🦘 Math Kangaroo ⇄ switch contest
All lessons / Think It Through

Think It Through — Use the clues, cross things out, and find the one answer that fits.

6 chapters Practice problems →

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

About this topic

Pretend you are a detective. A puzzle gives you clues. Your job is to find the one answer the clues point to.

The big secret is short. Cross out what cannot be true. When only one thing is left standing, that is your answer.

You do not have to be fast. Read a clue. Cross out. Read the next clue. Slow and steady wins.

CHAPTER 1

Cross out what cannot be

THEORY

Watch this. Five shapes are lined up. One of them is the secret shape.

A clue says: it is round. So we cross out the ones that are not round.

Clue: it is roundtwo round ones are still in

See the pattern? Each clue is like a little broom. It sweeps away the shapes that do not fit. We swept away the square and the triangle.

Two round shapes are still standing. Now we just need one more clue to pick between them. Clues do not hand you the answer — they tell you what to throw away.

🎯 Try it
There are 5 cups in a row. The secret cup is NOT cup 1. It is NOT cup 3, 4, or 5. Which cup is the secret one?
Here is how: Cross out cup 1. Cross out cups 3, 4, and 5. Only cup 2 is left. So it is cup 2.
THE TRICK

THE MOVE: one clue at a time. Sweep away what it kills. The last one standing wins.

WATCH OUT

Read the clue all the way. A tiny word like not flips everything. Not a square means throw the squares away.

WORKED EXAMPLE
PROBLEM · 2011 #5

Maria describes one of these five shapes in the following way: “It is not a square. It is grey. It is either round or three‑sided.” Which shape did she describe?

Figure for Math Kangaroo 2011 Problem 5
A) A B) B C) C D) D E) E

Maria gives three clues. We sweep with one clue at a time.

Clue 1: it is not a square. Cross out the two squares.

Clue 2: it is grey. Cross out the white shape that is left.

Clue 3: it is round or three‑sided. Cross out the grey rectangle, since it is neither.

One shape is left: the grey triangle. That is shape B.

I do not try to be clever. Each clue throws some shapes away. After three clues, only one can be standing. Whoever is left is my answer.

Answer: B — B
RULE OF THUMB

Clues do not tell you the answer. They tell you what to throw away.

MORE LIKE THIS
2016 · #11 Peter wants to guess Paul's password. He already knows the following: the three last characters are digits, and there are at most three...

Peter wants to guess Paul's password. He already knows the following: the three last characters are digits, and there are at most three capital letters in the password. Which of the following passwords could be Paul's?

Show answer
Answer: C — 1234LLuuaapp4321
Show hints
Hint 1 of 2
Check two rules for each option: the last three characters are digits, and there are at most three capital letters.
Still stuck? Show hint 2 →
Hint 2 of 2
Eliminate any password that breaks either rule.
Show solution
Approach: test each option against both rules
  1. PAUL123 has four capitals (P, A, U, L) — too many.
  2. P0a1u2L3 and 123PAUL do not end in three digits, and Paulin3 also fails the last-three-digits rule.
  3. 1234LLuuaapp4321 ends in 321 (three digits) and has only two capitals (L, L), so it satisfies both rules.
2021 · #20 Mia throws darts at balloons worth 3, 9, 13, 14 and 18 points. She scores 30 points in total. Which balloon does Mia definitely hit?

Mia throws darts at balloons worth 3, 9, 13, 14 and 18 points. She scores 30 points in total. Which balloon does Mia definitely hit?

Show answer
Answer: A — 3
Show hints
Hint 1 of 2
Find every set of balloon values that adds up to 30.
Still stuck? Show hint 2 →
Hint 2 of 2
The balloon that appears in all of those sets is the one she definitely hit.
Show solution
Approach: list the subsets summing to 30 and find the common balloon
  1. Balloons are 3, 9, 13, 14, 18. The sums making 30 are 3+9+18 and 3+13+14.
  2. Both winning sets include the balloon worth 3.
  3. So whichever way she scored 30, she hit the 3.
  4. She definitely hit balloon 3 (A).
2024 · #1 Which number is in the triangle and also in the square and also in the circle?

Which number is in the triangle and also in the square and also in the circle?

Figure for Math Kangaroo 2024 Problem 1
Show answer
Answer: C — 5
Show hints
Hint 1 of 3
We want a number that lives inside all three shapes, not just one or two of them.
Still stuck? Show hint 2 →
Hint 2 of 3
Find the little spot in the middle where the triangle, the square, and the circle all cover each other.
Still stuck? Show hint 3 →
Hint 3 of 3
Read the number sitting in that shared middle spot.
Show solution
Approach: find the number in the shared middle spot
  1. The three shapes overlap in the middle, making a small spot that is inside the triangle AND the square AND the circle.
  2. Only one number sits in that shared middle spot.
  3. That number is 5, so the answer is 5.
CHAPTER 2

Who is taller? Line them up

THEORY

Imagine three friends standing back to back. You are told who is taller, two at a time. Tom is taller than Sam. Mia is taller than Tom.

Trying to hold that in your head is hard. So do not. Just line them up, shortest at one end. Slide each new friend into the right spot.

short → tallSamTomMia→

Once they all stand in a line, the answer is easy to see. The tallest is at one end. The shortest is at the other. You do not have to remember anything — the line remembers for you.

🎯 Try it
Ann is taller than Bob. Bob is taller than Cal. Counting the shortest as number 1, what number is Ann (1, 2, or 3)?
Here is how: Cal is shortest (1). Bob is in the middle (2). Ann is tallest (3).

A balance scale tells you who is heavier

A balance scale is like a see‑saw. The heavier side goes down. The lighter side goes up.

25up = lighterdown = heavier

The 5 went down, so 5 is heavier than 2. If the two sides sit level, then both sides weigh the same.

🎯 Try it
A balance is level. The left side holds an 8‑kg block. The right side holds a 2‑kg block and one mystery block. How heavy is the mystery block, in kg? (Hint: level means both sides weigh the same.)
Here is how: Level means equal. The left side is 8. The right side is 2 plus the mystery. So 2 plus the mystery is 8. The mystery is 6 kg.

Worked puzzle (2023, #10): Six weights are 1, 2, 3, 4, 5, and 6 kg. You put five of them on the two pans so the scale is level. One weight is left out. Which one?

Level means both pans weigh the same. So the five weights must split into two equal piles. Add all six: 1+2+3+4+5+6 = 21. That is odd, so all six cannot split evenly. We take one away to fix it.

