30 problems — one per position, pulled from random authored years. Hints and solutions are locked until you submit. Retake as often as you want — every attempt is saved to your test history (if you're logged in).
Pick a paper and hit Start to take it as a timed test — or browse past papers untimed to read the original questions, figures and answers at your own pace.
← BackTip: real Math Kangaroo is about 75 minutes — set a timer if you want the full pressure.
Problem 1 · 2016 Math Kangaroo
Easy
Spatial & Visual Reasoningsymmetry
Which of the following road signs has the most axes of symmetry?
Show answer
Answer: C — The no-entry sign.
Show hints
Hint 1 of 3
Imagine folding each sign along a straight line so the two halves land exactly on top of each other.
Still stuck? Show hint 2 →
Hint 2 of 3
Try both a left-right fold and a top-bottom fold on every sign, then count how many folds work.
Still stuck? Show hint 3 →
Hint 3 of 3
A plain horizontal bar inside a circle matches itself for both folds.
Show solution
Approach: fold each sign and count the lines that match
An axis of symmetry is a fold line where one half lands perfectly on the other half.
The arrow signs match only one fold (or none, once an arrowhead points a direction), and the car shape matches just its up-down fold.
The no-entry sign (a horizontal bar in a circle) matches a left-right fold AND a top-bottom fold, so it has 2 folds.
Two is the most of any sign, so the answer is the no-entry sign, choice (C).
In the square you can see the digits from 1 to 9. A number is created by starting at the star, following the line and writing down the digits along the line while passing. For example, the line shown represents the number 42685. Which of the following lines represents the largest number?
Show answer
Answer: E
Show hints
Hint 1 of 2
Read off the digit string each path makes, then compare them as numbers.
Still stuck? Show hint 2 →
Hint 2 of 2
The biggest number starts with the largest leading digit; break ties by the next digit.
Show solution
Approach: trace each path into a number and compare
Each option traces a path from the star across cells of the 1-9 grid, writing the digit of every cell it passes.
Convert each path to its number and compare digit by digit from the left.
In an enclosure there is a group of kangaroos. If you add up the ages of all the kangaroos you get 36 years. In two years all the kangaroos together will be 60 years old. How many kangaroos are in the enclosure?
Show answer
Answer: A — 12
Show hints
Hint 1 of 3
In two years every single kangaroo gets exactly 2 years older.
Still stuck? Show hint 2 →
Hint 2 of 3
So the whole total grows by 2 for each kangaroo there is.
Still stuck? Show hint 3 →
Hint 3 of 3
The total grew from 36 to 60; ask how many 2s fit into that growth.
Show solution
Approach: the total gains 2 years per kangaroo
In two years the combined age goes from 36 to 60, so it grows by 60 − 36 = 24 years.
Each kangaroo is responsible for 2 of those extra years, so there must be 24 ÷ 2 = 12 kangaroos.
Grey and white pearls are threaded on a string (see picture). Monika wants 5 grey pearls, but she can only pull pearls off from an end of the string, so she has to pull off some white pearls too. What is the smallest number of white pearls she has to pull off to get 5 grey pearls?
Show answer
Answer: B — 3
Show hints
Hint 1 of 2
She can only take pearls from one of the two ends, so compare the two ends.
Still stuck? Show hint 2 →
Hint 2 of 2
Pick the end where the fifth grey pearl is reached after passing the fewest white pearls, and count just those whites.
Show solution
Approach: scan inward from the better end and count the white pearls passed
Pearls come off only from an end, so check each end and stop once 5 grey pearls have come off.
Reading in from the end that reaches the fifth grey pearl soonest, only a few white pearls sit among those first five greys.
Counting just those white pearls gives a minimum of 3.
Andrea has 4 equally long strips of paper. When she glues two together with an overlap of 10 cm, she gets a strip 50 cm long. With the other two she wants to make a 56 cm long strip. How long must the overlap be?
Show answer
Answer: A — 4 cm
Show hints
Hint 1 of 2
Two strips glued together lose exactly one overlap from their combined length.
Still stuck? Show hint 2 →
Hint 2 of 2
First find the length of one strip from the 50 cm result, then use it for the 56 cm strip.
Show solution
Approach: find one strip length, then solve for the new overlap
Two strips glued with a 10 cm overlap measure 50 cm, so the two full strips total 50 + 10 = 60 cm, meaning each strip is 30 cm.
The other two strips also total 60 cm; gluing them to make 56 cm loses 60 − 56 = 4 cm to overlap.
The number 5021972970 is written on a sheet of paper. Julian cuts the sheet twice, so he gets 3 numbers. What is the smallest sum he can get by adding these 3 numbers?
