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All lessons / Shapes You Can See in Your Head

Shapes You Can See in Your Head — Fold it, stack it, flip it, trace it, and turn it — all in your imagination

8 chapters Practice problems →

Showing the Grades 1–4 version. See the Grades 5–8 version →

About this topic

Close your eyes. Can you picture your own bedroom? The bed, the window, the door?

That little movie in your head is your superpower for these puzzles.

Some puzzles give you a picture and ask you to change it in your mind. Fold a flat shape into a box. Peek at a block tower from the top. Flip a picture like a mirror. Trace a path with your finger.

You can do all of this. The one big secret: go slow, and watch one little piece at a time.

CHAPTER 1

Fold the flat shape into a box

THEORY

Take a paper box and snip the edges so it falls open flat. What do you get? Six squares, all stuck together!

That flat shape is called a net. Fold it back up and each square swings into one side of the box.

flat net: 6 squares→folded box

Now here is the pattern to notice. Look at a straight strip of three squares in a row. When you fold them, they wrap around the box like a belt.

The two ends of the belt meet on opposite sides. So in a strip of three, the two end squares end up across from each other — with one square sitting between them.

1231 and 3 landopposite (2 sits between)
🎯 Try it
Four squares sit in one straight line: A B C D. Fold them into a belt. Which square ends up opposite A? Type 2 if you think C, or 3 if you think D.
Here's how: Opposite squares have exactly one square between them. Between A and C sits B — that is one square. So A is opposite C. Pick 2.
THE TRICK

THE MOVE: find the straight strip of three squares. The two ends of that strip are opposite faces. Two squares with one square between them sit across from each other on the box.

WATCH OUT

Six squares is not always enough! Some shapes made of 6 squares still will not fold into a box. So fold them in your head to be sure — do not trust the count alone.

WORKED EXAMPLE
PROBLEM · 2015 #10

Julia folds the paper net pictured on the right into a cube. Which number is on the face that is opposite to the face with the number 3?

Figure for Math Kangaroo 2015 Problem 10
A) 1 B) 2 C) 4 D) 5 E) 6

Julia's net folds into a cube. We want the number that lands across from the 3.

First, find the 3 in the picture. Now look for the straight strip of three squares it sits in: that strip holds 3, then 5, then 6 in a line.

Fold that strip into a belt. The 5 in the middle wraps around the box. The two ends, 3 and 6, meet on opposite sides.

So the face opposite the 3 is the 6. That is choice E.

Answer: E — 6
RULE OF THUMB

Opposite faces never touch. In the net, they are the two ends of a strip of three.

MORE LIKE THIS
2020 · #14 (figure problem)
Figure for Math Kangaroo 2020 Problem 14
Show answer
Answer: A
Show hints
Hint 1 of 2
Folding the net, work out which colored faces end up opposite each other.
Still stuck? Show hint 2 →
Hint 2 of 2
A correct die must show three faces that meet at a corner without contradicting the net.
Show solution
Approach: fold the net and check each candidate die
  1. Folding the cross-shaped net fixes which faces are opposite and which share a corner.
  2. Only die A shows three faces that can meet at one corner consistently with the net.
2023 · #7 Susi folds a piece of paper in the middle. She stamps 2 holes. What does the piece of paper look like when she unfolds it again?

Susi folds a piece of paper in the middle. She stamps 2 holes. What does the piece of paper look like when she unfolds it again?

Figure for Math Kangaroo 2023 Problem 7
Show answer
Answer: B
Show hints
Hint 1 of 3
The punch goes through both layers, so each hole really makes two holes.
Still stuck? Show hint 2 →
Hint 2 of 3
When you open the paper, every hole gets a twin on the other side of the fold.
Still stuck? Show hint 3 →
Hint 3 of 3
Each twin sits the same distance from the fold line, like a mirror.
Show solution
Approach: each punched hole appears twice, mirrored across the fold
  1. Because the paper is folded, the punch cuts through both halves at once.
  2. So when Susi opens it, each of the 2 holes has a matching twin mirrored across the fold line.
  3. That gives 4 holes in mirror-image positions, which is the picture in option B.
2010 · #22 Lines are drawn on a piece of paper and some of the lines are numbered. The paper is cut along some of these lines and then folded as...

Lines are drawn on a piece of paper and some of the lines are numbered. The paper is cut along some of these lines and then folded as shown in the picture. What is the total of the numbers on the lines that were cut?

Figure for Math Kangaroo 2010 Problem 22
Show answer
Answer: D — 20
Show hints
Hint 1 of 2
A line is cut only if the fold could not bring its two sides together; uncut lines are the fold creases.
Still stuck? Show hint 2 →
Hint 2 of 2
Figure out which numbered lines stayed as folds, and add up the rest.
Show solution
Approach: separate fold creases from cut lines, then add the cut numbers
  1. Match the folded result to the flat sheet to see which lines were creases and which were cut.
  2. Add the numbers on the lines that were cut.
  3. That total is 20.
CHAPTER 2

Peek from the top

THEORY

Imagine you are a tiny bird hovering right above a block tower, looking straight down.

From way up there, you cannot tell how tall the tower is! A tower 5 blocks high and a tower 1 block high look exactly the same: a single square.

from the side: tall stack + short one↓from the top: just 2 squares

See the pattern? Tall and short look the same from above. The height vanishes.

So the top view is really just the footprint — the floor spots that have a block on them. It is like the tower's shadow on the floor.

