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Shapes & Sides — Count the sides. Fold to match. Walk all the way around.

6 chapters Practice problems →

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

About this topic

Look around the room right now.

A window is a square. A door is a tall rectangle. A slice of pizza is a triangle. A clock is a circle.

Shapes are everywhere. In this lesson you will name them, count their sides, fold them to find matching halves, walk all the way around them, and count the little squares inside.

One trick runs through the whole lesson: go slow and point with your finger. Touch each side, each corner, each square, and say a number out loud.

CHAPTER 1

Sides and corners

THEORY

Look at these shapes. Count the straight edges on each one.

trianglesquarehexagon

A side is one straight edge. A corner is the pointy spot where two sides meet.

Now count both for each shape and look for the pattern:

  • Triangle: 3 sides, 3 corners.
  • Square: 4 sides, 4 corners.
  • Hexagon: 6 sides, 6 corners.

Did you see it? The two numbers are always the same!

The number of sides always equals the number of corners.

So count one, and you already know the other. Here are a square's corners marked with red dots. Count the dots.

4 dots = 4 corners
🎯 Try it
A starfish shape has 5 corners (5 pointy arms). How many sides does it have?
Here's how: Sides always match corners. 5 corners means 5 sides.
THE TRICK

Touch each side with your finger and say a number out loud: one, two, three. The last number you say is how many sides there are. And that is how many corners there are too.

WATCH OUT

Do not count a side twice. Pick a starting corner and walk one way around, so you stop right where you began.

WORKED EXAMPLE
PROBLEM · 2024 #4

Bruno builds a big triangle out of small triangles that are all the same size. Some are already placed (shown grey). How many more small triangles does he need so that the big triangle is completely filled?

Figure for Math Kangaroo 2024 Problem 4
A) 5 B) 6 C) 7 D) 8 E) 9

Bruno is filling a big triangle with little triangles. The grey ones are already in. The white spots are the holes still to fill.

You do not need the grey ones at all. Each hole needs one new triangle. So count the white holes.

Point at each white triangle and count out loud: 1, 2, 3, 4, 5, 6.

Bruno needs 6 more little triangles.

The grey ones are done. I only count what is still empty. Touch each white hole once so I do not skip one or count it twice.

Answer: B — 6
RULE OF THUMB

Sides = corners, every time. Count one, and you know the other.

MORE LIKE THIS
2025 · #4 Felix forms a square out of 16 small, grey tiles. He then removes some of the small tiles — see picture. How many small tiles did he remove?

Felix forms a square out of 16 small, grey tiles. He then removes some of the small tiles — see picture. How many small tiles did he remove?

Figure for Math Kangaroo 2025 Problem 4
Show answer
Answer: C — 6
Show hints
Hint 1 of 3
The empty spots are where tiles used to be — you can count those directly.
Still stuck? Show hint 2 →
Hint 2 of 3
A full 4-by-4 square holds 16 tiles in all.
Still stuck? Show hint 3 →
Hint 3 of 3
Count the grey tiles that are still there, then see how many are missing from 16.
Show solution
Approach: count what remains, then subtract from the total
  1. The full square has 4 × 4 = 16 small tiles.
  2. Count the grey tiles still in place: there are 10 of them.
  3. Removed tiles = 16 − 10 = 6.
2017 · #3 How many blocks are missing in this igloo?

How many blocks are missing in this igloo?

Figure for Math Kangaroo 2017 Problem 3
Show answer
Answer: C — 10
Show hints
Hint 1 of 2
Look only at the white hole in the igloo, not the whole dome.
Still stuck? Show hint 2 →
Hint 2 of 2
Count the empty brick spaces one row at a time and add them up.
Show solution
Approach: count the empty brick spaces in the hole
  1. Look at the white hole in the brick dome.
  2. Count the missing bricks in each row that the hole passes through.
  3. Adding the gaps row by row gives 10 empty brick spaces.
  4. So 10 blocks are missing.
CHAPTER 2

Folding to match

THEORY

Take a butterfly. Fold it right down the middle. The two wings land on top of each other. They match!

Some shapes have a special line like that. Fold along it and the two halves cover each other exactly. That line is called the line of symmetry. Think of it as the fold line.

fold here — halves match

Fold this triangle down the dotted line. The left half drops right onto the right half.

Now look for fold lines on a square. A square is special. It has 4 different fold lines!

4 fold lines

Two go straight across (up-down and side-to-side). Two go corner to corner. Each one makes the halves match.

If the two halves match when you fold, that line is a line of symmetry.

Peek: which letters fold?
The letter A folds down the middle into two matching halves. So do B, C, D, E. But F and G do not match if you fold them. Try drawing each one and folding in your head!
🎯 Try it
The letter A looks like a tent. How many fold lines does it have, where the two halves match?
Here's how: Fold the A straight down the middle, top to bottom. The left side matches the right side. That is 1 fold line.
THE TRICK

Pretend the shape is paper. Ask: could I fold it so one half lands right on top of the other? If yes, that fold is a line of symmetry.