Try leaving out the 1. The other five add to 20, which is 10 and 10. Put 6+4 on one pan and 5+3+2 on the other — both pans are 10. It balances! The left‑out weight is 1 kg.

THE TRICK

THE MOVE: draw a line. Put the shortest at one end. Slide each friend in until they all fit.

WATCH OUT

Go slow. Taller than and shorter than are easy to flip. Point with your finger as you read each clue.

WORKED EXAMPLE
PROBLEM · 2014 #6

Anita has built fewer sandcastles than Hans but more than Stefan. Fabian has built more sandcastles than Anita and more than Hans. Bruno has built more sandcastles than Hans but fewer than Fabian. Who has built the most sandcastles?

A) Hans B) Anita C) Stefan D) Bruno E) Fabian

Make one line, from fewest sandcastles to most. Add each kid where the clue says.

Anita made more than Stefan but fewer than Hans. So the line so far is Stefan, Anita, Hans.

Fabian made more than Anita and more than Hans. So Fabian goes on the far end — the most.

The kid on the “most” end is Fabian.

I do not hold all the names in my head. I build one line, fewest to most, and slide each kid in. Whoever ends up on the tall end made the most.

Answer: E — Fabian
RULE OF THUMB

One long line beats trying to hold every clue in your head.

MORE LIKE THIS
2010 · #6 Four friends each eat some ice cream. Mike eats more than Franz, Jaroslav eats more than Veit, and Jaroslav eats less than Franz. Put...

Four friends each eat some ice cream. Mike eats more than Franz, Jaroslav eats more than Veit, and Jaroslav eats less than Franz. Put the friends in order by how much ice cream they ate, starting with the largest amount.

Show answer
Answer: C — Mike, Franz, Jaroslav, Veit
Show hints
Hint 1 of 2
Turn each clue into a simple "more than" comparison and chain them together.
Still stuck? Show hint 2 →
Hint 2 of 2
Jaroslav eats less than Franz but more than Veit, and Mike eats more than Franz.
Show solution
Approach: chain the inequalities into one order
  1. Mike > Franz, Jaroslav > Veit, and Jaroslav < Franz.
  2. Combine: Mike is biggest, then Franz, then Jaroslav, then Veit.
  3. Largest first: Mike, Franz, Jaroslav, Veit — option C.
2021 · #6 The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which...

The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which tower is the tallest?

Show answer
Answer: D — silver tower
Show hints
Hint 1 of 3
Line the towers up from shortest to tallest using the clues.
Still stuck? Show hint 2 →
Hint 2 of 3
Green is taller than pink and red; now see how silver compares to green.
Still stuck? Show hint 3 →
Hint 3 of 3
Whoever is taller than green must be the tallest of all.
Show solution
Approach: order the towers by height
  1. The first clues say red is shortest, then pink, then green is taller than both.
  2. The last clue says silver is taller than green, so silver beats green and everyone below it.
  3. The tallest is the silver tower.
2014 · #10 Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle?

Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle?

Figure for Math Kangaroo 2014 Problem 10
Show answer
Answer: B — 3
Show hints
Hint 1 of 3
Where two sticks cross, the one you can see all of is on top and the one that gets hidden is underneath.
Still stuck? Show hint 2 →
Hint 2 of 3
Build the pile order from the bottom stick (2) up to the top stick (6) by checking each crossing.
Still stuck? Show hint 3 →
Hint 3 of 3
With seven sticks in a pile, the exact middle one is the 4th counting from the bottom.
Show solution
Approach: read each crossing to stack the sticks from bottom to top, then take the 4th one
  1. At every place two sticks cross, the unbroken stick is the one lying on top.
  2. Using that, build the order of the pile from stick 2 at the very bottom up to stick 6 at the very top.
  3. Seven sticks make a pile, and the middle of seven is the 4th one from the bottom.
  4. That 4th, middle stick is stick 3.
2018 · #24 Each of the four balls weighs either 10, 20, 30 or 40 grams. The two balances are shown. Which ball weighs 30 grams?

Each of the four balls weighs either 10, 20, 30 or 40 grams. The two balances are shown. Which ball weighs 30 grams?

Figure for Math Kangaroo 2018 Problem 24
Show answer
Answer: C — C
Show hints
Hint 1 of 3
The four weights 10, 20, 30, 40 add up to 100 grams altogether — keep that total handy.
Still stuck? Show hint 2 →
Hint 2 of 3
The second scale is level, so the two balls on one side weigh exactly as much as the single ball C on the other side.
Still stuck? Show hint 3 →
Hint 3 of 3
Try out which single weight can be split into two of the other weights, and check it against the tilted first scale.
Show solution
Approach: use the level scale to say C equals two other balls added, then test which weight that can be
  1. The second scale balances, so B and D together weigh the same as C alone — that means C is made by adding two of the other weights.
  2. Among 10, 20, 30 and 40, the only weight that is the sum of two of the others is 30 = 10 + 20, so C must be 30 grams (with B and D being 10 and 20, leaving A as 40).
  3. Check the first scale: A and B together (40 + 20 = 60) are heavier than C and D (30 + 10 = 40), and that side does tip down, so everything fits — the 30 gram ball is C.
2025 · #24 To compare the weights of a red square, a star and a green circle, Mona uses a beam balance (see picture). The lower pan holds the...

To compare the weights of a red square, a star and a green circle, Mona uses a beam balance (see picture). The lower pan holds the heavier side. Each shape always has the same weight, different shapes have different weights, and every weight is 1, 2, 3, 4 or 5 kg. How many kilograms does the red square weigh?

Figure for Math Kangaroo 2025 Problem 24
Show answer
Answer: C — 3 kg
Show hints
Hint 1 of 2
The lower pan is heavier: one square outweighs two stars, and two circles outweigh three squares.
Still stuck? Show hint 2 →
Hint 2 of 2
Try weights 1 to 5 for the square: it must be more than two stars yet light enough that two circles beat three of it.
Show solution
Approach: read both balances, then test the square's weight
  1. Left balance: the square pan is lower, so one square is heavier than two stars.
  2. Right balance: the two-circle pan is lower, so two circles are heavier than three squares.
  3. One square beating two stars means the square is at least 3 (two different weights of 1 and 2 already make 3).
  4. If the square is 4, three squares weigh 12, but two circles can be at most 2×5 = 10 - too light. So the square is 3, with stars 1 and circles 5: 3 > 2 and 10 > 9 both hold, giving 3 kg, option C.
★ MINI-QUIZ

Clues and lining up

Two puzzles: cross out what cannot be, and put friends in order.