🎯 Try it
You have 3 stacks of blocks in a row. One is 4 tall, one is 2 tall, one is 1 tall. Looking straight down from above, how many squares do you see?
Here's how: From the top, each stack is one square — tall or short does not matter. There are 3 stacks, so you see 3 squares.
THE TRICK

THE MOVE: from the top, count floor spots, not heights. A tall stack and a short stack both look like one square from above.

WATCH OUT

Do not count a stack more times because it is tall. Looking down, you cannot even see how tall it is — one stack is one square.

WORKED EXAMPLE
PROBLEM · 2018 #5

Theodor has built this tower out of discs. He looks at the tower from above. How many discs does he see?

Figure for Math Kangaroo 2018 Problem 5
A) 1 B) 2 C) 3 D) 4 E) 5

Theodor stacks round discs and looks down from above. We count how many discs he can see.

From straight up top, a disc shows only if its rim pokes out past the disc sitting on top of it. So hunt for the edges that stick out.

Start at the top and work down. The top disc shows. Below it, one wider disc has a rim peeking out. Below that, one more rim pokes out. That makes 3 rims you can see.

So Theodor sees 3 discs. That is choice C.

Answer: C — 3
RULE OF THUMB

Top view = the shadow on the floor. Heights vanish; only the footprint stays.

MORE LIKE THIS
2017 · #17 Max builds this construction using some small equally big cubes. If he looks at his construction from above, the plan on the right tells...

Max builds this construction using some small equally big cubes. If he looks at his construction from above, the plan on the right tells the number of cubes in every tower. How big is the sum of the numbers covered by the two hearts?

Figure for Math Kangaroo 2017 Problem 17
Show answer
Answer: C — 5
Show hints
Hint 1 of 2
The plan number in each square is the height of the tower standing there.
Still stuck? Show hint 2 →
Hint 2 of 2
Read the two hidden tower heights off the 3-D picture, then add them.
Show solution
Approach: read the two covered tower heights from the construction and add
  1. Each square of the plan shows how many cubes are stacked there.
  2. The two hearts cover two of these tower heights.
  3. Reading those two towers from the picture and adding gives the total.
  4. Their sum is 5.
2020 · #9 (figure problem)
Figure for Math Kangaroo 2020 Problem 9
Show answer
Answer: B
Show hints
Hint 1 of 2
Looking straight down, each tower of cubes shows up as one colored square in the grid.
Still stuck? Show hint 2 →
Hint 2 of 2
Match the footprint and the light/dark pattern to one option.
Show solution
Approach: project the construction onto a top-view grid
  1. From above you see one square per stack, colored by the top cube (light or dark).
  2. The footprint shape and the light/dark squares match option B.
★ MINI-QUIZ

Fold and peek-from-above check

One folding puzzle and one block tower. Go slow and watch one piece.

2010 · #10 Maria folds a square piece of paper so that the two kangaroos land exactly on top of each other. Along how many of the lines shown is...

Maria folds a square piece of paper so that the two kangaroos land exactly on top of each other. Along how many of the lines shown is this possible?

Figure for Math Kangaroo 2010 Problem 10
Show answer
Answer: C — 2
Show hints
Hint 1 of 2
A fold line works only if the two halves are mirror images across it.
Still stuck? Show hint 2 →
Hint 2 of 2
Check each drawn line and count how many are true lines of symmetry for the four kangaroos.
Show solution
Approach: count the lines of symmetry of the figure
  1. A fold makes the kangaroos overlap exactly only along a line of symmetry.
  2. Test the drawn lines: only two of them reflect the figure onto itself.
  3. So the answer is 2 lines.
2018 · #5 Theodor has built this tower out of discs. He looks at the tower from above. How many discs does he see?

Theodor has built this tower out of discs. He looks at the tower from above. How many discs does he see?

Figure for Math Kangaroo 2018 Problem 5
Show answer
Answer: C — 3
Show hints
Hint 1 of 3
Imagine your eye is a bird looking straight down on the top of the tower.
Still stuck? Show hint 2 →
Hint 2 of 3
A disc hides under any wider disc that sits above it — you only see the edges that poke out.
Still stuck? Show hint 3 →
Hint 3 of 3
Look from the top disc downward and count each rim that sticks out past the ones above it.
Show solution
Approach: look straight down and count the rims that poke out
  1. Looking straight down, a disc is hidden if a wider disc sits on top of it.
  2. Starting from the top, you can see the top disc, plus each lower disc whose edge sticks out beyond everything above it.
  3. Exactly three rims poke out, so from above Theodor sees 3 discs.
CHAPTER 3

Mirrors and flips

THEORY

Hold your left hand up to a mirror. Look! The hand in the mirror looks like a right hand.

A mirror does one funny thing: it swaps left and right. That swap is called a flip.

letter Lflipped Lmirror line

Now how do you tell a flip from a plain turn? Watch one part — like which way a foot or a hook points.

If that part points the other way in the twin, it is a mirror flip. If it points the same way after you spin it, it was only turned.

start: hook faces rightturned: same hookflipped: hook swapped
🎯 Try it
Flip the letter b left-to-right in a mirror. Which letter do you get? Type 1 for d, or 2 for p.
Here's how: A left-right mirror keeps top and bottom the same but swaps the sides. The bump on b moves to the other side and you get d. So type 1.
THE TRICK

THE MOVE: to spot a mirror, watch which way one part points. A mirror points it the other way. A plain turn keeps it the same.

WATCH OUT

A flip is not the same as a turn. Turning a shape keeps left on the left. A mirror swaps left and right. Do not call them the same thing.