WATCH OUT

Not every line is a fold line. Test it in your head first: do the two halves really land on top of each other? A slanted line through a rectangle does not match. Only the straight middle lines do.

WORKED EXAMPLE
PROBLEM · 2025 #15

Tom wants to cut the pizza into two halves so that each half has the same number of tomatoes. There are two ways to do this. Along which lines can he cut?

Figure for Math Kangaroo 2025 Problem 15
A) 1 or 3 B) 1 or 4 C) 2 or 3 D) 2 or 4 E) 3 or 4

Tom wants to cut the pizza into two halves with the same number of tomatoes on each side.

This works the same way as folding. A good cut splits the pizza so each side has an equal share of tomatoes.

Look at the lines one at a time. For each one, count the tomatoes on the left, then count the tomatoes on the right.

Line 2: the two sides have the same number. It works. Line 4: same on both sides too. It works.

Lines 1 and 3 leave more tomatoes on one side. They do not work. So the two good cuts are 2 and 4.

I do not need a fancy middle line. I only need the same tomato count on each side. So I count one side, count the other, and compare.

Answer: D — 2 or 4
RULE OF THUMB

Fold it. If the halves match, it is a line of symmetry.

MORE LIKE THIS
2009 · #4 In the picture you see the number 930. How many small squares must be changed so that the number becomes 806?

In the picture you see the number 930. How many small squares must be changed so that the number becomes 806?

Figure for Math Kangaroo 2009 Problem 4
Show answer
Answer: B — 6
Show hints
Hint 1 of 3
Look at one digit at a time: the 9 changing to 8, the 3 changing to 0, and the 0 changing to 6.
Still stuck? Show hint 2 →
Hint 2 of 3
For each digit, lay the new shape on top of the old one and watch only the little squares that have to switch from dark to light or light to dark.
Still stuck? Show hint 3 →
Hint 3 of 3
Add up the switches from all three digits.
Show solution
Approach: compare the two numbers one digit at a time and count the little squares that flip
  1. Look at each digit on its own: 9 becomes 8, then 3 becomes 0, then 0 becomes 6.
  2. For the first digit, just one little square switches to turn the 9 into an 8.
  3. For the middle digit, two little squares switch to turn the 3 into a 0; for the last digit, three little squares switch to turn the 0 into a 6.
  4. Counting the switches: 1 + 2 + 3 = 6, so 6 small squares must be changed.
2024 · #2 Five pencils labelled A, B, C, D and E lie on a grid of lines. Which pencil is the longest?

Five pencils labelled A, B, C, D and E lie on a grid of lines. Which pencil is the longest?

Figure for Math Kangaroo 2024 Problem 2
Show answer
Answer: D — D
Show hints
Hint 1 of 3
The straight lines in the background are spaced evenly, like the marks on a ruler.
Still stuck? Show hint 2 →
Hint 2 of 3
For each pencil, count how many spaces it covers from its flat back end to its pointy tip.
Still stuck? Show hint 3 →
Hint 3 of 3
The pencil that crosses the most spaces is the longest one.
Show solution
Approach: use the evenly spaced lines like a ruler
  1. The background lines are all the same distance apart, so we can count spaces to compare lengths.
  2. Count the spaces each pencil covers from its back end to its tip; pencil D reaches across more spaces than any other.
  3. So pencil D is the longest.
★ MINI-QUIZ

Quick check: shapes and folding

Two warm-ups. Count carefully, then find the matching halves.

2024 · #2 Five pencils labelled A, B, C, D and E lie on a grid of lines. Which pencil is the longest?

Five pencils labelled A, B, C, D and E lie on a grid of lines. Which pencil is the longest?

Figure for Math Kangaroo 2024 Problem 2
Show answer
Answer: D — D
Show hints
Hint 1 of 3
The straight lines in the background are spaced evenly, like the marks on a ruler.
Still stuck? Show hint 2 →
Hint 2 of 3
For each pencil, count how many spaces it covers from its flat back end to its pointy tip.
Still stuck? Show hint 3 →
Hint 3 of 3
The pencil that crosses the most spaces is the longest one.
Show solution
Approach: use the evenly spaced lines like a ruler
  1. The background lines are all the same distance apart, so we can count spaces to compare lengths.
  2. Count the spaces each pencil covers from its back end to its tip; pencil D reaches across more spaces than any other.
  3. So pencil D is the longest.
2009 · #4 In the picture you see the number 930. How many small squares must be changed so that the number becomes 806?

In the picture you see the number 930. How many small squares must be changed so that the number becomes 806?

Figure for Math Kangaroo 2009 Problem 4
Show answer
Answer: B — 6
Show hints
Hint 1 of 3
Look at one digit at a time: the 9 changing to 8, the 3 changing to 0, and the 0 changing to 6.
Still stuck? Show hint 2 →
Hint 2 of 3
For each digit, lay the new shape on top of the old one and watch only the little squares that have to switch from dark to light or light to dark.
Still stuck? Show hint 3 →
Hint 3 of 3
Add up the switches from all three digits.
Show solution
Approach: compare the two numbers one digit at a time and count the little squares that flip
  1. Look at each digit on its own: 9 becomes 8, then 3 becomes 0, then 0 becomes 6.
  2. For the first digit, just one little square switches to turn the 9 into an 8.
  3. For the middle digit, two little squares switch to turn the 3 into a 0; for the last digit, three little squares switch to turn the 0 into a 6.
  4. Counting the switches: 1 + 2 + 3 = 6, so 6 small squares must be changed.
CHAPTER 3

Taller, shorter, and how much more

THEORY

Two friends stand back to back to see who is taller. They line up their feet on the floor first. Then they look at the tops of their heads.

Shapes work the same way. To compare two shapes, line them up at the bottom. Then look at the tops.

shorttallextra

Both blocks sit on the same line. The tall one pokes up higher. The bit poking up above the short one is how much taller it is.

To find that gap, do not add the two heights. Count up from the short top to the tall top. That count is the difference.

Fitting inside is the same idea. A small shape fits in a big one when it is both shorter and thinner. A coin fits in a cup. A little square fits in a big square.

🎯 Try it
One tower is 7 blocks tall. Another is 4 blocks tall. How many blocks taller is the first tower?
Here's how: Start at 4. Count up to 7: 5, 6, 7. That is 3 steps. So it is 3 blocks taller.
THE TRICK

Line the shapes up at the same bottom. Then count the steps from the short top up to the tall top. That gap is the difference.

WATCH OUT

Difference does not mean add. If you hear how much taller or how much more, you count the gap between the tops. You take the small away from the big.

WORKED EXAMPLE
PROBLEM · 2021 #2

The picture shows 2 mushrooms. What is the difference between their heights?

Figure for Math Kangaroo 2021 Problem 2
A) 4 B) 5 C) 6 D) 11 E) 17

The picture shows two mushrooms. You want the difference between their heights.

Both mushrooms grow up from the ground, so they share the same bottom line. Good.

Read the tall mushroom on the scale: it reaches 11. Read the short one: it reaches 6.

Now count up from 6 to the tall top: 7, 8, 9, 10, 11. That is 5 steps. So the mushrooms differ by 5.

If I added 11 and 6 I would get 17, which is one of the wrong answers on purpose! Difference means the gap, so I count up from the small to the big.

Answer: B — 5
RULE OF THUMB

Difference = how far apart the two tops are. Count the gap, do not add.

MORE LIKE THIS
2019 · #11 The giants Tim and Tom build a sandcastle and decorate it with a flag. They push half the flagpole into the highest point of the...

The giants Tim and Tom build a sandcastle and decorate it with a flag. They push half the flagpole into the highest point of the sandcastle. The highest point of the flagpole is now 16 m above the floor, and the lowest is 6 m (see diagram). How high is the sandcastle?

Figure for Math Kangaroo 2019 Problem 11
Show answer
Answer: A — 11 m
Show hints
Hint 1 of 2
Find the whole length of the flagpole from its top and bottom heights.
Still stuck? Show hint 2 →
Hint 2 of 2
Half the pole is buried, so the sand reaches halfway up the pole; that halfway height is the castle.
Show solution
Approach: find pole length, then take the midpoint where the sand reaches
  1. The pole's top is 16 m up and its bottom is 6 m up, so the pole is 16 − 6 = 10 m long.
  2. Half the pole (5 m) is buried in the castle, starting from its bottom at 6 m.
  3. So the sand reaches up to 6 + 5 = 11 m, which is the top of the castle.
  4. The answer is A.
2018 · #21 A belt can be closed in 5 different holes (see picture). How many cm longer is the belt if it is closed in the first hole instead of in...

A belt can be closed in 5 different holes (see picture). How many cm longer is the belt if it is closed in the first hole instead of in the fifth (last) hole?

Figure for Math Kangaroo 2018 Problem 21
Show answer
Answer: B — 8 cm
Show hints
Hint 1 of 2
The holes are spaced 2 cm apart; using the first hole leaves more belt than using the fifth.
Still stuck? Show hint 2 →
Hint 2 of 2
The extra length is the distance from the first hole to the fifth.
Show solution
Approach: measure the gap between the first and last hole
  1. The five holes are 2 cm apart, so the first and fifth holes are 4 × 2 = 8 cm apart.
  2. Closing in the first hole instead of the fifth leaves that extra 8 cm of belt.
  3. So the belt is 8 cm longer.
CHAPTER 4

The walk around (perimeter)

THEORY

A tiny ant walks all the way around a shape. It starts at one corner, follows every side, and comes back to where it began.

How far did the ant walk? That total is called the perimeter. It means the distance all the way around.

To find it, add up the length of every side.

6644start at the red dot

Walk around this rectangle from the red dot. Add the sides: 6 + 4 + 6 + 4.

That is 6 + 4 = 10, and 10 + 10 = 20. The walk around is 20.

A square is easy. All four sides are the same. So add that one side four times. A square with side 5 has a walk of 5 + 5 + 5 + 5 = 20.

Perimeter is the steps all the way around. Add every side.

Peek (for big kids)
For a rectangle there is a fast way: add one long side and one short side, then double it. (6 + 4) × 2 = 20. Adding all four sides gives the same answer, so use whichever you like.
🎯 Try it
A triangle has sides 3, 4, and 5. What is the walk all the way around?
Here's how: Add the three sides: 3 + 4 = 7, then 7 + 5 = 12. The walk around is 12.
THE TRICK

Start at one corner. Walk around and add each side as you pass it. Stop when you get back to the start.

WATCH OUT

Do not forget a side. Mark your starting corner with your finger so you know exactly when you have gone all the way around.

WORKED EXAMPLE
PROBLEM · 2009 #13

The length of a rectangle is 8 cm. The width is half as long. How long are the sides of a square that has the same perimeter as the rectangle?

A) 4 cm B) 6 cm C) 8 cm D) 12 cm E) 24 cm

The rectangle is 8 cm long. The width is half as long, so the width is 4 cm.

First walk around the rectangle. Add all four sides: 8 + 4 + 8 + 4 = 24.

Now we want a square with that same walk of 24. A square has 4 equal sides. So share 24 into 4 equal parts.

24 shared into 4 equal parts is 6. So each side of the square is 6 cm.

Same perimeter means same total walk. The rectangle walk is 24. A square has 4 equal sides, so each side is 24 shared by 4, which is 6.

Answer: B — 6 cm
RULE OF THUMB

Perimeter = add up every side. For a square, that is one side four times.

MORE LIKE THIS
2018 · #14 Susi makes this pattern using ice-lolly sticks. Each stick is 5 cm long and 1 cm wide. How long is Susi's pattern?

Susi makes this pattern using ice-lolly sticks. Each stick is 5 cm long and 1 cm wide. How long is Susi's pattern?

Figure for Math Kangaroo 2018 Problem 14
Show answer
Answer: B — 21 cm
Show hints
Hint 1 of 3
Look only at how far the pattern reaches from left to right — that is its length.
Still stuck? Show hint 2 →
Hint 2 of 3
Along that length, five 5 cm sticks reach across, but they cross each other and overlap.
Still stuck? Show hint 3 →
Hint 3 of 3
Each place two sticks overlap, they share 1 cm (the width of a stick), so that 1 cm only counts once.
Show solution
Approach: add the stick lengths along the row, then subtract the overlaps
  1. Measure the pattern left to right: five 5 cm sticks reach across its length, which alone would be 5 × 5 = 25 cm.
  2. But the sticks cross and overlap at 4 places, and each overlap is 1 cm wide (a stick's width), so 4 cm is counted twice.
  3. Take those 4 cm away once: 25 − 4 = 21 cm.
2022 · #21 Ahmed and Sara start at point A and walk in the directions shown, at the same speed. Ahmed walks around the square garden and Sara walks...

Ahmed and Sara start at point A and walk in the directions shown, at the same speed. Ahmed walks around the square garden and Sara walks around the rectangular garden. How many rounds must Ahmed walk to meet Sara at point A again for the first time?

Figure for Math Kangaroo 2022 Problem 21
Show answer
Answer: C — 3
Show hints
Hint 1 of 3
Work out how far one lap is for each child by adding up the sides of their garden.
Still stuck? Show hint 2 →
Hint 2 of 3
Each child is back at A after 1 lap, 2 laps, 3 laps… so skip-count the total distance for each.
Still stuck? Show hint 3 →
Hint 3 of 3
Look for the first distance that shows up in both lists — that is when they meet at A.
Show solution
Approach: skip-count each child's distances until they match
  1. Ahmed's square garden is 5 + 5 + 5 + 5 = 20 m around; Sara's rectangle is 10 + 5 + 10 + 5 = 30 m around.
  2. Ahmed is back at A after 20, 40, 60… metres; Sara is back at A after 30, 60, 90… metres.
  3. The first distance in both lists is 60 m, so that is when they meet at A again.
  4. Ahmed has gone 60 ÷ 20 = 3 laps, so he walks 3 rounds (choice C).
2019 · #18 Anna uses 32 small grey squares to frame a 7 cm by 7 cm big picture. How many small grey squares does she have to use to frame a 10 cm...

Anna uses 32 small grey squares to frame a 7 cm by 7 cm big picture. How many small grey squares does she have to use to frame a 10 cm by 10 cm big picture?

Figure for Math Kangaroo 2019 Problem 18
Show answer
Answer: C — 44
Show hints
Hint 1 of 2
The grey squares make a ring one square thick all the way around the picture.
Still stuck? Show hint 2 →
Hint 2 of 2
Picture the four sides of the ring, and be careful not to count the four corner squares twice.
Show solution
Approach: count the four sides of the grey ring
  1. Around the 7×7 picture, each side of the grey ring is 9 squares long (the 7 picture squares plus one corner at each end), and 4 sides of 9 with the 4 corners counted once give 32 — matching the picture.
  2. Around the 10×10 picture, each side of the ring is 12 squares long.
  3. Four sides of 12 is 48, but the 4 corners were each counted twice, so take 4 away: 48 − 4 = 44.
  4. So she needs 44 (C) grey squares.
★ MINI-QUIZ

Quick check: compare and walk around

Find a difference, then walk all the way around a shape.

2018 · #21 A belt can be closed in 5 different holes (see picture). How many cm longer is the belt if it is closed in the first hole instead of in...

A belt can be closed in 5 different holes (see picture). How many cm longer is the belt if it is closed in the first hole instead of in the fifth (last) hole?

Figure for Math Kangaroo 2018 Problem 21
Show answer
Answer: B — 8 cm
Show hints
Hint 1 of 2
The holes are spaced 2 cm apart; using the first hole leaves more belt than using the fifth.
Still stuck? Show hint 2 →
Hint 2 of 2
The extra length is the distance from the first hole to the fifth.
Show solution
Approach: measure the gap between the first and last hole
  1. The five holes are 2 cm apart, so the first and fifth holes are 4 × 2 = 8 cm apart.
  2. Closing in the first hole instead of the fifth leaves that extra 8 cm of belt.
  3. So the belt is 8 cm longer.
2018 · #14 Susi makes this pattern using ice-lolly sticks. Each stick is 5 cm long and 1 cm wide. How long is Susi's pattern?

Susi makes this pattern using ice-lolly sticks. Each stick is 5 cm long and 1 cm wide. How long is Susi's pattern?

Figure for Math Kangaroo 2018 Problem 14
Show answer
Answer: B — 21 cm
Show hints
Hint 1 of 3
Look only at how far the pattern reaches from left to right — that is its length.
Still stuck? Show hint 2 →
Hint 2 of 3
Along that length, five 5 cm sticks reach across, but they cross each other and overlap.
Still stuck? Show hint 3 →
Hint 3 of 3
Each place two sticks overlap, they share 1 cm (the width of a stick), so that 1 cm only counts once.
Show solution
Approach: add the stick lengths along the row, then subtract the overlaps
  1. Measure the pattern left to right: five 5 cm sticks reach across its length, which alone would be 5 × 5 = 25 cm.
  2. But the sticks cross and overlap at 4 places, and each overlap is 1 cm wide (a stick's width), so 4 cm is counted twice.
  3. Take those 4 cm away once: 25 − 4 = 21 cm.
CHAPTER 5

Counting little squares

THEORY

A big rectangle can be built from little squares, like a chocolate bar or a window with panes.

You could count the little squares one by one. But that is slow and you might lose your place. Look at the rows instead.

4 in each row3 rows

This bar has 3 rows. Each row holds 4 little squares.

Skip-count, one jump for each row: 4, then 8, then 12. So there are 12 little squares.

Count one row, then skip-count once for each row. The last number is the total.

Peek (for big kids)
Counting equal rows is the same as multiplying. 3 rows of 4 is 3 × 4 = 12. Skip-counting and multiplying always land on the same answer.
🎯 Try it
A window has 5 rows of little panes. Each row has 3 panes. How many panes in all?
Here's how: Skip-count by 3, one jump per row: 3, 6, 9, 12, 15. Five rows gives 15 panes.

Cut an odd shape into easy pieces

Some shapes are not a neat rectangle. They bend, like a letter L or a set of stairs.

An odd shape is hard to count all at once. So do not try. Cut it into easy pieces first.

Draw a line that splits the shape into neat rectangles. Count each piece. Then add the pieces up.

tall partflat partcut on the dotted line

This L is cut into two easy pieces. A tall part and a flat part.

Say the tall part holds 6 little squares and the flat part holds 8. Add them: 6 + 8 = 14 little squares in all.

Cut the odd shape into easy pieces. Count each piece. Add them up.

🎯 Try it
An L shape is cut into two pieces. One piece has 9 little squares. The other has 5. How many little squares in all?
Here's how: Add the two pieces: 9 + 5 = 14 little squares.
THE TRICK

Count how many are in one row. Then skip-count, one jump for each row. The last number is the total.

WATCH OUT

Do not count the same square twice. Go row by row, neat and tidy. And when a row crosses a column, that shared corner square belongs to both, so count it only once.

WORKED EXAMPLE
PROBLEM · 2024 #3

The rectangle on the right has 4 rows and 7 columns, so it is made of 28 white squares. Ira paints 2 whole rows and 1 whole column. How many squares are still white?

Figure for Math Kangaroo 2024 Problem 3
A) 8 B) 10 C) 12 D) 14 E) 17

The rectangle has 4 rows and 7 columns, so 28 little white squares to start. The problem hands us that number.

Here is the light way: a square stays white only if its row was not painted and its column was not painted.

Ira paints 2 of the 4 rows, so 2 rows stay white. She paints 1 of the 7 columns, so 6 columns stay white.

The white squares fill those 2 leftover rows across those 6 leftover columns. Skip-count: 6, then 12. That is 12 white squares.

Counting the painted squares is messy, because the row and the column overlap. Counting the leftover white rows and columns avoids the overlap completely. 2 rows of 6 is 12.

Answer: C — 12
RULE OF THUMB

Rows of equal size: count one row, then skip-count by that number for each row.

MORE LIKE THIS
2009 · #10 Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for...

Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for his sister. How many pieces were there in the entire bar of chocolate?

Figure for Math Kangaroo 2009 Problem 10
Show answer
Answer: D — 40
Show hints
Hint 1 of 2
A 'row of five' tells you how many columns the bar has; a 'column of seven' tells you how many rows.
Still stuck? Show hint 2 →
Hint 2 of 2
Be careful: the column he breaks off is from what is LEFT after the first row is gone.
Show solution
Approach: recover the grid dimensions, then multiply
  1. A row holds 5 pieces, so the bar is 5 columns wide.
  2. After removing that row, a full column still has 7 pieces, so the bar has 7 + 1 = 8 rows.
  3. The whole bar is 8 rows × 5 columns = 40 pieces.
2012 · #7 A wall was tiled alternately with grey and striped tiles. Some tiles have fallen from the wall. How many grey tiles have fallen off?

A wall was tiled alternately with grey and striped tiles. Some tiles have fallen from the wall. How many grey tiles have fallen off?

Figure for Math Kangaroo 2012 Problem 7
Show answer
Answer: C — 7
Show hints
Hint 1 of 2
The wall is a checkerboard, so each missing square was either grey or striped.
Still stuck? Show hint 2 →
Hint 2 of 2
Find every empty square and decide its colour from the alternating pattern.
Show solution
Approach: use the checkerboard pattern to colour each gap
  1. The tiles alternate grey/striped like a chessboard, so a square's colour is fixed by its position.
  2. There are 13 empty squares in the hole.
  3. Working through the checkerboard, 7 of the gaps sit on grey positions and 6 on striped positions.
  4. So 7 grey tiles fell off.
2018 · #17 The big rectangle is made up of several squares of different sizes. Each of the three smallest squares has area 1. What is the area of...

The big rectangle is made up of several squares of different sizes. Each of the three smallest squares has area 1. What is the area of the big rectangle?

Figure for Math Kangaroo 2018 Problem 17
Show answer
Answer: C — 77
Show hints
Hint 1 of 3
Each tiny square has area 1, so its side is just 1 little step — use that step as your ruler for everything else.
Still stuck? Show hint 2 →
Hint 2 of 3
The three tiny squares sit in a row, so the square resting on top of them is 3 steps wide, and you can keep measuring the bigger squares the same way.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you know how many steps wide and how many steps tall the whole rectangle is, multiply those two numbers.
Show solution
Approach: use the side of a tiny square as a measuring step, find each bigger square's side, then multiply the rectangle's width by its height
  1. Each smallest square has area 1, so its side is 1 step long; the three of them in a row make the bottom-left part 3 steps wide.
  2. The square sitting on top of those three is 3 steps on each side, and together with the row of tiny squares the left part is 3 + 1 = 4 steps tall; the square next to it is 4 steps on each side, so the bottom strip is 3 + 4 = 7 steps wide.
  3. The big square on top is as wide as the whole strip, 7 steps, so the rectangle is 7 steps wide and 4 + 7 = 11 steps tall, giving an area of 7 × 11 = 77, answer C.
CHAPTER 6

Hidden shapes

THEORY

Here is a fun one. How many squares can you see in this picture?

Most kids say 4. But look again!

There are 4 small squares. And the big outside square counts too. That is 5 squares, not 4.

The big one is made of the 4 little ones stuck together. It is easy to miss because it is hiding in plain sight.

So count in two passes:

  • First pass: count the smallest shapes, one at a time.
  • Second pass: hunt for bigger shapes made of smaller ones joined together.

Then add the two counts.

Big shapes hide inside little ones. Count the small ones, then hunt for the big ones, then add.

🎯 Try it
A big triangle is split into 2 small triangles by one line down the middle. Counting the small ones AND the big outside triangle, how many triangles are there in all?
Here's how: 2 small triangles, plus the 1 big triangle they make together. 2 + 1 = 3.
THE TRICK

Count in two passes. First the smallest shapes. Then the bigger shapes built from joining small ones. Add the two counts together.

WATCH OUT

The trap is stopping after the small ones. Always ask: can I glue a few small ones into a bigger shape of the same kind? If yes, that is one more to count.

WORKED EXAMPLE
PROBLEM · 2013 #9

How many triangles can be seen in the picture on the right? (Be careful! A triangle can also be made by joining several smaller triangles together.)

Figure for Math Kangaroo 2013 Problem 9
A) 9 B) 10 C) 11 D) 13 E) 12

The question asks how many triangles you can see. The clue warns that a triangle can also be made by joining smaller ones. So we count in two passes.

First pass: point at each smallest triangle, one at a time. There are 6 of them. Count: 1, 2, 3, 4, 5, 6.

Second pass: now hunt for bigger triangles made of 2 or 3 small ones joined together. There are 4 of those. Keep counting: 7, 8, 9, 10.

Add the two passes: 6 small and 4 big. 6 + 4 = 10 triangles.

If I stop after the 6 little ones I get the trap answer. The clue is shouting at me to look for the big ones too. So I do a careful second pass before I answer.

Answer: B — 10
RULE OF THUMB

Hidden shapes: count small, then count big, then add. Never stop after the little ones.

MORE LIKE THIS
2017 · #2 Into how many pieces will the string be cut?

Into how many pieces will the string be cut?

Figure for Math Kangaroo 2017 Problem 2
Show answer
Answer: E — 9
Show hints
Hint 1 of 2
Put your finger on the cut line and touch every spot where it crosses the string.
Still stuck? Show hint 2 →
Hint 2 of 2
A cut always makes one more piece than the number of times it crosses the string.
Show solution
Approach: count crossings along the cut
  1. The string is one long curve, and the dashed line is one straight cut across it.
  2. Walk along the dashed line and count the dots where it crosses the string: there are 8 crossings.
  3. Each crossing makes one extra piece, so 8 crossings give 8 + 1 = 9 pieces.
  4. So the string is cut into 9 pieces.
2015 · #22 In this square there are 9 dots. The distance between the points is always the same. You can draw a square by joining 4 points. How many...

In this square there are 9 dots. The distance between the points is always the same. You can draw a square by joining 4 points. How many different sizes can such squares have?

Figure for Math Kangaroo 2015 Problem 22
Show answer
Answer: D — 3
Show hints
Hint 1 of 2
Squares can sit straight on the dots, but they can also be tilted like a diamond.
Still stuck? Show hint 2 →
Hint 2 of 2
Hunt for a tiny straight square, a big straight square, and one slanted square.
Show solution
Approach: find every square size, straight ones and the tilted one
  1. On the 3-by-3 dots you can make a tiny straight square (1 step on each side) and a big straight square (2 steps on each side).
  2. You can also make a slanted square shaped like a diamond, joining the four middle dots of the edges.
  3. That is three squares of three different sizes.
  4. The answer is 3.
2013 · #20 If I join the midpoints of the sides of the large triangle in the picture, a small triangle is formed. If I join the midpoints of the...

If I join the midpoints of the sides of the large triangle in the picture, a small triangle is formed. If I join the midpoints of the sides of this small triangle, a tiny triangle is formed. How many of these tiny triangles can fit into the largest triangle at the same time?

Figure for Math Kangaroo 2013 Problem 20
Show answer
Answer: D — 16
Show hints
Hint 1 of 3
When you join the midpoints of a triangle, it splits into 4 equal little triangles just like it.
Still stuck? Show hint 2 →
Hint 2 of 3
So the big triangle holds 4 small triangles, and each small triangle holds 4 tiny ones.
Still stuck? Show hint 3 →
Hint 3 of 3
Count the tiny ones: 4 small triangles, each made of 4 tiny ones.
Show solution
Approach: each midpoint-join splits a triangle into 4 copies, so do it twice
  1. Joining the midpoints cuts the big triangle into 4 equal small triangles.
  2. Doing it again cuts each of those small triangles into 4 tiny ones.
  3. That is 4 groups of 4 tiny triangles, so 4 × 4 = 16 fit in the big triangle, which is answer D.
⬢ FINAL TEST

Shapes & Sides: final test

Six problems, easy to medium. Go slow. Point with your finger and count.

2017 · #3 How many blocks are missing in this igloo?

How many blocks are missing in this igloo?

Figure for Math Kangaroo 2017 Problem 3
Show answer
Answer: C — 10
Show hints
Hint 1 of 2
Look only at the white hole in the igloo, not the whole dome.
Still stuck? Show hint 2 →
Hint 2 of 2
Count the empty brick spaces one row at a time and add them up.
Show solution
Approach: count the empty brick spaces in the hole
  1. Look at the white hole in the brick dome.
  2. Count the missing bricks in each row that the hole passes through.
  3. Adding the gaps row by row gives 10 empty brick spaces.
  4. So 10 blocks are missing.
2021 · #2 The picture shows 2 mushrooms. What is the difference between their heights?

The picture shows 2 mushrooms. What is the difference between their heights?