2010 · #6 Four friends each eat some ice cream. Mike eats more than Franz, Jaroslav eats more than Veit, and Jaroslav eats less than Franz. Put...

Four friends each eat some ice cream. Mike eats more than Franz, Jaroslav eats more than Veit, and Jaroslav eats less than Franz. Put the friends in order by how much ice cream they ate, starting with the largest amount.

Show answer
Answer: C — Mike, Franz, Jaroslav, Veit
Show hints
Hint 1 of 2
Turn each clue into a simple "more than" comparison and chain them together.
Still stuck? Show hint 2 →
Hint 2 of 2
Jaroslav eats less than Franz but more than Veit, and Mike eats more than Franz.
Show solution
Approach: chain the inequalities into one order
  1. Mike > Franz, Jaroslav > Veit, and Jaroslav < Franz.
  2. Combine: Mike is biggest, then Franz, then Jaroslav, then Veit.
  3. Largest first: Mike, Franz, Jaroslav, Veit — option C.
2021 · #6 The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which...

The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which tower is the tallest?

Show answer
Answer: D — silver tower
Show hints
Hint 1 of 3
Line the towers up from shortest to tallest using the clues.
Still stuck? Show hint 2 →
Hint 2 of 3
Green is taller than pink and red; now see how silver compares to green.
Still stuck? Show hint 3 →
Hint 3 of 3
Whoever is taller than green must be the tallest of all.
Show solution
Approach: order the towers by height
  1. The first clues say red is shortest, then pink, then green is taller than both.
  2. The last clue says silver is taller than green, so silver beats green and everyone below it.
  3. The tallest is the silver tower.
CHAPTER 3

Who is telling the truth?

THEORY

In some puzzles, people say things. Some are telling the truth. Some are fibbing. Your job is to find who is fibbing.

Here is the move: try one guess and check it. Pick one person to be the answer. Then read every sentence. Do they all fit the rule? If one sentence breaks the rule, your guess was wrong. Try the next.

✓ this guess fits× this guess breaksTry a guess.Check every line.Keep the guessthat never breaks.

It is like trying on shoes. You try one pair. If it pinches, take it off and try the next. You keep the pair that fits just right. The right answer never makes a rule break.

🎯 Try it
Two kids. Kid 1 says: I did not do it. Kid 2 says: Kid 1 is telling the truth. Only one kid did it, and the kids who did NOT do it told the truth. Which kid number did it?
Here is how: Try Kid 1 did it. Then Kid 2 did not, so Kid 2 must tell the truth. But Kid 2 says Kid 1 is honest, and Kid 1 (who said “I did not do it”) is really fibbing — so Kid 2 ends up wrong. A non‑doer cannot be wrong, so that breaks. Now try Kid 2 did it. Then Kid 1 did not, and Kid 1 truly says “I did not do it” — the truth. That fits, so it was Kid 2.
THE TRICK

THE MOVE: guess one answer. Read every sentence. Keep the guess that never breaks the rule.

WATCH OUT

Do not stop at the first guess that seems okay. Read all the sentences. The right guess must fit every single one.

WORKED EXAMPLE
PROBLEM · 2019 #23

One of the 5 children Alex, Bartek, Cora, Dani and Emil has eaten a cake. Alex says: “I did not eat a cake.” Bartek says: “I ate a cake.” Cora says: “Emil has not eaten a cake.” Dani says: “I did not eat a cake.” Emil says: “Alex has eaten a cake.” One of the children lies. Which child has eaten a cake?

A) Alex B) Bartek C) Cora D) Dani E) Emil

One child ate the cake. The rule is that just one child fibs. Try a guess and check it.

Pretend Bartek ate it. Bartek said “I ate a cake” — that is true.

Now check the rest. Alex, Cora, and Dani all say true things. Only Emil, who said “Alex ate it,” is wrong.

That is exactly one fib, just like the rule wants. So the child who ate the cake is Bartek.

I do not have to be clever. I pick a child, read every sentence, and count the fibs. When my guess leaves exactly one fib, I have found the eater.

Answer: B — Bartek
RULE OF THUMB

You do not have to be clever. You can try each one and check.

MORE LIKE THIS
2020 · #20 Five friends decided to spend their vacation together. In a conversation, Adam said, “Yesterday was Wednesday.” Beto said, “Tomorrow...

Five friends decided to spend their vacation together. In a conversation, Adam said, “Yesterday was Wednesday.” Beto said, “Tomorrow will be Friday.” Carlos said, “The day before yesterday was Tuesday.” David said, “The day after tomorrow is Saturday.” Finally, Eli said, “Today is Monday.” One of them was wrong. Who was wrong?

Show answer
Answer: E — Eli
Show hints
Hint 1 of 2
Translate each statement into 'today is ___' and see if they agree.
Still stuck? Show hint 2 →
Hint 2 of 2
Four friends point to the same day; the odd one out is wrong.
Show solution
Approach: convert each clue to today's weekday
  1. Adam (yesterday Wed), Beto (tomorrow Fri), Carlos (day-before-yesterday Tue) and David (day-after-tomorrow Sat) all mean today is Thursday.
  2. Eli says today is Monday, disagreeing with the other four, so Eli was wrong.
2010 · #23 Andrew, Stefan, Robert and Marko meet at a concert in Zagreb. They come from different cities — Paris, Dubrovnik, Rome and Berlin (not...

Andrew, Stefan, Robert and Marko meet at a concert in Zagreb. They come from different cities — Paris, Dubrovnik, Rome and Berlin (not necessarily in this order).

• Andrew and the friend from Berlin arrive first in Zagreb. Neither of these two has ever been to Paris or Rome.
• Robert is not from Berlin, but he arrives together with the friend from Paris.
• Marko and the friend from Paris enjoyed the concert very much.

Which city does Marko come from?

Show answer
Answer: D — Berlin
Show hints
Hint 1 of 2
Andrew is not from Paris, Rome, or Berlin, so his city is forced.
Still stuck? Show hint 2 →
Hint 2 of 2
Place each person's city one clue at a time until only Marko's is left.
Show solution
Approach: eliminate cities person by person
  1. Andrew and the Berlin friend are two people, and Andrew hasn't been to Paris or Rome, so Andrew is from Dubrovnik.
  2. Robert isn't from Berlin and arrives with the Paris friend, so Robert is from Rome.
  3. That leaves Paris and Berlin for Stefan and Marko; since Marko isn't the Paris friend, Marko is from Berlin.
CHAPTER 4

What comes next? Find the rule

THEORY

The days of the week go round and round, like a wheel. Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday — then Monday comes back.

To find what comes next, first find the rule. Then follow it, one tiny step.

yesterday → today → tomorrowSunday→Monday→Tuesday?

The rule here is: tomorrow is always one day after today. So step forward, one day at a time.

🎯 Try it
A pattern goes 2, 4, 6, 8, and keeps adding 2 each time. What number comes next?
Here is how: Each number is 2 more than the one before. 8 plus 2 is 10.

Follow the path, one step at a time

Some puzzles show a path: a maze, a road, or ladders. You have to get from the start to the end. The move is to trace it with your finger and count each step as you go.

start123end

Count the steps as you trace: 1, 2, 3. Do not jump ahead. If you skip a step, your count goes wrong.

🎯 Try it
You climb a ladder from the ground to a roof. The ladder has 4 rungs and you step on every rung. How many steps up do you take?
Here is how: Count each rung as one step: 1, 2, 3, 4. That is 4 steps.

Worked puzzle (2024, #4): A firefighter wants to reach a fire. He can only climb ladders, never jump. What is the fewest ladders he must climb to get there?

Start at the firefighter on the ground. Trace one ladder up to the next platform. Then the next ladder, then the next. Count one ladder for each climb.

Follow the route that reaches the fire with the fewest ladders. Counting each ladder, the shortest route uses 6 ladders.

THE TRICK

THE MOVE: find the rule from what you see. Then take one step at a time.

WATCH OUT

Do not leap to the end. Take the steps one by one with your finger, or you may skip one and lose count.

WORKED EXAMPLE
PROBLEM · 2019 #3

Yesterday it was Sunday. Which day will it be tomorrow?

A) Saturday B) Thursday C) Wednesday D) Tuesday E) Monday

The clue tells you about yesterday. Walk forward to today, then to tomorrow.

Yesterday was Sunday. So today is the day after Sunday: Monday.

Tomorrow is the day after Monday. Step forward one more day: Tuesday.

I take it one day at a time, never jumping. Yesterday Sunday, so today Monday, so tomorrow Tuesday. The answer is Tuesday.

Answer: D — Tuesday
RULE OF THUMB

Find the rule first. Then the rule does the work for you.

MORE LIKE THIS
2018 · #13 Felix the rabbit has 20 carrots. Every day he eats 2 of them. He has eaten the 12th carrot on a Wednesday. On which day of the week did...

Felix the rabbit has 20 carrots. Every day he eats 2 of them. He has eaten the 12th carrot on a Wednesday. On which day of the week did he start eating the carrots?

Show answer
Answer: E — Friday
Show hints
Hint 1 of 2
Carrots come in pairs each day: day 1 is carrots 1 and 2, day 2 is carrots 3 and 4, and so on.
Still stuck? Show hint 2 →
Hint 2 of 2
Find which day number holds the 12th carrot, then hop backwards on the calendar to day 1.
Show solution
Approach: pair the carrots into days to reach day 6, then count back on the days of the week
  1. Two carrots each day means: day 1 = carrots 1,2; day 2 = 3,4; day 3 = 5,6; day 4 = 7,8; day 5 = 9,10; day 6 = 11,12.
  2. So the 12th carrot is eaten on day 6, which is the Wednesday given.
  3. Count back 5 days from Wednesday: Tue, Mon, Sun, Sat, Friday — that is day 1, answer E.
2023 · #17 An underground line has the six stations A, B, C, D, E and F. The train stops at every station. After reaching the end of the line (A or...

An underground line has the six stations A, B, C, D, E and F. The train stops at every station. After reaching the end of the line (A or F) the train continues in the opposite direction. The train conductor starts his journey in station B. His first stop is in station C. In which station will be his 46th stop?

Figure for Math Kangaroo 2023 Problem 17
Show answer
Answer: D — D
Show hints
Hint 1 of 2
List the stops in order; the train bounces off the ends A and F.
Still stuck? Show hint 2 →
Hint 2 of 2
The sequence of stops repeats, so find the cycle length and use the remainder.
Show solution
Approach: list the bouncing sequence, find its period, take 46 mod period
  1. Starting at B and first stopping at C, the stops run C,D,E,F,E,D,C,B,A,B and then repeat with period 10.
  2. Since 46 = 4×10 + 6, the 46th stop matches the 6th stop in the cycle.
  3. The 6th stop is D.
2023 · #20 Five clocks are hanging on the wall. One clock is one hour ahead. Another one is one hour late and one is correct. Two clocks have...

Five clocks are hanging on the wall. One clock is one hour ahead. Another one is one hour late and one is correct. Two clocks have stopped working. Which clock shows the correct time?

Figure for Math Kangaroo 2023 Problem 20
Show answer
Answer: D — D
Show hints
Hint 1 of 2
The fast, slow and correct clocks differ by exactly one hour from each other in a chain.
Still stuck? Show hint 2 →
Hint 2 of 2
Find three clocks whose times are consecutive hours; the middle one is correct.
Show solution
Approach: locate the +1, correct and -1 trio of consecutive-hour clocks
  1. The accurate clock, the one an hour fast, and the one an hour slow show three times spaced one hour apart.
  2. Only one set of three clock faces forms such a consecutive-hour chain.
  3. The middle time of that chain is the correct clock, shown in D.
2025 · #22 The calendar shows the days of a month, with the columns running Monday, Tuesday, …, Sunday, but the dates are missing. The two dark...

The calendar shows the days of a month, with the columns running Monday, Tuesday, …, Sunday, but the dates are missing. The two dark grey boxes are a Thursday and a Wednesday. If you add the two dates in the dark grey boxes, you get 29. What day of the week is the 1st of the month?

Figure for Math Kangaroo 2025 Problem 22
Show answer
Answer: D — Thursday
Show hints
Hint 1 of 3
Count the days from the grey Thursday box down to the grey Wednesday box - it is exactly 13 days later.
Still stuck? Show hint 2 →
Hint 2 of 3
So the two grey dates are 13 apart and add up to 29; find two such numbers.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you know the grey Thursday's date, count back by 7s to reach the 1st.
Show solution
Approach: use the 13-day gap and the sum of 29, then walk back to the 1st
  1. Going down two rows and one box to the left, the grey Wednesday lands 13 days after the grey Thursday.
  2. Take away those extra 13 days from the sum 29, and 16 is left for two equal Thursday dates, so each is 16 ÷ 2 = 8 - the grey Thursday is the 8th.
  3. Counting back a week, the 1st of the month is also a Thursday (8, then 1).
  4. So the 1st is a Thursday, option D.
2020 · #13 Amelia built a crown using 10 copies of the small piece shown. The pieces were joined so that touching sides always show the same...

Amelia built a crown using 10 copies of the small piece shown. The pieces were joined so that touching sides always show the same number, as in the picture, where four pieces are filled in. What number appears in the coloured triangle?

Figure for Math Kangaroo 2020 Problem 13
Show answer
Answer: A — 1
Show hints
Hint 1 of 3
All ten pieces are exact copies, so the same little numbers repeat as you go around the ring.
Still stuck? Show hint 2 →
Hint 2 of 3
Where two pieces touch, the touching numbers match, which lines the pieces up the same way each time.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you know how one piece sits, every piece sits the same, so jump that pattern around to the colored triangle.
Show solution
Approach: use the repeating pattern of identical pieces around the ring
  1. Every piece is the same copy, and because touching sides must show equal numbers, each piece is placed in the very same way as its neighbour.
  2. So the numbers repeat in the same order all the way around the crown, like a pattern that copies itself ten times.
  3. Reading that repeating pattern from the filled-in pieces around to the colored triangle, the number landing there is 1, choice A.
★ MINI-QUIZ

Truth and patterns

Who is telling the truth, and what comes next.

2020 · #20 Five friends decided to spend their vacation together. In a conversation, Adam said, “Yesterday was Wednesday.” Beto said, “Tomorrow...

Five friends decided to spend their vacation together. In a conversation, Adam said, “Yesterday was Wednesday.” Beto said, “Tomorrow will be Friday.” Carlos said, “The day before yesterday was Tuesday.” David said, “The day after tomorrow is Saturday.” Finally, Eli said, “Today is Monday.” One of them was wrong. Who was wrong?

Show answer
Answer: E — Eli
Show hints
Hint 1 of 2
Translate each statement into 'today is ___' and see if they agree.
Still stuck? Show hint 2 →
Hint 2 of 2
Four friends point to the same day; the odd one out is wrong.
Show solution
Approach: convert each clue to today's weekday
  1. Adam (yesterday Wed), Beto (tomorrow Fri), Carlos (day-before-yesterday Tue) and David (day-after-tomorrow Sat) all mean today is Thursday.
  2. Eli says today is Monday, disagreeing with the other four, so Eli was wrong.
2018 · #13 Felix the rabbit has 20 carrots. Every day he eats 2 of them. He has eaten the 12th carrot on a Wednesday. On which day of the week did...

Felix the rabbit has 20 carrots. Every day he eats 2 of them. He has eaten the 12th carrot on a Wednesday. On which day of the week did he start eating the carrots?

Show answer
Answer: E — Friday
Show hints
Hint 1 of 2
Carrots come in pairs each day: day 1 is carrots 1 and 2, day 2 is carrots 3 and 4, and so on.
Still stuck? Show hint 2 →
Hint 2 of 2
Find which day number holds the 12th carrot, then hop backwards on the calendar to day 1.
Show solution
Approach: pair the carrots into days to reach day 6, then count back on the days of the week
  1. Two carrots each day means: day 1 = carrots 1,2; day 2 = 3,4; day 3 = 5,6; day 4 = 7,8; day 5 = 9,10; day 6 = 11,12.
  2. So the 12th carrot is eaten on day 6, which is the Wednesday given.
  3. Count back 5 days from Wednesday: Tue, Mon, Sun, Sat, Friday — that is day 1, answer E.
CHAPTER 5

Sort by a rule

THEORY

Look at a pile of shapes. Each shape has more than one thing about it: a color, a size, a shape. It is a lot to look at all at once.

So do not. The move is to look for one thing at a time. First find all the grey ones. Then the big ones. Then the round ones.

grey?big?

Sorting is making groups. One group for each rule. Here is the fun part: one shape can join more than one group at the same time. The big grey circle is in both groups.

🎯 Try it
You have 6 shapes. 4 of them are red. 2 of them are NOT red. How many red shapes are there?
Here is how: 4 are red and 2 are not. The red group has 4 shapes.

Fill a grid by a rule

A grid is a box of little squares, set out in rows and columns. A row goes across. A column goes up and down.

Some grid puzzles give a rule, like “every row needs exactly two dots.” The move is to count one row at a time, then one column at a time. Find the row with too many, and the row with too few.

212

The top row has 2 dots — good. The middle row has only 1 — too few. Now look down each column the same way. Where a too‑full row crosses a too‑full column, that is the dot to move.

🎯 Try it
A row should have exactly 3 stickers. Right now it has 2 stickers. How many more stickers do you add to that row?
Here is how: The row needs 3 and has 2. 3 take away 2 is 1. You add 1 more sticker.

Worked puzzle (2022, #3): Coins sit on a grid. You move just one coin to an empty square so every row and every column ends with exactly two coins. Which coin moves?

Count each row. One row has 3 coins (too many) and one row has only 1 (too few). Now count each column the same way: one has 3, one has 1.

The coin to move is the one sitting where the too‑full row crosses the too‑full column. That coin is C. Move it into the empty square of the short row and short column, and every line has two.

THE TRICK

THE MOVE: one thing at a time. Make a group for each rule. A shape can join more than one group.

WATCH OUT

Do not try to see color and size and shape all at once. Pick one thing, find them all, then move to the next thing.

WORKED EXAMPLE
PROBLEM · 2022 #17

Wanda chooses some of the shapes shown. She says: “I have chosen exactly 2 grey, 2 big and 2 round shapes.” What is the smallest number of shapes Wanda could have chosen?

Figure for Math Kangaroo 2022 Problem 17
A) 2 B) 3 C) 4 D) 5 E) 6

Wanda needs 2 grey, 2 big, and 2 round. The trick: one clever shape can count for more than one rule.

Pick a shape that is big AND grey. That ticks one grey box and one big box.

Pick a shape that is big AND round. That ticks the other big box and one round box.

Pick a shape that is grey AND round. That ticks the last grey box and the last round box.

