes are there altogether?
Show a hint
Turn each layer into a count of cubes before you add, and check you have used every layer.
9. Reflecting in the diagonal
Difficulty 2 of 5 · Straightforward
Two counters sit on a three-by-three board: one in the top row, middle column; one in the bottom row, left column. The pattern is reflected in the diagonal running from the top-left corner to the bottom-right corner. Where are the counters afterwards?
Show a hint
Give each cell a row number and a column number, then see what the diagonal does to the pair.
10. Two rows of boxes
Difficulty 2 of 5 · Straightforward
One rule runs along each row of the figure below. What belongs in the empty box?
Two rows of three boxes; the last box of the lower row is empty.
Two rows of three boxes. The upper row holds, from left to right, a box containing one circle, a box containing two circles, and a box containing three circles. The lower row holds a box containing one cross, a box containing two crosses, and an empty box to be filled.
Show a hint
Check the top row for what the count does; two terms alone can fit more than one rule.
11. The shape that will not fold
Difficulty 2 of 5 · Straightforward
Three of the shapes below can be folded along some straight line so that the two halves land exactly on each other. Which shape cannot?
Show a hint
For each shape, try to name the fold line out loud; the one you cannot name is the answer.
12. One fold, one punch
Difficulty 3 of 5 · Tricky
A square sheet is folded once, bringing the bottom edge up onto the top edge. A single hole is then punched through both layers, in the left half of the folded sheet and a little way above the fold. The sheet is unfolded flat. What do you see?
Show a hint
Count the layers the punch goes through; that is how many holes you will find.
13. Two things change at once
Difficulty 3 of 5 · Tricky
Each step along a row changes the figure in two ways at once. The top row runs: a plain triangle, a shaded square, a plain pentagon. The bottom row runs: a plain square, a shaded pentagon, then a gap. What belongs in the gap?
Show a hint
Write the two tracks as two short lists, one for sides and one for shading, then extend both.
14. Turning and swapping
Difficulty 3 of 5 · Tricky
A tile shows an arrow pointing up; its head is shaded and its tail is plain. Each move turns the tile a quarter turn clockwise and also swaps the shading, so whatever was shaded becomes plain and whatever was plain becomes shaded. What does the tile look like after three moves?
Show a hint
Handle direction and shading separately, and count how many steps each one needs to repeat.
15. Seen from the other side
Difficulty 3 of 5 · Tricky
Four posts stand in a straight line running east to west. Someone standing south of the line and facing it reads the posts from left to right as: striped, plain, notched, capped. Another person walks round to the north of the line and faces it. What do they read from left to right?
Show a hint
Nothing physically moves; ask only which end of the row is now on the viewer's left.
16. Every rectangle in the grid
Difficulty 3 of 5 · Tricky
How many rectangles of any size can be found in the figure below? Count every rectangle, not only the ones that happen to be squares.
A rectangle divided into eight equal cells arranged in two rows of four, so the figure is two cells tall and four cells wide. Five vertical lines and three horizontal lines make up the grid.
Show a hint
Count the vertical lines and the horizontal lines, then think about choosing two of each.
17. The arrow on the folded wall
Difficulty 3 of 5 · Tricky
A flat cross of five squares makes an open box: a base square with one square joined to each of its four sides, folded up to form walls. On the flat cross, the square to the left of the base carries an arrow pointing away from the base, towards the left. Once the walls are folded upright, where does that arrow point?
Show a hint
Find the hinge line first, then ask whether the arrow runs along it or away from it.
18. Two folds, one punch
Difficulty 4 of 5 · Hard
A square sheet is folded in half, bottom edge up onto the top edge, then folded in half again, left edge across onto the right edge. One hole is punched through the folded packet, at the centre of the small square that results. The sheet is unfolded flat. What do you see?
Show a hint
Work out how many layers the punch went through before you think about where anything lands.
19. Every shape once, every count once
Difficulty 4 of 5 · Hard
In the matrix below, each row contains each of the three shapes exactly once and each of the three dot counts exactly once; the same is true of each column. What belongs in the empty cell?
A three-by-three matrix. T is a triangle, S a square, C a circle; each full stop after the letter is one dot inside that shape.
T. S.. C...
S... C. T..
C.. T... ?
Describe this figure in words
A three-by-three matrix of shapes containing dots. Top row: a triangle with one dot, a square with two dots, a circle with three dots. Middle row: a square with three dots, a circle with one dot, a triangle with two dots. Bottom row: a circle with two dots, a triangle with three dots, and an empty cell to be filled.
Show a hint
Treat shape and dot count as two separate puzzles, and finish each row and column in turn.
20. Tipping and turning the cube
Difficulty 4 of 5 · Hard
A cube sits with a star on top, a cross underneath, a circle facing you, a square on the far face, a triangle on the left and an arrow on the right. Tip it forward so the top face swings down onto the front, then turn it a quarter turn clockwise as seen from above. Which face is on top, and which faces you?
Show a hint
Note which pair of faces each movement leaves untouched; that halves the bookkeeping.
21. Three lines through a rectangle
Difficulty 4 of 5 · Hard
A rectangle has both of its diagonals drawn, and also the straight line joining the midpoints of its left and right sides. Into how many separate regions is the inside of the rectangle divided?
Show a hint
Notice where all three lines meet, then add them one at a time and count what each one cuts.
22. Walking the grid
Difficulty 4 of 5 · Hard
You start on a square grid facing north. Walk forward three squares, turn right, walk two, turn right, walk three, turn left, walk one, turn left, walk two. Which way are you facing, and where are you relative to your starting square?
Show a hint
Track the heading and the two distance totals separately, and change the heading before you move.
23. The off-centre fold
Difficulty 5 of 5 · Toughest
A square sheet measures eight units by eight. Its left edge folds across to the right, with the crease three units from the left edge; the bottom edge then folds up onto the top edge. A hole is punched through every layer, two units in from the packet's left edge and two units above its foot. What do you see once the sheet is unfolded?
Show a hint
Reflect in the crease itself: a point two units past a crease lands two units short of it.
24. Rows one way, shading the other
Difficulty 5 of 5 · Toughest
In the matrix below, one rule runs along the rows and a different rule governs the shading. What belongs in the empty cell?
A three-by-three matrix. T is a triangle, S a square, C a circle; the digit is the number of dots inside the shape; a star means that shape is shaded.
T1* T2 T3
S2 S3* S4
C3 C4 ?
Describe this figure in words
A three-by-three matrix. Top row: a shaded triangle with one dot, a plain triangle with two dots, a plain triangle with three dots. Middle row: a plain square with two dots, a shaded square with three dots, a plain square with four dots. Bottom row: a plain circle with three dots, a plain circle with four dots, and an empty cell to be filled.
Show a hint
Two features fit the rows and one does not; trace the odd one out through the matrix on its own.
25. Turn then mirror
Difficulty 5 of 5 · Toughest
A tile shows an upright pole with a small flag attached at the top, sticking out to the right. The tile is first turned a quarter turn clockwise, and then reflected in a vertical mirror line. What does it show at the end?
Show a hint
Do the two moves in the stated order, and check what happens to the flag's side at each step.
Skills these puzzles practise
Each category maps onto the shared aptitude constructs, so a puzzle that interests you has a lesson and a larger practice bank behind it:
