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Topic

Counting & Probability

Careful counting and how likely something is.

19 problems 📖 Read the lesson
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Problem 3 · 2025 Math Kangaroo Easy
Counting & Probability careful-counting

How many pencils are in this picture?

Figure for Math Kangaroo 2025 Problem 3
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Answer: B — 8
Show hints
Hint 1 of 3
The pencils cross over each other, so counting the middles is tricky.
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Hint 2 of 3
Each pencil has one pointy tip and one pink eraser end — count the ends instead.
Still stuck? Show hint 3 →
Hint 3 of 3
Count just the pointy tips, or just the pink ends, and that is how many pencils there are.
Show solution
Approach: count by matching each tip to its eraser end
  1. The pencils overlap, so count the ends instead of the middles.
  2. Match each sharpened dark tip to one pink eraser end.
  3. There are 8 such pairs, so there are 8 pencils.
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Problem 1 · 2023 Math Kangaroo Easy
Counting & Probability careful-counting

Out of how many circles is the beaver made of?

Figure for Math Kangaroo 2023 Problem 1
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Answer: D — 8
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Hint 1 of 3
Touch each round circle with your finger as you count, so you don't miss one or count it twice.
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Hint 2 of 3
Don't forget the small circles — the eyes and the little circles inside the ears count too.
Still stuck? Show hint 3 →
Hint 3 of 3
Count the big circles first, then go back and count all the little circles.
Show solution
Approach: point to and count every circle, big and small
  1. Start with the big round parts: the face is 1 circle, and the two ears are 2 more.
  2. Now the small circles: each ear has a tiny circle inside (2), the two eyes (2), and the round mouth (1).
  3. Counting them all gives 8 circles, so the answer is D.
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Problem 1 · 2022 Math Kangaroo Easy
Counting & Probability careful-counting

In which box are the most triangles? (Each box holds a mix of triangles, circles, and squares.)

Figure for Math Kangaroo 2022 Problem 1
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Answer: B
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Hint 1 of 2
Triangles are the shapes with three pointy corners - ignore the round circles and the square boxes.
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Hint 2 of 2
Touch and count only the triangles in each box, then see which count is biggest.
Show solution
Approach: count only the triangles in each box
  1. Point to each triangle (three-corner shape) and skip the circles and squares.
  2. Counting triangles: box A has 1, box B has 4, box C has 1, box D has 3, and box E has 1.
  3. Box B has the most triangles, so the answer is B.
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Problem 5 · 2019 Math Kangaroo Easy
Counting & Probability careful-counting

Jörg is sorting his socks. Two socks with the same number make one pair. How many pairs can he find?

Figure for Math Kangaroo 2019 Problem 5
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Answer: C — 5
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Hint 1 of 2
A pair needs two socks showing the same number.
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Hint 2 of 2
Tally how many socks show each number, then count how many numbers appear twice.
Show solution
Approach: tally each number and count those that appear twice
  1. List the number on every sock and see which numbers show up two times.
  2. The numbers 1, 2, 3, 5 and 7 each appear on two socks, while 6 and 8 appear only once.
  3. That makes 5 matching pairs.
  4. So the answer is C.
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Problem 1 · 2016 Math Kangaroo Easy
Counting & Probability careful-countingpath-tracing

How many ropes can you see in this picture?

Figure for Math Kangaroo 2016 Problem 1
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Answer: B — 3
Show hints
Hint 1 of 2
Pick one loose end and follow the strand all the way to where it stops.
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Hint 2 of 2
Each rope has exactly two ends, so count the ends and pair them up.
Show solution
Approach: trace each strand from end to end
  1. Find a free end of a rope and trace the curve until you reach its other end; that is one rope.
  2. Cross it off and repeat with an untraced end.
  3. Following the strands this way separates the tangle into 3 distinct ropes.
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Problem 3 · 2014 Math Kangaroo Easy
Counting & Probability careful-countingcomplementary-counting

There are more grey squares than white. How many more?