WORKED EXAMPLE
PROBLEM · 2018 #10

How many of the hands in the picture show a right hand?

Figure for Math Kangaroo 2018 Problem 10
A) 3 B) 4 C) 5 D) 6 E) 7

Some hands in the picture are right hands, some are left. We count the right hands.

Use your own right hand as a checker. Try to turn it so it looks the same as a hand in the picture. If you can match it with only turning, that one is a right hand.

If you would have to flip it like a mirror to match, then it is a left hand. Go around the picture and tally just the right ones.

You find 5 right hands. So the answer is 5, which is choice C.

Answer: C — 5
RULE OF THUMB

Mirror = left and right trade places. Turn = nothing trades, the whole thing just spins.

MORE LIKE THIS
2016 · #4 (figure problem)
Figure for Math Kangaroo 2016 Problem 4
Show answer
Answer: A
Show hints
Hint 1 of 2
A mirror swaps left and right but keeps top and bottom the same.
Still stuck? Show hint 2 →
Hint 2 of 2
Flip the clown left-to-right and see which option matches.
Show solution
Approach: apply a left-right mirror flip to the clown
  1. In a mirror, everything on the clown's left moves to the right and vice versa.
  2. Up and down stay the same.
  3. The picture that is the exact left-right flip of the original is option A.
2023 · #6 Christoph folds a see-through piece of foil along the dashed line. What can he then see? (Choose from pictures A–E.)

Christoph folds a see-through piece of foil along the dashed line. What can he then see? (Choose from pictures A–E.)

Figure for Math Kangaroo 2023 Problem 6
Show answer
Answer: A
Show hints
Hint 1 of 2
Folding along the dashed line flips the figure like a mirror.
Still stuck? Show hint 2 →
Hint 2 of 2
Reflect each digit across the fold line and read the result.
Show solution
Approach: reflect the pattern across the fold line
  1. Folding the transparent foil mirrors the drawing over the dashed line.
  2. Each mark lands on its mirror image, turning the shapes into readable digits.
  3. The reflected result reads the pattern shown in option A.
CHAPTER 4

Find the SAME shape (turning is allowed)

THEORY

Here is a sticker shaped like an L. Spin it around on the table. Is it still the same sticker?

Yes! Turning a shape does not change what it is. It is the same shape facing a new way.

shape=same shape, turned

So when a puzzle asks which shapes are the same, do this: tilt your head, or turn the page in your mind, so the two shapes face the same way. Then check if they match.

But careful — a turn keeps it the same, a mirror flip makes a different shape. Turning is allowed; flipping is not.

🎯 Try it
You have one L-sticker. You see 4 other L's. Three of them are your L just turned a different way. One is a mirror flip. How many of the 4 are the SAME shape as yours?
Here's how: Turning keeps a shape the same, so the 3 turned L's all match. The mirror-flipped one is a different shape. That makes 3 the same.
THE TRICK

THE MOVE: turn the page in your head until the two shapes face the same way, then compare. If they match after only turning, they are the same shape.

WATCH OUT

Same color and same size is not enough. A mirror flip can look almost right. The bumps must match after only turning — no flipping allowed.

WORKED EXAMPLE
PROBLEM · 2014 #3

For which houses were exactly the same building blocks used?

Figure for Math Kangaroo 2014 Problem 3
A) House 1 and 4 B) House 3 and 4 C) House 1, 4 and 5 D) House 3, 4 and 5 E) House 1, 2, 4 and 5

Some toy houses are built from the exact same set of blocks. We find which houses match.

Pick one house and make a little list of its blocks: the roof, the squares, the door. Now do the same for each other house.

Look for two houses with the very same list — same pieces, even if stacked a bit differently. House 1 and House 4 use the same blocks.

So the answer is House 1 and 4, which is choice A.

Answer: A — House 1 and 4
RULE OF THUMB

Same after a turn = same shape. Same only after a flip = a different shape.

MORE LIKE THIS
2017 · #4 The left picture is rotated, and the right picture shows the new position after the rotation. Which footprints are missing after the rotation?

The left picture is rotated, and the right picture shows the new position after the rotation. Which footprints are missing after the rotation?

Figure for Math Kangaroo 2017 Problem 4
Show answer
Answer: C
Show hints
Hint 1 of 2
Turn the left picture in your mind and compare every footprint with the right one.
Still stuck? Show hint 2 →
Hint 2 of 2
List the kinds of prints in each picture; the one type that appears on the left but not the right is missing.
Show solution
Approach: compare the two pictures and find the footprint type that is absent after the turn
  1. Rotating does not change which footprints exist, only where they sit.
  2. Match each print in the right picture to one in the left.
  3. One footprint from the left has no partner in the right picture.
  4. That missing footprint is the one shown in C.
2021 · #20 The picture beside shows two cogs, each with a black tooth. Where will the black teeth be after the small cog has made one full turn?

The picture beside shows two cogs, each with a black tooth. Where will the black teeth be after the small cog has made one full turn?

Figure for Math Kangaroo 2021 Problem 20
Show answer
Answer: C
Show hints
Hint 1 of 3
When two cogs touch, they push the same number of teeth past the meeting point.
Still stuck? Show hint 2 →
Hint 2 of 3
The small cog makes one whole turn, so its black tooth comes right back to where it started.
Still stuck? Show hint 3 →
Hint 3 of 3
The big cog has more teeth, so it only turns part way around, not all the way.
Show solution
Approach: track each cog's black tooth after one small turn
  1. One full turn of the small cog brings its black tooth back to its starting spot.
  2. The big cog is pushed the same number of teeth, but since it has more teeth it turns only part way.
  3. Putting both black teeth in their new spots matches option C.
CHAPTER 5

Fit the pieces together

THEORY

A jigsaw piece clicks in when its bumps fill the gap's dents, and its dents make room for the gap's bumps. Bump meets dent, dent meets bump.

hole with a dent+piece with matching bumpfits!