Show answer
Answer: B — 3444
Show hints
Hint 1 of 2
Two cuts give three numbers; their sum is smallest when fewer digits sit in high place-value spots.
Still stuck? Show hint 2 →
Hint 2 of 2
Avoid leaving any single long piece with a large leading digit — spread the digits so the place values stay low.
Show solution
Approach: place the cuts to minimise total place value
The string is 5021972970; two cuts split it into three numbers.
Cutting as 502 | 1972 | 970 keeps the high place values small.
Their sum is 502 + 1972 + 970 = 3444, the smallest achievable.
A box-shaped water tank measures 4 m × 2 m × 1 m, and the water in it is 25 cm deep. The tank is then turned onto its side (see the picture on the right). How high is the water in the tank now?
Show answer
Answer: D — 1 m
Show hints
Hint 1 of 2
The amount of water does not change — only the shape of the space it fills.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the water's volume, then divide by the area of the new bottom face to get the new height.
Show solution
Approach: the volume stays the same, so divide by the new base
The water's volume is 4 × 2 × 0.25 = 2 m³ (using 25 cm = 0.25 m).
After tipping, the tank rests on a 1 m × 2 m face, so the new bottom has area 2 m².
On a distant island, 2020 kangaroos hold hands in a large circle. Each kangaroo is either brown (and always tells the truth) or grey (and always lies). Every one of them says, “One of my neighbours is brown and the other is grey.” How many of the kangaroos are brown?
Show answer
Answer: A — 0
Show hints
Hint 1 of 2
Suppose a brown (truthful) kangaroo exists: its statement about its neighbours would have to hold.
Still stuck? Show hint 2 →
Hint 2 of 2
Test whether any mix of brown and grey can sit in a circle when all say the same sentence - it collapses to one case.
Show solution
Approach: check the statement's consistency around the circle
A brown kangaroo tells the truth, so its two neighbours would be one brown and one grey.
Following that around the circle leads to a contradiction, so no truthful (brown) kangaroo can exist.
Every kangaroo is therefore grey and lying - consistent, since the statement is then false for each.
The year 2022 has three equal digits. This is the third time that Tortoise Eva has experienced a year where the same digit appears three times. What is the minimum age that Tortoise Eva can be this year?
Show answer
Answer: C — 23
Show hints
Hint 1 of 2
List recent years whose four digits include the same digit three times.
Still stuck? Show hint 2 →
Hint 2 of 2
She has lived through three such years; to make her as young as possible, pick the latest possible earlier two.
Show solution
Approach: find the three latest triple-digit years up to 2022
2022 has three 2s. The two latest earlier such years are 2000 (three 0s) and 1999 (three 9s).
If 2022 is her third, she was alive in 1999, so she was born by 1999.
In the last hockey game there were lots of goals. In the first half 6 goals were scored in total and the visiting team was leading. In the second half the home team scored another three goals and won the match. How many goals did the home team score in total?
Show answer
Answer: C — 5
Show hints
Hint 1 of 3
In the first half the two teams together scored 6, and the visitors were ahead.
Still stuck? Show hint 2 →
Hint 2 of 3
List the first-half scores where visitors lead: 6-0, 5-1 or 4-2.
Still stuck? Show hint 3 →
Hint 3 of 3
Then add the home team's 3 second-half goals and see which case lets them win.
Show solution
Approach: test first‑half splits that let home win
First half the goals add to 6 with visitors ahead, so the splits are visitors 6 home 0, visitors 5 home 1, or visitors 4 home 2.
The home team then scores 3 more; to win they need their total above the visitors' 6, 5, or 4.
Only 4-2 works: home ends with \(2 + 3 = 5\) against 4, a win, so the home team scored 5 in total, choice C.
Emily wants to write a number in each empty small triangle. The sum of the numbers in any two triangles that share a side should always be the same. Two of the numbers are already given. What is the sum of all the numbers in the figure?
Show answer
Answer: C — 21
Show hints
Hint 1 of 2
Equal sums on shared sides force neighbouring triangles to repeat values in a pattern.
Still stuck? Show hint 2 →
Hint 2 of 2
Use the two given numbers to fill the pattern, then add every triangle's number.
Show solution
Approach: propagate the equal-sum rule from the given numbers
Two triangles sharing a side must give the same total with each neighbour, which forces a repeating pattern of values across the figure.
Starting from the given 2 and 3, fill every small triangle by that rule.
Adding all the resulting numbers gives a total of 21.
3 green apples, 5 yellow apples, 7 green pears and 2 yellow pears are in a sack. Without looking, Sebastian takes either an apple or a pear out of the sack. How many pieces of fruit must he take out of the sack to be sure of having at least one apple and one pear of the same colour?