Figure for Math Kangaroo 2021 Problem 2
Show answer
Answer: B — 5
Show hints
Hint 1 of 3
Each mushroom has a number telling how tall it is.
Still stuck? Show hint 2 →
Hint 2 of 3
"Difference" means how much taller one is than the other.
Still stuck? Show hint 3 →
Hint 3 of 3
Take the smaller height away from the bigger height.
Show solution
Approach: read the two heights and subtract
  1. The taller mushroom is 11 tall and the shorter mushroom is 6 tall.
  2. To see how much taller, take away: 11 − 6 = 5.
  3. So their heights differ by 5.
2024 · #3 The rectangle on the right has 4 rows and 7 columns, so it is made of 28 white squares. Ira paints 2 whole rows and 1 whole column. How...

The rectangle on the right has 4 rows and 7 columns, so it is made of 28 white squares. Ira paints 2 whole rows and 1 whole column. How many squares are still white?

Figure for Math Kangaroo 2024 Problem 3
Show answer
Answer: C — 12
Show hints
Hint 1 of 3
Instead of counting the painted squares, count the ones that stay white.
Still stuck? Show hint 2 →
Hint 2 of 3
A square stays white only if its row was not painted AND its column was not painted.
Still stuck? Show hint 3 →
Hint 3 of 3
How many rows are left unpainted, and how many columns?
Show solution
Approach: count the white squares directly using the leftover rows and columns
  1. A square stays white only when both its row and its column are unpainted.
  2. Ira paints 2 of the 4 rows, so 2 rows stay white; she paints 1 of the 7 columns, so 6 columns stay white.
  3. The white squares fill those 2 leftover rows across those 6 leftover columns: 2 × 6 = 12.
  4. So 12 (C) squares are still white.
Another way:
2009 · #13 The length of a rectangle is 8 cm. The width is half as long. How long are the sides of a square that has the same perimeter as the rectangle?

The length of a rectangle is 8 cm. The width is half as long. How long are the sides of a square that has the same perimeter as the rectangle?

Show answer
Answer: B — 6 cm
Show hints
Hint 1 of 2
Find the rectangle's perimeter first.
Still stuck? Show hint 2 →
Hint 2 of 2
A square with the same perimeter has all four sides equal, so divide by 4.
Show solution
Approach: equal perimeters
  1. The rectangle is 8 cm long and half as wide, so 4 cm wide.
  2. Its perimeter is 2 × (8 + 4) = 24 cm.
  3. A square with perimeter 24 cm has side 24 ÷ 4 = 6 cm.
  4. So the square's side is 6 cm.
2009 · #10 Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for...

Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for his sister. How many pieces were there in the entire bar of chocolate?

Figure for Math Kangaroo 2009 Problem 10
Show answer
Answer: D — 40
Show hints
Hint 1 of 2
A 'row of five' tells you how many columns the bar has; a 'column of seven' tells you how many rows.
Still stuck? Show hint 2 →
Hint 2 of 2
Be careful: the column he breaks off is from what is LEFT after the first row is gone.
Show solution
Approach: recover the grid dimensions, then multiply
  1. A row holds 5 pieces, so the bar is 5 columns wide.
  2. After removing that row, a full column still has 7 pieces, so the bar has 7 + 1 = 8 rows.
  3. The whole bar is 8 rows × 5 columns = 40 pieces.
2022 · #21 Ahmed and Sara start at point A and walk in the directions shown, at the same speed. Ahmed walks around the square garden and Sara walks...

Ahmed and Sara start at point A and walk in the directions shown, at the same speed. Ahmed walks around the square garden and Sara walks around the rectangular garden. How many rounds must Ahmed walk to meet Sara at point A again for the first time?

Figure for Math Kangaroo 2022 Problem 21
Show answer
Answer: C — 3
Show hints
Hint 1 of 3
Work out how far one lap is for each child by adding up the sides of their garden.
Still stuck? Show hint 2 →
Hint 2 of 3
Each child is back at A after 1 lap, 2 laps, 3 laps… so skip-count the total distance for each.
Still stuck? Show hint 3 →
Hint 3 of 3
Look for the first distance that shows up in both lists — that is when they meet at A.
Show solution
Approach: skip-count each child's distances until they match
  1. Ahmed's square garden is 5 + 5 + 5 + 5 = 20 m around; Sara's rectangle is 10 + 5 + 10 + 5 = 30 m around.
  2. Ahmed is back at A after 20, 40, 60… metres; Sara is back at A after 30, 60, 90… metres.
  3. The first distance in both lists is 60 m, so that is when they meet at A again.
  4. Ahmed has gone 60 ÷ 20 = 3 laps, so he walks 3 rounds (choice C).
APPENDIX

Quick reference

Memorize these
  • Sides = corners. Always the same number.
  • Fold line: the two halves match when you fold.
  • Difference: count the gap between the tops. Do not add.
  • Perimeter: the walk all the way around. Add every side.
  • Little squares: count one row, then skip-count for each row.
  • Hidden shapes: count small, then count big, then add.
→ Practice Geometry & Measurement problems
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