When there is one extra piece, do not test every piece by guessing. Look at each gap and find the one piece that fills it. Fill all the gaps.

The piece that no gap needs is the leftover. That is your answer.

🎯 Try it
A picture needs 3 puzzle pieces to be finished. Your box has 4 pieces. After you finish the picture, how many pieces are left over?
Here's how: You have 4 pieces and you use 3. 4 take away 3 leaves 1. So 1 piece is left over.
THE TRICK

THE MOVE: match each gap to the piece that fills it. The piece nobody needs is the leftover.

WATCH OUT

Do not pick a piece just because it looks nice. Check that its bumps really meet the gap's dents. A near-fit is still a no-fit.

WORKED EXAMPLE
PROBLEM · 2018 #8

Lisa needs exactly 3 pieces to complete her jigsaw. Which of the 4 pieces is left over?

Figure for Math Kangaroo 2018 Problem 8
A) A B) B C) C D) D E) C or D

Lisa needs 3 pieces to finish her jigsaw, but the box has 4. We find the one left over.

Look at the empty frame and what shape each gap needs. Now match the pieces one by one. Three pieces have bumps and dents that fit a gap just right — they each find a home.

One piece has bumps poking out on every side, so it cannot slide into any gap. That is piece A.

So piece A is the leftover — choice A.

Answer: A — A
RULE OF THUMB

Fill the gaps first. The piece with no gap to fill is your answer.

MORE LIKE THIS
2023 · #4 Alice has the four jigsaw pieces 1, 2, 3 and 4 shown. Which two can she put together to form the square shown?

Alice has the four jigsaw pieces 1, 2, 3 and 4 shown. Which two can she put together to form the square shown?

Figure for Math Kangaroo 2023 Problem 4
Show answer
Answer: E — 1 and 4
Show hints
Hint 1 of 2
The big square is a block of small cells; each piece must cover part of it with no gaps.
Still stuck? Show hint 2 →
Hint 2 of 2
Look for two pieces whose notches and bumps fit together like a lock and key.
Show solution
Approach: match complementary outlines that combine to the full square
  1. The target square must be filled by the two chosen pieces with no overlap and no gap.
  2. Pieces 1 and 4 have matching step-shaped edges that fit exactly together.
  3. Together they form the full square, so the answer is 1 and 4.
2012 · #9 Anna has made two L-shapes, each out of 4 squares (8 squares in all). How many of the following 4 shapes can she make using both L-shapes?

Anna has made two L-shapes, each out of 4 squares (8 squares in all). How many of the following 4 shapes can she make using both L-shapes?

Figure for Math Kangaroo 2012 Problem 9
Show answer
Answer: E — 4
Show hints
Hint 1 of 2
Each L is made of 4 squares; two of them give 8 squares, so try to cover each shape with two L pieces.
Still stuck? Show hint 2 →
Hint 2 of 2
Turn and flip the L freely and see which target shapes it can fill.
Show solution
Approach: try to tile each shape with two L-pieces
  1. Each L-piece covers 4 squares, and every shape shown has 8 squares, so a fit is at least possible by area.
  2. Trying the two L-pieces (rotated or flipped) on each shape, every one of the four can be built.
  3. So Anna can make all of them.
  4. The answer is 4.
2020 · #4 (figure problem)
Figure for Math Kangaroo 2020 Problem 4
Show answer
Answer: C
Show hints
Hint 1 of 2
The same set of shapes must build the picture — check that the right number of each piece is used.
Still stuck? Show hint 2 →
Hint 2 of 2
One picture needs a piece that is not in the kit (or uses one twice).
Show solution
Approach: check each picture against the available pieces
  1. Four of the pictures can be assembled using exactly the given pieces, each used once.
  2. Picture C cannot be made from that set of pieces, so it is the answer.
★ MINI-QUIZ

Flip, match, and fit check

A mirror puzzle, a same-shape puzzle, and a jigsaw piece. Watch the flip; match the gap.

2023 · #4 Alice has the four jigsaw pieces 1, 2, 3 and 4 shown. Which two can she put together to form the square shown?

Alice has the four jigsaw pieces 1, 2, 3 and 4 shown. Which two can she put together to form the square shown?

Figure for Math Kangaroo 2023 Problem 4
Show answer
Answer: E — 1 and 4
Show hints
Hint 1 of 2
The big square is a block of small cells; each piece must cover part of it with no gaps.
Still stuck? Show hint 2 →
Hint 2 of 2
Look for two pieces whose notches and bumps fit together like a lock and key.
Show solution
Approach: match complementary outlines that combine to the full square
  1. The target square must be filled by the two chosen pieces with no overlap and no gap.
  2. Pieces 1 and 4 have matching step-shaped edges that fit exactly together.
  3. Together they form the full square, so the answer is 1 and 4.
2018 · #8 Using the two tiles shown, Robert makes different patterns. How many of the patterns shown below can he make?

Using the two tiles shown, Robert makes different patterns. How many of the patterns shown below can he make?