Show answer
Answer: E — 13
Show hints
Hint 1 of 2
Think of the unluckiest case: how many fruit could he pull out and still NOT have an apple and a pear of the same colour?
Still stuck? Show hint 2 →
Hint 2 of 2
He could keep drawing green pears and yellow apples forever without ever matching a colour across the two fruit types.
Show solution
Approach: find the largest unlucky collection with no same-colour apple+pear, then add one
He wins when he holds an apple and a pear of the same colour: green-with-green or yellow-with-yellow.
The biggest collection that still avoids this takes all 7 green pears and all 5 yellow apples — green pears with no green apple, yellow apples with no yellow pear — that's 12 fruit and still no match.
Any 13th fruit must be a green apple or a yellow pear, which completes a same-colour pair, so he needs to take 13.
The map shows the seven subway lines of a city. The stations are shown by circles. Martin wants to colour in the subway lines on the plan. If two lines share a common station, they must have different colours. What is the smallest number of different colours he can use?
Show answer
Answer: A — 3
Show hints
Hint 1 of 3
Two lines need different colours only when they share a station, so first hunt for lines that all meet one another.
Still stuck? Show hint 2 →
Hint 2 of 3
If you can find three lines where every pair shares a station, those three already need three different colours — so you can never do it with just two.
Still stuck? Show hint 3 →
Hint 3 of 3
After that, try to colour the rest of the lines reusing only those three colours; if it works, three is the answer.
Show solution
Approach: find three lines that all meet (needs 3 colours), then colour everything with 3
Lines that cross at a shared station must get different colours.
On the map there are three lines that each share a station with the other two, so those three lines need three different colours — two colours can never be enough.
Going line by line, every remaining line shares stations with only lines you have already coloured, so it can always reuse one of the three colours.
Three colours are both needed and enough, so the smallest number is 3.
Maria pours 4 litres of water into vase I, 3 litres into vase II and 4 litres into vase III, as shown. Seen from the front, the three vases look the same size. Which of the following pictures can show the three vases seen from above?
Show answer
Answer: A
Show hints
Hint 1 of 2
Same water heights from the front but different amounts means the vases have different base areas.
Still stuck? Show hint 2 →
Hint 2 of 2
Vase II holds less (3 L vs 4 L) at the same height, so II has the smaller top - match the top-view sizes.
Show solution
Approach: use volume = base area x height to rank the tops
From the front the vases look the same size, so the shown heights reflect base area, not real width.
Vases I and III hold 4 L and II holds 3 L; with the heights shown, the top-view areas differ accordingly.
The top view giving I and III equal larger tops and II a smaller top is option A.
A tower is built from bricks labelled 1 to 50, from bottom to top. Bob builds a new tower: each time he takes the top two bricks off the old tower (keeping their order) and places them on top of the new tower (see picture). When he is finished, which two bricks lie directly on top of each other?
Show answer
Answer: E — 27 and 30
Show hints
Hint 1 of 2
Each move peels the top two bricks (keeping their order) and drops them on the new tower, so the new tower forms in pairs.
Still stuck? Show hint 2 →
Hint 2 of 2
Track the new stack pair by pair and look for which two of the listed bricks end up directly one above the other.
Show solution
Approach: simulate the pair-by-pair transfer onto the new tower
The old tower has 50 on top of 49 on top of 48 … down to 1; each move lifts the top two as a pair and stacks them on the new tower.
So the new tower is built bottom-up as 49,50, then 47,48, then 45,46, and so on, in steps of two.
Following this all the way down, the pair 30 then 27 lands one directly above the other in the finished new tower.
So the two bricks on top of each other are 27 and 30, answer E.
ABCD is a square with side length 10 cm. The distance from N to M is 6 cm. Every part that is not shaded grey is either a square or an isosceles triangle. What is the grey shaded area?
Show answer
Answer: C — 48 cm²
Show hints
Hint 1 of 2
The corner cut-offs and the centred segment NM = 6 fix the small white pieces.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the total white area (corner squares plus isosceles triangles) and subtract from 100.
Show solution
Approach: whole minus white pieces
The square ABCD has area 10 × 10 = 100.
The unshaded parts are small squares at the corners and isosceles triangles, fixed by NM = 6 and the side 10.
We want to colour each square in the grid with one of the colours A, B, C and D so that neighbouring squares always have different colours. (Squares that share a corner also count as neighbouring.) Some squares are already coloured. Which colour(s) could the grey square be?
Show answer
Answer: A — A
Show hints
Hint 1 of 2
Neighbours include diagonal touches, so the grey square clashes with every square around its corner.