Figure for Math Kangaroo 2014 Problem 3
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Answer: D — 9
Show hints
Hint 1 of 3
The picture is a 5-by-5 board, so there are 25 little squares in all.
Still stuck? Show hint 2 →
Hint 2 of 3
The white squares are easy to count because they sit together in the middle — count those first.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you know how many are white, the rest are grey, and you compare the two piles.
Show solution
Approach: count the few white squares, then the grey ones are all the rest
  1. The big square is 5 across and 5 down, so it has 25 small squares altogether.
  2. Count the white squares in the middle: 3 on top, 2 in the middle row, 3 on the bottom, which is 8 white.
  3. The grey squares are all the others: 25 − 8 = 17 grey.
  4. Grey has 17 and white has 8, so there are 17 − 8 = 9 more grey squares.
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Problem 4 · 2014 Math Kangaroo Easy
Counting & Probability careful-counting

A big square is made from 25 small squares put together. A few of the small squares have been lost. How many have been lost?

Figure for Math Kangaroo 2014 Problem 4
Show answer
Answer: D — 10
Show hints
Hint 1 of 3
A full big square is 5 rows of 5, which is 25 small squares.
Still stuck? Show hint 2 →
Hint 2 of 3
Don't try to count the holes — count the squares that are still there.
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Hint 3 of 3
Take the number that is still there away from 25 to find how many are missing.
Show solution
Approach: count the squares still present, then take that away from the full 25
  1. A whole big square is made of 5 × 5 = 25 small squares.
  2. Carefully count the small squares that are still in the picture: there are 15 of them.
  3. The lost ones are the ones missing from 25, so 25 − 15 = 10 squares were lost.
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Problem 24 · 2020 Math Kangaroo Stretch
Counting & Probability careful-countingcasework

Maia the bee can only walk on coloured houses. In how many ways can you colour exactly three white houses, all the same colour, so that Maia can walk from A to B?

Figure for Math Kangaroo 2020 Problem 24
Show answer
Answer: B — 16
Show hints
Hint 1 of 2
You need a connected colored path linking A and B using exactly three newly-colored white houses.
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Hint 2 of 2
Count the distinct sets of three white houses that connect A to B.
Show solution
Approach: count valid 3-house connecting paths
  1. Maia needs a connected run of colored houses from A to B.
  2. Colour exactly three white houses so that, together with A and B, they form a connected walk.
  3. Carefully listing the choices of three white houses that complete a path gives 16 ways.
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Problem 13 · 2016 Math Kangaroo Stretch
Counting & Probability careful-countingcasework

Five sparrows are sitting on a rope (see picture). Some of them are looking to the left, some of them are looking to the right. Every sparrow whistles as many times as the number of sparrows it can see sitting in front of it. For example, the third sparrow whistles exactly twice. How many times do all the sparrows whistle altogether?

Figure for Math Kangaroo 2016 Problem 13
Show answer
Answer: D — 10
Show hints
Hint 1 of 2
Each sparrow only sees the birds in the direction its beak is pointing.
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Hint 2 of 2
Count how many birds are in front of each sparrow, then add all five counts.
Show solution
Approach: count each sparrow's forward view and add them up
  1. Look at which way each beak points, then count the sparrows in front of it.
  2. From left to right the sparrows look right, left, right, left, right, so they see 4, 1, 2, 3, and 0 birds in front.
  3. Adding the whistles: 4 + 1 + 2 + 3 + 0 = 10.
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Problem 11 · 2014 Math Kangaroo Hard
Counting & Probability careful-counting

How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used more than once in the numbers.

Show answer
Answer: D — 7
Show hints
Hint 1 of 3
Bigger than 10 and smaller than 32 means the number has two digits and starts with 1, 2, or 3.
Still stuck? Show hint 2 →
Hint 2 of 3
Be neat: write all the numbers that start with 1, then all that start with 2, then those that start with 3.
Still stuck? Show hint 3 →
Hint 3 of 3
Remember each digit can only be 1, 2, or 3, and don't forget to stop before 32.
Show solution
Approach: list the two-digit numbers in order, using only the digits 1, 2, 3
  1. We want two-digit numbers made only from 1, 2, 3 that are bigger than 10 and smaller than 32.
  2. Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3 (but under 32): just 31.
  3. Count the list: 11, 12, 13, 21, 22, 23, 31.
  4. That makes 7 numbers.
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Problem 11 · 2025 Math Kangaroo Medium
Counting & Probability careful-counting

Alex threads white and black beads alternately onto a piece of string. Twice, 5 beads are hidden — see picture. How many white beads are hidden in total?

Figure for Math Kangaroo 2025 Problem 11
Show answer
Answer: C — 6
Show hints
Hint 1 of 3
The beads always go white, black, white, black — never two of the same colour together.
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Hint 2 of 3
Look at the visible bead right before each hidden part to know what colour comes next.
Still stuck? Show hint 3 →
Hint 3 of 3
Fill in each hidden run of 5 by carrying on the colours, then count just the white ones.
Show solution
Approach: carry on the alternating colour pattern through each hidden run
  1. The beads keep switching: white, black, white, black, and so on.
  2. Each hidden group of 5 carries on that pattern, and in each one 3 of the 5 beads come out white.
  3. There are two hidden groups, so 3 and 3 make 6 white beads in total. The answer is C.
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Problem 10 · 2024 Math Kangaroo Medium
Counting & Probability careful-countingsum-constraint

The numbers 1, 2, 3, 4 and 5 are written on the board. Ali chooses 2 of them and adds them together. How many different results are possible?

Figure for Math Kangaroo 2024 Problem 10
Show answer
Answer: C — 7
Show hints
Hint 1 of 3
The very smallest answer comes from adding the two smallest numbers, 1 and 2.
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Hint 2 of 3
The very biggest answer comes from adding the two biggest numbers, 4 and 5.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you know the smallest and biggest answers, count the numbers in between — but only count each answer once.
Show solution
Approach: find the smallest and biggest sum, then count in between
  1. The smallest sum is 1 + 2 = 3, and the biggest sum is 4 + 5 = 9.
  2. Every number from 3 up to 9 can be made, and none is skipped: 3, 4, 5, 6, 7, 8, 9.
  3. Counting those answers gives 7 different results.
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Problem 9 · 2019 Math Kangaroo Medium
Counting & Probability grid-countingcareful-counting

The picture shows a mouse and a piece of cheese. The mouse is only allowed to move to the neighbouring fields in the direction of the arrows. How many paths are there from the mouse to the cheese?

Figure for Math Kangaroo 2019 Problem 9
Show answer
Answer: E — 6
Show hints
Hint 1 of 2
The arrows only let the mouse go forward toward the cheese, never back.
Still stuck? Show hint 2 →
Hint 2 of 2
Trace one path with your finger, then carefully find every different way without repeating one.
Show solution
Approach: trace every allowed path one at a time and count them
  1. Put your finger on the mouse and follow the arrows toward the cheese.
  2. Each time you reach a spot with two arrows, you can pick a different way to go.
  3. Carefully trace each different route all the way to the cheese without repeating one.
  4. Counting all the different routes gives 6, so the answer is E.
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Problem 12 · 2018 Math Kangaroo Medium
Counting & Probability careful-counting

You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two different digits. How many different numbers do you get?

Show answer
Answer: A — 4
Show hints
Hint 1 of 3
The number is between 10 and 25, so it must start with a 1 or a 2 — try each.
Still stuck? Show hint 2 →
Hint 2 of 3
For a number starting with 1, the other digit comes from 2, 0, 8 (a 1 would repeat).
Still stuck? Show hint 3 →
Hint 3 of 3
Write out every number you can make, then cross off any below 11 or 25 and up, and any with two equal digits.
Show solution
Approach: list every allowed number and count them
  1. The number is bigger than 10 and smaller than 25, so it starts with 1 or 2.
  2. Starting with 1, the other digit (from 2, 0, 8) gives 12, 10, 18 — but 10 is not bigger than 10, so keep 12 and 18.
  3. Starting with 2, staying under 25, gives 20 and 21. All together: 12, 18, 20, 21, which is 4 numbers.
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Problem 8 · 2017 Math Kangaroo Medium
Counting & Probability careful-countingratio

In which picture are there half as many circles as triangles and twice as many squares as triangles? (The five pictures are shown as choices A, B, C, D, E.)

Figure for Math Kangaroo 2017 Problem 8
Show answer
Answer: E
Show hints
Hint 1 of 2
For each picture, count the circles, the triangles, and the squares.
Still stuck? Show hint 2 →
Hint 2 of 2
You want the triangles to be in the middle: half as many circles, and double as many squares.
Show solution
Approach: count the shapes in each picture
  1. Count the three kinds of shape in each picture.
  2. We need a picture where the circles are the small group, the triangles are double the circles, and the squares are double the triangles.
  3. For example 1 circle, 2 triangles, 4 squares fits: circles are half the triangles and squares are twice the triangles.
  4. The picture that matches both rules is E.
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Problem 7 · 2016 Math Kangaroo Medium
Counting & Probability caseworkspatial-reasoning

A hen lays white and brown eggs. Lisa takes six of them and puts them in a box as shown. The brown eggs are not allowed to touch each other. What is the largest number of brown eggs Lisa can put in the box?

Figure for Math Kangaroo 2016 Problem 7
Show answer
Answer: C — 3
Show hints
Hint 1 of 2
Two round eggs touch only when their cups are right next to each other (side by side or one above the other).
Still stuck? Show hint 2 →
Hint 2 of 2
Try putting brown eggs in cups that skip a space, like a checkerboard pattern.
Show solution
Approach: spread the brown eggs out so none are next to each other
  1. The box has 6 cups in 2 rows of 3. Eggs touch only when their cups are side by side or one directly above the other.
  2. Put brown eggs in the two top corners and the middle cup of the bottom row — none of these three cups touch.
  3. A fourth brown egg would have to sit next to one of them, so the most Lisa can place is 3.
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Problem 8 · 2015 Math Kangaroo Medium
Counting & Probability careful-counting

How many numbers are outside the square?

Figure for Math Kangaroo 2015 Problem 8
Show answer
Answer: E — 2
Show hints
Hint 1 of 2
The square holds some of the numbers; the rest are outside it.
Still stuck? Show hint 2 →
Hint 2 of 2
Count only the numbers that are NOT inside the square.
Show solution
Approach: count the numbers that lie outside the square
  1. Draw a finger around the edge of the square: numbers touching the inside are 'in', the rest are 'out'.
  2. Point at each number outside the square and count them.
  3. Exactly 2 numbers sit outside the square, choice E.
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Problem 21 · 2021 Math Kangaroo Stretch
Counting & Probability careful-counting

3 girls and 2 boys were dancing. They danced in pairs so that each girl danced with each boy for exactly 1 minute. At any time, there was only one pair on the dance floor. For how many minutes did they dance?

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Answer: B — 6
Show hints
Hint 1 of 3
Only one pair dances at a time, so the total minutes equals the number of pairs.
Still stuck? Show hint 2 →
Hint 2 of 3
Each girl needs to dance once with each boy.
Still stuck? Show hint 3 →
Hint 3 of 3
Count all the different girl-and-boy pairs you can make.
Show solution
Approach: count the pairs
  1. There are 3 girls and 2 boys, giving 3 × 2 = 6 different pairs.
  2. Each pair dances for 1 minute, one pair at a time.
  3. So they dance for 6 minutes, option B.
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Problem 22 · 2021 Math Kangaroo Stretch
Counting & Probability careful-counting

Each participant in a cooking contest baked one tray of cookies like the one shown beside. What is the smallest number of trays of cookies needed to make the following plate?

Figure for Math Kangaroo 2021 Problem 22
Show answer
Answer: C — 3
Show hints
Hint 1 of 3
Count how many of each kind of cookie the big plate needs.
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Hint 2 of 3
Now see how many of each kind you get from just one tray.
Still stuck? Show hint 3 →
Hint 3 of 3
The kind of cookie you need the most of decides how many trays you must bake.
Show solution
Approach: compare each cookie type to what one tray gives
  1. Count each kind of cookie on the plate, and count how many of that kind one tray makes.
  2. For each kind, see how many trays it would take, then pick the biggest of those numbers.
  3. Three trays are enough to cover every kind, so the answer is 3.
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