Figure for Math Kangaroo 2018 Problem 8
Show answer
Answer: D — 4
Show hints
Hint 1 of 2
Robert may turn the tiles any way he likes, so try to trace the two tile shapes onto each pattern.
Still stuck? Show hint 2 →
Hint 2 of 2
A pattern works only if you can colour it in completely using copies of the tiles with no gaps and no overlaps.
Show solution
Approach: try to cover each shown pattern by tracing copies of the two tiles, turning them as needed
  1. Take the two tile shapes and imagine laying copies of them (turned any way) onto each pattern.
  2. A pattern is buildable only if the tiles fill it exactly — no empty squares and no overlaps.
  3. Four of the five patterns can be made this way, so the answer is 4, choice D.
2016 · #4 (figure problem)
Figure for Math Kangaroo 2016 Problem 4
Show answer
Answer: A
Show hints
Hint 1 of 2
A mirror swaps left and right but keeps top and bottom the same.
Still stuck? Show hint 2 →
Hint 2 of 2
Flip the clown left-to-right and see which option matches.
Show solution
Approach: apply a left-right mirror flip to the clown
  1. In a mirror, everything on the clown's left moves to the right and vice versa.
  2. Up and down stay the same.
  3. The picture that is the exact left-right flip of the original is option A.
CHAPTER 6

Count the blocks you cannot see

THEORY

Here is a sneaky thing about a pile of blocks: some blocks hide behind and under the others. If you count only the ones you can see, you will count too few.

a block hides behindthe top-left one!

The safe way: count layer by layer. Do the bottom floor first. Then the next floor up. Add them together.

On each floor, count every spot, even the hidden ones at the back. Ask yourself: what is holding up the blocks I can see? Those hidden helpers count too.

bottom floor = 4 blocks
🎯 Try it
A pile has 2 floors. The bottom floor is a square of 2 blocks by 2 blocks. The top floor is also 2 by 2. How many blocks in all?
Here's how: Bottom floor: 2 rows of 2 = 4 blocks. Top floor: 4 blocks too. 4 + 4 = 8 blocks.
THE TRICK

THE MOVE: count one floor at a time, then add the floors. On each floor, count the hidden back blocks too.

WATCH OUT

The blocks at the back and underneath are easy to forget. They are not visible, but they are still there. Ask what is holding the visible blocks up.

WORKED EXAMPLE
PROBLEM · 2009 #14

Thomas has made a table out of small cubes. How many small cubes did he use?

Figure for Math Kangaroo 2009 Problem 14
A) 24 B) 26 C) 28 D) 32 E) 36

Thomas builds a little table out of small cubes. We count them all — including the hidden ones. (This is a tricky one, so go slow.)

Split the table into two easy parts: the flat top, and the four legs.

The top is a flat slab, 4 cubes long and 4 cubes wide. That is 4 × 4 = 16 cubes, counting the hidden ones at the back.

Now the legs. There are 4 legs, one in each corner, and each leg is a stack of 4 cubes. So 4 legs × 4 cubes = 16 cubes.

Add the parts: 16 in the top + 16 in the legs = 32 cubes. That is choice D.

Answer: D — 32
RULE OF THUMB

Floors add up. Always include the blocks you cannot see — they are still holding the pile up.

MORE LIKE THIS
2013 · #7 Nathalie wanted to build a large cube out of lots of small cubes, just like in Picture 1. How many cubes are missing from Picture 2 that...

Nathalie wanted to build a large cube out of lots of small cubes, just like in Picture 1. How many cubes are missing from Picture 2 that would be needed to build the large cube?

Figure for Math Kangaroo 2013 Problem 7
Show answer
Answer: C — 7
Show hints
Hint 1 of 3
A full big cube like Picture 1 is 3 cubes wide, 3 deep and 3 tall, so it needs 27 small cubes.
Still stuck? Show hint 2 →
Hint 2 of 3
Count how many small cubes are really in Picture 2, layer by layer.
Still stuck? Show hint 3 →
Hint 3 of 3
The missing number is 27 take away the cubes you counted in Picture 2.
Show solution
Approach: subtract the cubes present from a full cube
  1. A complete large cube is 3 × 3 × 3 = 27 small cubes.
  2. Counting the cubes in picture 2 layer by layer gives 20 cubes.
  3. So 27 − 20 = 7 cubes are missing.
2011 · #22 The picture shows a fortress made from cubes. How many cubes were used to make it?

The picture shows a fortress made from cubes. How many cubes were used to make it?

Figure for Math Kangaroo 2011 Problem 22
Show answer
Answer: A — 56
Show hints
Hint 1 of 2
The fortress is a square wall with a hollow middle — count the wall, not a solid block.
Still stuck? Show hint 2 →
Hint 2 of 2
Count the cubes layer by layer, remembering the inside is empty.
Show solution
Approach: count the hollow square wall
  1. The base is a 5×5 ring of cubes plus its floor, and the wall rises with battlements on top.
  2. Adding the floor, the surrounding wall cubes and the raised corner/battlement cubes gives a total of 56 cubes.
CHAPTER 7

Trace the path with your finger

THEORY

Pretend your finger is a little bug. Put it on the START and let it walk along the line, never lifting up.

Watch what the bug bumps into, in order. First this, then that. Do not jump ahead and do not skip!

startfollow the line, meet the shapes in order

This trick works for mazes, roads, and lines too. In a maze, follow only the open gaps and never cross a wall.

🎯 Try it
A path goes past a star, then a heart, then a moon, then a sun. How many shapes does your finger pass before it reaches the sun?
Here's how: Before the sun, the finger passes the star, the heart, and the moon. That is 3 shapes.