Still stuck? Show hint 2 →
Hint 2 of 2
List the colours already used by all squares touching the grey one; whatever is left is the answer.
Show solution
Approach: eliminate neighbour colours
The grey square touches several painted squares, including diagonally.
Those neighbours already use the colours B, C and D.
The only colour left that differs from every neighbour is A.
My little brother has a 4-digit bike lock with the digits 0 to 9 on each part of the lock as shown. He started on the correct combination and turned each part the same amount in the same direction and now the lock shows the combination 6348. Which of the following CANNOT be the correct combination of my brother's lock?
Show answer
Answer: C
Show hints
Hint 1 of 2
Turning every wheel by the same amount in the same direction shifts each digit by the same value (mod 10).
Still stuck? Show hint 2 →
Hint 2 of 2
Subtract 6348 digit-by-digit (mod 10); a real start must give the same shift on all four digits.
Show solution
Approach: check for a constant per-digit shift mod 10
Since each wheel turned the same amount, the true combination differs from 6348 by the same shift in every digit (mod 10).
For 8560 the shifts are 2,2,2,2 and for the others they are equal too — except 4906, whose shifts 8,6,2,8 are not all the same.
So 4906 cannot be the correct combination: choice C.
Each shelf holds a total of 64 deciliters of apple juice. The bottles come in three different sizes: large, medium and small (see picture). How many deciliters of apple juice does a medium bottle contain?
Show answer
Answer: D — 10
Show hints
Hint 1 of 2
Let large, medium, small bottles hold L, M, S deciliters; each shelf's bottles total 64.
Still stuck? Show hint 2 →
Hint 2 of 2
Write one equation per shelf and solve the system for M.
Show solution
Approach: set up one equation per shelf and solve
Counting bottles, the three shelves give 3L + 4S = 64, 2L + 2M + 3S = 64, and 4M + 6S = 64.
Solving the system gives S = 4, L = 16 and M = 10.
Dirce built the sculpture shown by gluing together cubic boxes that are half a metre on each side. She then painted the whole sculpture except the base it rests on, using a special paint sold in cans. Each can covers 4 square metres. How many cans of paint did she have to buy?
Show answer
Answer: B — 4
Show hints
Hint 1 of 2
Each cube edge is 0.5 m, so a small face is 0.25 m^2; count painted faces of the stepped solid, skipping the base.
Still stuck? Show hint 2 →
Hint 2 of 2
Total painted area / 4 m^2 per can, then round up to whole cans.
Show solution
Approach: count exposed faces, convert to area, divide by can coverage
Each cube is 0.5 m on a side, so one face is 0.5x0.5 = 0.25 m^2.
Count every exposed face of the stepped solid except the bottom support; multiplying by 0.25 gives the painted area.
Dividing by 4 m^2 per can and rounding up, she needs 4 cans.
In a group of 10 elves and trolls, each was given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token, and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group?
Show answer
Answer: B — 3
Show hints
Hint 1 of 2
If everyone told the truth the answers would total 1 + 2 + … + 10 = 55; the actual total is only 36.
Still stuck? Show hint 2 →
Hint 2 of 2
Each troll replaces its own token number with a smaller answer; how much total drop can just a few trolls create?
Show solution
Approach: cover the shortfall with the fewest liars
Honest answers would total 1 + 2 + … + 10 = 55, but the answers added to 36, so the trolls' lies pulled the total down by 19.
Each troll's biggest possible drop is from token 10 down to answer 1, a drop of 9; one troll can drop at most 9 and two trolls at most 9 + 8 = 17, both short of 19.
Three trolls can manage it — for example tokens 10, 9, 8 answering 1, 1, 1 drops the total by 9 + 8 + 7 = 24, and other choices hit exactly 19 — so the smallest number of trolls is 3.
There are rectangular cards divided into 4 equal cells with different shapes drawn in each cell. Cards can be placed side by side only if the same shapes appear in adjacent cells on their common side. 9 cards are used to form a rectangle as shown in the figure. Which of the following cards was definitely NOT used to form this rectangle?
Show answer
Answer: E
Show hints
Hint 1 of 2
Cards join only when the touching cells match, so trace the shape sequence along each row and column of the assembled rectangle.
Still stuck? Show hint 2 →
Hint 2 of 2
Read the forced shapes from the given grid; one listed card has a cell pattern that can never fit.
Show solution
Approach: match each card against the forced grid pattern
The assembled rectangle fixes which shapes sit in each cell because adjacent cards must agree on their shared edge.
Reading those forced shapes, four of the candidate cards can occur somewhere in the layout.
Card E has a cell arrangement that cannot fit anywhere, so it was definitely not used.