Now check: 2 grey, 2 big, 2 round — all done with just 3 shapes. So the fewest is 3.

If each shape does double duty, I need fewer shapes. With three shapes that each cover two rules, I fill all six boxes. Three is the smallest number that works.

Answer: B — 3
RULE OF THUMB

When one item fits two rules, you need fewer items.

MORE LIKE THIS
2016 · #16 Hannes has a game board with 11 spaces (see picture). He places one coin each on eight spaces that lie next to each other. He can choose...

Hannes has a game board with 11 spaces (see picture). He places one coin each on eight spaces that lie next to each other. He can choose on which space to place his first coin. No matter where Hannes starts, some spaces will definitely be filled. How many spaces will definitely be filled?

Figure for Math Kangaroo 2016 Problem 16
Show answer
Answer: D — 5
Show hints
Hint 1 of 2
The 8 coins can start at space 1, 2, 3, or 4 (and run forward 8 in a row).
Still stuck? Show hint 2 →
Hint 2 of 2
Find the spaces that are covered no matter which of those starts is chosen.
Show solution
Approach: intersect every possible block of 8 consecutive spaces
  1. Eight in a row on an 11-space board can begin at space 1, 2, 3, or 4.
  2. Those blocks are 1–8, 2–9, 3–10, and 4–11; the spaces common to all of them are 4, 5, 6, 7, 8.
  3. So 5 spaces are always filled.
2018 · #7 Mike sets the table for 8 people: the fork has to lie to the left and the knife to the right of the plate. For how many people is the...

Mike sets the table for 8 people: the fork has to lie to the left and the knife to the right of the plate. For how many people is the cutlery set correctly?

Figure for Math Kangaroo 2018 Problem 7
Show answer
Answer: A — 5
Show hints
Hint 1 of 2
Sit in each chair in turn; the setting is right only if the fork is on your left hand and the knife on your right hand.
Still stuck? Show hint 2 →
Hint 2 of 2
Remember that someone sitting on the far side of the table faces you, so their left and right are flipped from yours.
Show solution
Approach: stand in each person's place and check fork-on-left, knife-on-right from their seat
  1. Picture yourself sitting in each of the 8 chairs, facing the plate.
  2. From that seat the fork must be by your left hand and the knife by your right hand — tick the places that match.
  3. Exactly five places pass the test, so the cutlery is set correctly for 5 people, answer A.
2022 · #3 One of the five coins A, B, C, D or E should be moved to an empty square so that each row and each column ends up with exactly two...

One of the five coins A, B, C, D or E should be moved to an empty square so that each row and each column ends up with exactly two coins. Which coin should be moved?

Figure for Math Kangaroo 2022 Problem 3
Show answer
Answer: C — C
Show hints
Hint 1 of 2
Count how many coins are in each row and in each column right now.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the one row that has too many and the one column that has too many; the coin sitting where they cross is the one to move.
Show solution
Approach: balance rows and columns
  1. Counting coins, one row has three coins (too many) and one row has only one (too few).
  2. Likewise one column has three coins and another has only one.
  3. The coin that sits in BOTH the overloaded row and the overloaded column is the one to move.
  4. That coin is C; moving it to the empty cell of the short row and short column fixes every count to two.
2018 · #10 Albert places these 5 figures on a 5×5 grid. Each figure is only allowed to appear once in every column and once in every row. Which...

Albert places these 5 figures on a 5×5 grid. Each figure is only allowed to appear once in every column and once in every row. Which figure does Albert have to place on the field with the question mark?

Figure for Math Kangaroo 2018 Problem 10
Show answer
Answer: A
Show hints
Hint 1 of 2
Each of the five figures shows up exactly once in every row and exactly once in every column.
Still stuck? Show hint 2 →
Hint 2 of 2
Look along the question mark's row and down its column, and cross out every figure you already see there.
Show solution
Approach: cross out every figure already in the marked cell's row and column; one figure is left
  1. Because no figure repeats in a row or a column, the missing one must be a figure not yet in that row or column.
  2. Read across the marked cell's row and down its column and cross off each figure that already appears.
  3. Exactly one figure is never crossed off, and that is the one that belongs in the question-mark cell, answer A.
2017 · #24 Leonie has hidden a Smiley behind some of the grey boxes. The numbers state how many Smileys there are in the neighbouring boxes. Two...

Leonie has hidden a Smiley behind some of the grey boxes. The numbers state how many Smileys there are in the neighbouring boxes. Two boxes are neighbouring if they have one side or one corner in common. How many Smileys has Leonie hidden?

Figure for Math Kangaroo 2017 Problem 24
Show answer
Answer: B — 5
Show hints
Hint 1 of 2
A number in a white box counts the Smileys in its grey neighbour boxes (boxes touching it by a side or a corner).
Still stuck? Show hint 2 →
Hint 2 of 2
Begin with a number whose grey neighbours are few: it may force every one of them to hold a Smiley.
Show solution
Approach: use each clue box to decide which grey boxes hold Smileys
  1. Each white number tells how many of its touching grey boxes hide a Smiley.
  2. A clue near an edge has only a few grey neighbours, so a large number there fills them all; a small number elsewhere then forces nearby boxes to stay empty.
  3. Working clue by clue, every grey box is decided as either holding a Smiley or empty.
  4. Counting the boxes forced to hold a Smiley gives 5 in total.
CHAPTER 6

If this, then that

THEORY

Some puzzles work like a chain. If one thing is true, then another thing must be true too.

Here is a friendly move for these. First pretend the easiest thing is true. Then count what is left over, and fix it up. Watch.

pretend all camels have 1 hump+each extra hump= one 2-hump camel

If every camel had just 1 hump, counting is easy. Then any hump left over belongs to a 2‑hump camel. The leftover tells you the answer.

🎯 Try it
There are 5 bikes. Some have 2 wheels and some have 3 wheels. There are 12 wheels in all. How many bikes have 3 wheels?
Here is how: Pretend all 5 bikes have 2 wheels: that is 10 wheels. You have 12, so 2 are left over. Each 3‑wheel bike adds 1 extra wheel. 2 left over means 2 bikes have a third wheel. Check: 2 bikes × 3 = 6, plus 3 bikes × 2 = 6, total 12.
THE TRICK

THE MOVE: pretend the easy case first. Then count the leftover and fix it up.

WATCH OUT

When you have an answer, check it by counting forward. Put the numbers back in and make sure they really add up.