On a grid: count right and down

Sometimes the path lives on a grid — a board of little squares. To go from a start spot to an end spot, count how many squares right and how many squares down.

startend2 right,2 down

From start to end here, you go 2 squares right and 2 squares down. Match those moves to the arrows. Count the squares; do not jump.

THE TRICK

THE MOVE: put your finger on the start and walk the line slowly, naming each thing you meet. On a grid, count the right-moves and down-moves you need.

WATCH OUT

Do not lift your finger and guess where the line ends up. Lines love to curve back. Slide your finger the whole way, every wiggle.

WORKED EXAMPLE
PROBLEM · 2014 #2

Theresa moves a pencil along the line. She starts at the arrow shown. In which order will she go past the shapes?

Figure for Math Kangaroo 2014 Problem 2
A) triangle, square, circle B) triangle, circle, square C) circle, triangle, square D) square, triangle, circle E) square, circle, triangle

Theresa drags a pencil along a line from the arrow. We list the shapes in the order she passes them.

Put your finger on the arrow — that is where the trip starts. Now slide it along the line without lifting it.

The first shape the line runs into is the triangle. Keep going: next comes the square. Last is the circle.

So the order is triangle, square, circle — choice A.

Answer: A — triangle, square, circle
RULE OF THUMB

Slow finger, no lifting. Meet the shapes one at a time, in order.

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2022 · #1 The bee wants to get to the flower. Each arrow shows a move to one neighbouring square. Which path can the bee fly to reach the flower?

The bee wants to get to the flower. Each arrow shows a move to one neighbouring square. Which path can the bee fly to reach the flower?

Figure for Math Kangaroo 2022 Problem 1
Show answer
Answer: E — → ↓ → ↓ ↓ →
Show hints
Hint 1 of 2
Compare where the bee starts with where the flower is, and count how many squares right and how many down.
Still stuck? Show hint 2 →
Hint 2 of 2
Every correct path needs the same number of right-moves and down-moves; check which arrow string has exactly that many of each.
Show solution
Approach: match the net movement (right vs down) to a path
  1. The bee must move several squares to the right and several squares down to reach the flower.
  2. Count the squares: the flower is the same number of steps right and down from the bee.
  3. Only one arrow sequence keeps the bee on the board and uses exactly those right and down moves.
  4. That sequence is E.
2016 · #3 Which point in the labyrinth can we get to, starting at point O?

Which point in the labyrinth can we get to, starting at point O?

Figure for Math Kangaroo 2016 Problem 3
Show answer
Answer: C — C
Show hints
Hint 1 of 2
Put your finger on O and follow the open corridors without crossing any walls.
Still stuck? Show hint 2 →
Hint 2 of 2
Only one labelled point connects to O through gaps in the walls.
Show solution
Approach: follow the open path through the maze
  1. Starting at O, move only through the openings, never crossing a drawn wall.
  2. The corridor from O leads outward to exactly one labelled point.
  3. That reachable point is C.
2013 · #6 Anna starts walking in the direction of the arrow. At each crossing she turns either right or left. She turns right, then left, then...

Anna starts walking in the direction of the arrow. At each crossing she turns either right or left. She turns right, then left, then left again, then right, then left, then left again. What will she find at the next crossing she reaches?

Figure for Math Kangaroo 2013 Problem 6
Show answer
Answer: A
Show hints
Hint 1 of 3
Put your finger on the start and point it the way the arrow points.
Still stuck? Show hint 2 →
Hint 2 of 3
Make the turns one at a time in order: right, left, left, right, left, left.
Still stuck? Show hint 3 →
Hint 3 of 3
After the last turn, look at the very next crossing to see which pictured item is there.
Show solution
Approach: walk the path one turn at a time and read off the item at the final crossing
  1. Start at the arrow and trace the path, turning right, left, left, right, left, then left.
  2. Keep your finger moving along the streets so you do not skip a crossing.
  3. The item waiting at the next crossing is the one shown in option A.
CHAPTER 8

What comes next in the pattern?

THEORY

Look at a row of pictures where each one is a tiny bit different from the one before. Like a flip-book that grows.

Your job: find what changes each step. Then you can draw the next picture, or count what is in it.

1 dot2 dots3 dots4?

See it? Each step adds one more dot. So the next picture has 4 dots.

The big trick is to look at two pictures sitting next to each other and ask: what is new? A new dot? A bigger shape? A turn? Then keep doing that one change.

🎯 Try it
A pattern grows: 1 star, then 3 stars, then 5 stars, then 7 stars. Each step adds 2 more stars. How many stars are in the next picture?
Here's how: The change each step is +2 stars. After 7 comes 7 + 2 = 9 stars.
THE TRICK

THE MOVE: compare two pictures that touch. Find the one thing that changes. Do that change once more to get the next picture.

WATCH OUT

Do not guess from a single picture. You need at least two side by side to see what is changing.

WORKED EXAMPLE
PROBLEM · 2020 #1

A mushroom grows a little bigger every day. Over five days Maria took a photo of this mushroom, but she put the photos in the wrong order (see picture). Which order of the photos shows the mushroom growing, from left to right?

Figure for Math Kangaroo 2020 Problem 1
A) 2-5-3-1-4 B) 2-3-4-5-1 C) 5-4-3-2-1 D) 1-2-3-4-5 E) 2-3-5-1-4

Five photos of a mushroom got mixed up. A mushroom only ever grows bigger. We put the photos in order from smallest to biggest.

The change is simple: every day the mushroom is a little taller, with a wider cap. So the right order goes from the tiniest cap to the biggest cap.

Hunt for the smallest mushroom — that photo goes first. Then the next-smallest, and so on, reading the photo numbers as the mushroom grows.

Smallest to biggest gives the order 2-5-3-1-4. That is choice A.

Answer: A — 2-5-3-1-4
RULE OF THUMB

Find what changes each step. Then keep that change going.

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2018 · #4 Peter has drawn this pattern. He draws exactly the same pattern once more, right after it. Which point is on his drawing?

Peter has drawn this pattern. He draws exactly the same pattern once more, right after it. Which point is on his drawing?

Figure for Math Kangaroo 2018 Problem 4
Show answer
Answer: D — D
Show hints
Hint 1 of 3
Find the spot where Peter's pattern ends — the second copy starts right there.
Still stuck? Show hint 2 →
Hint 2 of 3
Draw the very same up-and-down line again, counting grid squares so it matches exactly.
Still stuck? Show hint 3 →
Hint 3 of 3
When the second copy is drawn, see which lettered dot the line touches.
Show solution
Approach: copy the pattern onto the grid and see which dot it hits
  1. The new copy begins where the first pattern stops and uses the same grid squares.
  2. Redraw the same zig-zag, going up and down by the same number of squares as before.
  3. The corner of the repeated pattern lands exactly on point D.
2012 · #1 (figure problem)
Figure for Math Kangaroo 2012 Problem 1
Show answer
Answer: D
Show hints
Hint 1 of 2
Each square has the same total area, so just compare how much is shaded versus blank.
Still stuck? Show hint 2 →
Hint 2 of 2
Look for the picture where the white region is more than half of the square.
Show solution
Approach: compare shaded vs unshaded as fractions of the square
  1. In four of the squares the grey and white parts each make up exactly half.
  2. In one square a corner-to-corner diagonal plus a centre line leave only two small grey triangles, each a quarter of a half — grey is just one quarter.
  3. There the white part is three quarters, which is bigger than the grey.
  4. That picture is D.
2025 · #2 Kenny the Kangaroo hops from his school to the zoo. He hops like this: up 2, up-left 2, down-left 1, left 4 (see picture). From the zoo,...

Kenny the Kangaroo hops from his school to the zoo. He hops like this: up 2, up-left 2, down-left 1, left 4 (see picture). From the zoo, Kenny hops like this: right 3, up-right 2, up 2. Which house does Kenny land at?

Figure for Math Kangaroo 2025 Problem 2
Show answer
Answer: A
Show hints
Hint 1 of 2
Start at the Zoo dot and follow the new hops one by one on the grid.
Still stuck? Show hint 2 →
Hint 2 of 2
Track the arrows right 3, up-right 2, up 2 step by step until you land on a house.
Show solution
Approach: trace the hops on the grid from the zoo
  1. Begin at the Zoo marker and move right 3 squares.
  2. Then move diagonally up-right 2 squares, then straight up 2 squares.
  3. The landing square sits at the house labelled A.
  4. So Kenny lands at house A.
⬢ FINAL TEST

Shapes-in-your-head test

Eight puzzles, easy to a little harder. Take your time and picture each one.

2014 · #2 Theresa moves a pencil along the line. She starts at the arrow shown. In which order will she go past the shapes?

Theresa moves a pencil along the line. She starts at the arrow shown. In which order will she go past the shapes?

Figure for Math Kangaroo 2014 Problem 2
Show answer
Answer: A — triangle, square, circle
Show hints
Hint 1 of 2
Put your finger on the arrow — that is where the trip begins.
Still stuck? Show hint 2 →
Hint 2 of 2
Slide your finger along the line without lifting it and watch which shape you bump into first, second, and last.
Show solution
Approach: trace the path and list shapes in the order met
  1. Put your finger at the arrow and follow the line without lifting it.
  2. The first shape the path runs into is the triangle.
  3. Next it reaches the square, and last the circle.
  4. So the order is triangle, square, circle — choice A.
2016 · #7 Part of a rectangle is hidden behind a curtain (see picture). The hidden part is a

Part of a rectangle is hidden behind a curtain (see picture). The hidden part is a

Figure for Math Kangaroo 2016 Problem 7
Show answer
Answer: A — triangle
Show hints
Hint 1 of 2
The whole shape is a rectangle; picture its straight edges where the curtain hides them.
Still stuck? Show hint 2 →
Hint 2 of 2
The slanted corner of the curtain cuts off only a small corner of the rectangle.
Show solution
Approach: reconstruct the hidden corner of the rectangle
  1. The curtain hangs over the rectangle and its lower edge slants across one corner.
  2. The piece left hidden between that slanted edge and the rectangle's corner has three sides.
  3. So the hidden part is a triangle.
2010 · #5 Six coins make a triangle (see the picture). What is the smallest number of coins that must be moved to make the circle?

Six coins make a triangle (see the picture). What is the smallest number of coins that must be moved to make the circle?

Figure for Math Kangaroo 2010 Problem 5
Show answer
Answer: B — 2
Show hints
Hint 1 of 2
Picture the 6-coin ring on top of the triangle and see which coins already sit in the right spots.
Still stuck? Show hint 2 →
Hint 2 of 2
Count how many coins are NOT already where the ring needs them.
Show solution
Approach: keep the coins already in place, move only the rest
  1. Lay the target ring of 6 coins over the triangle.
  2. Four of the coins already sit where the ring needs them; only two are out of place.
  3. So the smallest number of coins to move is 2.
2018 · #8 Lisa needs exactly 3 pieces to complete her jigsaw. Which of the 4 pieces is left over?