WORKED EXAMPLE
PROBLEM · 2019 #7

There are two kinds of camels: bactrian camels that have 2 humps, and dromedaries that have 1 hump. Exactly 10 camels live in a certain zoo. Together they have 14 humps. How many bactrian camels are there in this zoo?

A) 1 B) 2 C) 3 D) 4 E) 5

Pretend every camel has just 1 hump. Then count the leftover humps to find the 2‑hump camels.

There are 10 camels. If each had 1 hump, that is 10 humps. But there are 14 humps. So there are 4 humps left over. Each 2‑hump camel gives exactly 1 extra hump. So there are 4 two‑hump (bactrian) camels.

Answer: D — 4
RULE OF THUMB

Start with the easy guess. The leftover tells you the answer.

MORE LIKE THIS
2024 · #14 Andrew throws arrows at a target. He starts with 10 arrows. Each time he hits the target, he gets 2 more arrows. In total Andrew throws...

Andrew throws arrows at a target. He starts with 10 arrows. Each time he hits the target, he gets 2 more arrows. In total Andrew throws 20 arrows, and then he has run out of arrows. How many times did Andrew hit the target?

Show answer
Answer: B — 5
Show hints
Hint 1 of 3
He only started with 10 arrows, but he threw 20, so where did the extra arrows come from?
Still stuck? Show hint 2 →
Hint 2 of 3
Every hit is like a little gift of 2 more arrows.
Still stuck? Show hint 3 →
Hint 3 of 3
Figure out how many extra arrows he earned, then see how many hits that took.
Show solution
Approach: count the extra arrows the hits gave him
  1. He started with 10 arrows but threw 20 in total, so 10 extra arrows must have come from hitting the target.
  2. Each hit gives 2 extra arrows, so we count by twos: 2, 4, 6, 8, 10 — that takes 5 hits to reach 10 extra arrows.
  3. So Andrew hit the target 5 times.
2009 · #11 A farmer has 30 cows, some chickens and no other animals. The total number of chicken legs is equal to the total number of cow legs. How...

A farmer has 30 cows, some chickens and no other animals. The total number of chicken legs is equal to the total number of cow legs. How many animals does the farmer have?

Show answer
Answer: B — 90
Show hints
Hint 1 of 2
Count the cow legs first.
Still stuck? Show hint 2 →
Hint 2 of 2
Chicken legs match cow legs, so work out how many chickens that is.
Show solution
Approach: match leg counts, then total the animals
  1. 30 cows have 30 × 4 = 120 legs.
  2. The chickens have the same number of legs: 120.
  3. Each chicken has 2 legs, so there are 120 ÷ 2 = 60 chickens.
  4. Altogether: 30 + 60 = 90 animals.
2018 · #18 To slay a dragon, Mathias has to cut off all of its heads. As soon as he has cut off 3 heads, one new head grows back right away. After...

To slay a dragon, Mathias has to cut off all of its heads. As soon as he has cut off 3 heads, one new head grows back right away. After Mathias has cut off 13 heads, the dragon is dead. How many heads did the dragon have at the start?

Show answer
Answer: B — 9
Show hints
Hint 1 of 2
A new head grows after every 3rd cut, so cuts at 3, 6, 9, 12 each add one back.
Still stuck? Show hint 2 →
Hint 2 of 2
Net heads removed = cuts minus regrowths.
Show solution
Approach: count regrowths during the 13 cuts and subtract
  1. Cutting 13 heads triggers a regrowth after cuts 3, 6, 9 and 12 — that is 4 new heads.
  2. For the dragon to end with no heads, start = 13 − 4.
  3. So it had 9 heads initially.
⬢ FINAL TEST

Think-it-through test

Six puzzles, easy to a little harder. Read each clue. Cross out. Think it through.

2019 · #3 Yesterday it was Sunday. Which day will it be tomorrow?

Yesterday it was Sunday. Which day will it be tomorrow?

Show answer
Answer: D — Tuesday
Show hints
Hint 1 of 2
If yesterday was Sunday, what is today?
Still stuck? Show hint 2 →
Hint 2 of 2
Step forward one day at a time: today, then tomorrow.
Show solution
Approach: step through the days
  1. Yesterday was Sunday, so today is Monday.
  2. Tomorrow is the day after Monday, which is Tuesday (D).
2009 · #15 Three squirrels Anni, Asia and Elli have collected 7 nuts. They have all collected a different amount of nuts, and everybody has...

Three squirrels Anni, Asia and Elli have collected 7 nuts. They have all collected a different amount of nuts, and everybody has collected at least one nut. Anni has collected the least and Asia the most. How many nuts has Elli collected?

Show answer
Answer: B — 2
Show hints
Hint 1 of 2
All three counts are different whole numbers, each at least 1, adding to 7.
Still stuck? Show hint 2 →
Hint 2 of 2
With Anni smallest and Asia largest, try the smallest possible values for Anni.
Show solution
Approach: find the only set of three distinct positive numbers summing to 7
  1. The three counts are different and each at least 1, with Anni least and Asia most.
  2. The smallest Anni can be is 1; the three must still differ and sum to 7.
  3. 1 + 2 + 4 = 7 is the only way, so Anni = 1, Elli = 2, Asia = 4.
  4. Elli collected 2 nuts.
2024 · #4 Firefighter Fred wants to put out the fire. In the picture on the right, what is the smallest number of ladders he has to climb to reach...

Firefighter Fred wants to put out the fire. In the picture on the right, what is the smallest number of ladders he has to climb to reach the fire without jumping?

Figure for Math Kangaroo 2024 Problem 4
Show answer
Answer: C — 6
Show hints
Hint 1 of 2
Start at the firefighter and find every platform you can reach by climbing whole ladders, never jumping.
Still stuck? Show hint 2 →
Hint 2 of 2
Trace the route to the fire that uses the fewest ladders — each ladder climbed counts as one.
Show solution
Approach: trace the shortest ladder path from the firefighter up to the fire
  1. Begin at the firefighter on the ground and only move along complete ladders between platforms.
  2. Follow the chain of ladders upward that reaches the fire's platform.
  3. Counting each ladder used, the shortest such route uses 6 ladders.
  4. Answer: 6 (C).
2021 · #2 How many fish will have their heads pointing towards the ring when we straighten the line?

How many fish will have their heads pointing towards the ring when we straighten the line?