Lisa needs exactly 3 pieces to complete her jigsaw. Which of the 4 pieces is left over?

Figure for Math Kangaroo 2018 Problem 8
Show answer
Answer: A — A
Show hints
Hint 1 of 3
Look at the empty hole: where does it have bumps sticking in, and where does it have gaps?
Still stuck? Show hint 2 →
Hint 2 of 3
A bump on a piece must meet a gap, and a gap must meet a bump — they have to fit like puzzle pieces.
Still stuck? Show hint 3 →
Hint 3 of 3
Three of the pieces fit the hole together; find the one shape that just won't go in.
Show solution
Approach: fit pieces into the hole and find the odd one out
  1. The empty hole is a long rectangle with bumps and gaps along its edges.
  2. Try each piece in the hole — its bumps must meet the hole's gaps, and its gaps must meet the hole's bumps.
  3. Three pieces fill the hole exactly, but piece A (with bumps sticking out on every side) cannot fit, so it is left over.
2022 · #1 The bee wants to get to the flower. Each arrow shows a move to one neighbouring square. Which path can the bee fly to reach the flower?

The bee wants to get to the flower. Each arrow shows a move to one neighbouring square. Which path can the bee fly to reach the flower?

Figure for Math Kangaroo 2022 Problem 1
Show answer
Answer: E — → ↓ → ↓ ↓ →
Show hints
Hint 1 of 2
Compare where the bee starts with where the flower is, and count how many squares right and how many down.
Still stuck? Show hint 2 →
Hint 2 of 2
Every correct path needs the same number of right-moves and down-moves; check which arrow string has exactly that many of each.
Show solution
Approach: match the net movement (right vs down) to a path
  1. The bee must move several squares to the right and several squares down to reach the flower.
  2. Count the squares: the flower is the same number of steps right and down from the bee.
  3. Only one arrow sequence keeps the bee on the board and uses exactly those right and down moves.
  4. That sequence is E.
2015 · #10 Julia folds the paper net pictured on the right into a cube. Which number is on the face that is opposite to the face with the number 3?

Julia folds the paper net pictured on the right into a cube. Which number is on the face that is opposite to the face with the number 3?

Figure for Math Kangaroo 2015 Problem 10
Show answer
Answer: E — 6
Show hints
Hint 1 of 2
Find the straight strip of three squares that contains the 3.
Still stuck? Show hint 2 →
Hint 2 of 2
In a strip of three faces, the two end faces fold to opposite sides.
Show solution
Approach: use the straight three-square strip of the net
  1. The faces 3, 5 and 6 lie in a straight vertical strip of the net.
  2. When a strip of three faces is folded, the two ends become opposite faces.
  3. So 3 and 6 end up on opposite faces of the cube.
  4. The face opposite 3 carries 6.
2020 · #1 A mushroom grows a little bigger every day. Over five days Maria took a photo of this mushroom, but she put the photos in the wrong...

A mushroom grows a little bigger every day. Over five days Maria took a photo of this mushroom, but she put the photos in the wrong order (see picture). Which order of the photos shows the mushroom growing, from left to right?

Figure for Math Kangaroo 2020 Problem 1
Show answer
Answer: A — 2-5-3-1-4
Show hints
Hint 1 of 2
A mushroom only gets bigger from one day to the next, so order the photos from smallest cap to largest cap.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the tiniest mushroom and read the labels in growing order.
Show solution
Approach: order the photos from smallest to largest mushroom
  1. The mushroom grows every day, so the correct order goes from the smallest cap to the biggest, fully opened cap.
  2. Reading the photo labels from smallest to largest gives the sequence in choice A: 2-5-3-1-4.
2013 · #7 Nathalie wanted to build a large cube out of lots of small cubes, just like in Picture 1. How many cubes are missing from Picture 2 that...

Nathalie wanted to build a large cube out of lots of small cubes, just like in Picture 1. How many cubes are missing from Picture 2 that would be needed to build the large cube?

Figure for Math Kangaroo 2013 Problem 7
Show answer
Answer: C — 7
Show hints
Hint 1 of 3
A full big cube like Picture 1 is 3 cubes wide, 3 deep and 3 tall, so it needs 27 small cubes.
Still stuck? Show hint 2 →
Hint 2 of 3
Count how many small cubes are really in Picture 2, layer by layer.
Still stuck? Show hint 3 →
Hint 3 of 3
The missing number is 27 take away the cubes you counted in Picture 2.
Show solution
Approach: subtract the cubes present from a full cube
  1. A complete large cube is 3 × 3 × 3 = 27 small cubes.
  2. Counting the cubes in picture 2 layer by layer gives 20 cubes.
  3. So 27 − 20 = 7 cubes are missing.
APPENDIX

Things to remember

Memorize these

SHAPE TRICKS

  • A box has 6 flat sides. A net is the box opened up flat. The ends of a strip of three are opposite.
  • From the top, tall and short look the same — count floor spots.
  • A mirror swaps left and right. A turn does not.
  • Turning keeps a shape the same; a mirror flip makes a different one.
  • Fill each gap with its piece. The leftover piece is the answer.
  • Count blocks floor by floor, and never forget the hidden ones.
  • Trace a path with a slow finger — no lifting, no jumping.
  • In a pattern, find what changes each step and keep it going.
→ Practice Spatial & Visual Reasoning problems
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