Figure for Math Kangaroo 2021 Problem 2
Show answer
Answer: C — 6
Show hints
Hint 1 of 2
Pretend you grab the ring and pull the string straight; each fish stays facing the same way it was tied on.
Still stuck? Show hint 2 →
Hint 2 of 2
Start at the ring and look at each fish in turn, putting a tick on every fish whose head points back toward the ring.
Show solution
Approach: trace the line and count fish whose heads face the ring
  1. Follow the wiggly string starting from the ring and look at each fish one at a time.
  2. Tick the fish whose nose points back toward the ring, and skip the ones whose nose points away.
  3. Counting the ticked fish gives 6.
  4. So the answer is C.
2023 · #10 The six weights of a scale weigh 1 kg, 2 kg, 3 kg, 4 kg, 5 kg and 6 kg. Rosi places five weights on the two scale pans so that they are...

The six weights of a scale weigh 1 kg, 2 kg, 3 kg, 4 kg, 5 kg and 6 kg. Rosi places five weights on the two scale pans so that they are balanced. The sixth weight is left aside. Which weight is left aside?

Figure for Math Kangaroo 2023 Problem 10
Show answer
Answer: A — 1 kg
Show hints
Hint 1 of 2
Add up all six weights first, then notice the five used ones split into two equal piles.
Still stuck? Show hint 2 →
Hint 2 of 2
For the two piles to be equal, the weight left aside must make the rest share out evenly.
Show solution
Approach: the leftover weight must leave an amount you can split into two equal piles
  1. Count all six weights together: 1 + 2 + 3 + 4 + 5 + 6 = 21 kg.
  2. The five weights on the scale make two equal piles, so the leftover must leave an amount you can split in half evenly.
  3. Leaving the 1 kg aside leaves 20 kg, which shares out as 6 + 4 on one pan and 5 + 3 + 2 on the other — both 10 kg — so the leftover weight is 1 kg.
2014 · #19 Elisabeth sorts the cards shown above. With each move she is allowed to swap any two cards with each other. What is the smallest number...

Elisabeth sorts the cards shown above. With each move she is allowed to swap any two cards with each other. What is the smallest number of moves she needs in order to get the word KANGAROO?

Figure for Math Kangaroo 2014 Problem 19
Show answer
Answer: B — 3
Show hints
Hint 1 of 3
Write KANGAROO under the cards and mark every letter that is already in the right place.
Still stuck? Show hint 2 →
Hint 2 of 3
Only the wrong letters need to move, so look at just those.
Still stuck? Show hint 3 →
Hint 3 of 3
One swap trades two cards, so it can drop two wrong letters into the right spots at once.
Show solution
Approach: count the swaps that fix two misplaced letters at a time
  1. O A R G O N K A must become K A N G A R O O; the A and the G are already in place.
  2. The six remaining letters split into three pairs that each swap into place: (K↔O), (N↔R), (A↔O).
  3. Three swaps fix all six, and fewer is impossible.
  4. Answer: 3.
2019 · #23 One of the 5 children Alex, Bartek, Cora, Dani and Emil has eaten a cake. Alex says: “I did not eat a cake.” Bartek says: “I ate a...

One of the 5 children Alex, Bartek, Cora, Dani and Emil has eaten a cake. Alex says: “I did not eat a cake.” Bartek says: “I ate a cake.” Cora says: “Emil has not eaten a cake.” Dani says: “I did not eat a cake.” Emil says: “Alex has eaten a cake.” One of the children lies. Which child has eaten a cake?

Show answer
Answer: B — Bartek
Show hints
Hint 1 of 2
Exactly one statement is false; test who the eater could be and count the lies.
Still stuck? Show hint 2 →
Hint 2 of 2
Suppose Bartek is the eater and check whether only one child ends up lying.
Show solution
Approach: test the eater so that exactly one lies
  1. Assume Bartek ate the cake.
  2. Then Alex, Cora and Dani all speak truthfully, and Bartek's 'I ate' is true.
  3. Only Emil's 'Alex ate' is false — exactly one liar, as required.
  4. So the child who ate the cake is Bartek (B).
2013 · #19 Andi, Betti, Clara and Dani were born in the same year. Their birthdays are on 20 February, 12 April, 12 May and 25 May, but not...

Andi, Betti, Clara and Dani were born in the same year. Their birthdays are on 20 February, 12 April, 12 May and 25 May, but not necessarily in that order. Betti and Andi were born in the same month. Andi and Clara were born on the same day, in different months. Who is the oldest?

Show answer
Answer: D — Dani
Show hints
Hint 1 of 2
'Same month' must be the month that has two of the dates.
Still stuck? Show hint 2 →
Hint 2 of 2
'Same day, different months' must use the day number that appears twice.
Show solution
Approach: match the pairs of dates by the clues
  1. Betti and Andi share a month, and only May has two dates (12 May, 25 May).
  2. Andi and Clara share a day in different months, and only the 12th repeats (12 April, 12 May), so Andi = 12 May, Clara = 12 April; then Betti = 25 May.
  3. Dani gets the leftover 20 February, the earliest date, so Dani is the oldest.
2022 · #17 Wanda chooses some of the shapes shown. She says: “I have chosen exactly 2 grey, 2 big and 2 round shapes.” What is the smallest number...

Wanda chooses some of the shapes shown. She says: “I have chosen exactly 2 grey, 2 big and 2 round shapes.” What is the smallest number of shapes Wanda could have chosen?

Figure for Math Kangaroo 2022 Problem 17
Show answer
Answer: B — 3
Show hints
Hint 1 of 2
You want a shape to count toward more than one of the requirements at once.
Still stuck? Show hint 2 →
Hint 2 of 2
Pick shapes that are grey-and-big, big-and-round, or grey-and-round to overlap the three needs.
Show solution
Approach: make each shape cover two requirements
  1. She needs exactly 2 grey, 2 big and 2 round.
  2. Choose a big grey shape (grey+big), a big round shape (big+round) and a small grey round shape (grey+round).
  3. These three give exactly 2 grey, 2 big and 2 round.
  4. So the minimum is 3 shapes.
APPENDIX

Remember this

Memorize these

THINK-IT-THROUGH MOVES

  • Cross out what cannot be true. The last one left is the answer.
  • Line them up from smallest to biggest when clues say who is more.
  • Try a guess and check every sentence; keep the one that never breaks.
  • Find the rule first, then take one step at a time.
  • One thing at a time when sorting; a thing can be in two groups.
  • Pretend the easy case, then count the leftover to fix it.
→ Practice Logic & Word Problems problems
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