40 math brain teasers with answers

Math puzzles from easy to hard, grouped by topic with the difficulty marked on each. Every answer shows the working, so you can check the method as well as the result.

40 puzzles · Math brain teasers with answers · answers included

Numbers and arithmetic

1
Three whole numbers in a row add up to 72. What are they?

easy

Answer
23, 24 and 25.
The middle number is the average: 72 / 3 = 24. The other two are one less and one more. Check: 23 + 24 + 25 = 72.
2
Think of any number. Double it, add 10, halve the result, then take away the number you started with. What is left, whatever number you chose?

easy

Answer
5.
Call the number n. Doubling gives 2n, adding 10 gives 2n + 10, halving gives n + 5, and taking away n leaves 5. The starting number cancels out.
3
A brick weighs 1 kg plus half a brick. How much does the whole brick weigh?

easy

Answer
2 kg.
The brick is 1 kg plus half of itself, so the other half of the brick weighs 1 kg. Two halves make 2 kg. Check: 1 kg + half of 2 kg = 2 kg.
4
Place the numbers 1 to 9 in a 3 by 3 grid so that every row, column and diagonal has the same total. Which number must go in the center?

medium

Answer
5.
The numbers 1 to 9 add to 45, so each of the three rows totals 15. Four lines pass through the center: the middle row, the middle column and both diagonals. Together they total 4 x 15 = 60, and they cover every cell once plus the center three extra times. So 45 + 3 x center = 60, and the center is 5.
5
Write any three-digit number twice to make a six-digit number, such as 482482. Why can you always divide the result by 7 with nothing left over?

medium

Answer
Because the six-digit number is the three-digit number times 1001, and 1001 = 7 x 11 x 13.
482482 = 482 x 1000 + 482 = 482 x 1001. Since 1001 = 7 x 11 x 13, the result also divides exactly by 11 and by 13. For example, 482482 / 7 = 68926.
6
Multiply fifty 2s together: 2 x 2 x 2 and so on. What is the last digit of the answer?

hard

Answer
4.
The last digits of 2, 4, 8, 16, 32, 64 and so on repeat in a cycle of four: 2, 4, 8, 6. 50 = 12 x 4 + 2, so fifty 2s end the same way as two 2s: 2 x 2 = 4. The last digit is 4.
7
How many zeros are at the end of the product 1 x 2 x 3 x ... x 100?

hard

Answer
24.
Each final zero comes from a factor of 10, which needs a 2 and a 5. There are plenty of 2s, so count the 5s. Twenty of the numbers from 1 to 100 are multiples of 5, and four of them (25, 50, 75 and 100) hold a second 5. That makes 20 + 4 = 24 fives, so 24 zeros.
8
A post office has only 3-cent and 5-cent stamps, as many as you like. What is the largest postage in whole cents that you cannot make exactly?

hard

Answer
7 cents.
You cannot make 1, 2, 4 or 7 cents. You can make 8 (3 + 5), 9 (3 + 3 + 3) and 10 (5 + 5), and adding 3-cent stamps to those three covers every larger amount. So 7 is the largest amount you cannot make.
9
You want to weigh any whole number of kilograms from 1 to 40 on a balance scale, and weights may go on either pan. What is the fewest weights you need, and what should they weigh?

hard

Answer
Four weights: 1, 3, 9 and 27 kg.
Each weight can go on the item's pan, on the other pan or stay off, so three weights give at most 13 different amounts you can weigh, too few for 40. With 1, 3, 9 and 27 kg every amount works: for example 2 = 3 - 1, with the 1 kg weight beside the item, and 5 = 9 - 3 - 1. The four weights add up to exactly 40 kg.

Money and percentages

10
A shop has a deal on $12 T-shirts: buy 2, get 1 free. How much do 7 T-shirts cost?

easy

Answer
$60.
Every group of 3 shirts costs the price of 2. Seven shirts are two full groups of 3, where you pay for 4 shirts, plus one more shirt at full price. That is 5 shirts paid for: 5 x $12 = $60.
11
On a bus, a child's ticket costs half as much as an adult's. Two adults and three children pay $21 in total. How much is an adult ticket?

easy

Answer
$6.
Three child tickets cost the same as one and a half adult tickets, so the group pays for 3.5 adult tickets. $21 / 3.5 = $6, and a child ticket is $3. Check: 2 x $6 + 3 x $3 = $21.
12
A game's price rises by 50%, then the new price falls by 50%. Compared with the original price, where does it end up?

medium

Answer
25% lower than it started.
Try a price of $100. Up 50% gives $150, and half of $150 is $75. The second change is a percentage of a bigger number, so the two changes do not cancel out.
13
A farmer leaves 17 goats to three children: half to the eldest, a third to the middle child and a ninth to the youngest. Nobody wants to split a goat, so a neighbor lends them one. How many goats does each child get, and what happens to the borrowed goat?

medium

Answer
9, 6 and 2 goats. The borrowed goat goes back to the neighbor.
The loan changes what the fractions are taken of: with 18 goats, half is 9, a third is 6 and a ninth is 2. That is 17 goats, so one is left over and returned. Taken of 17, the shares would be 8 1/2, 5 2/3 and 1 8/9 goats, so each child ends up with a little more than the will says. It works because 1/2 + 1/3 + 1/9 = 17/18: the three shares never added up to the whole herd.
14
You are paid 1 cent on day 1, 2 cents on day 2 and 4 cents on day 3, and your pay keeps doubling every day. How much are you paid on day 30 alone?

medium

Answer
$5,368,709.12.
Day 30 comes after 29 doublings, so the pay is twenty-nine 2s multiplied together: 536,870,912 cents, which is $5,368,709.12. The daily pay first passes $1 on day 8 (128 cents) and $1,000 on day 18 (131,072 cents).
15
A crate holds 100 kg of cucumbers that are 99% water. After some days in the sun, they are 98% water. What do they weigh now?

hard

Answer
50 kg.
Only the water changes. At first 1% of 100 kg, which is 1 kg, is not water. Afterward that same 1 kg is 2% of the total, so the total is 1 / 0.02 = 50 kg.

Ages and unknowns

16
A cat and a dog weigh 10 kg together. The dog and a rabbit weigh 13 kg together. The cat and the rabbit weigh 5 kg together. How much does each animal weigh?

easy

Answer
Cat 1 kg, dog 9 kg, rabbit 4 kg.
Add the three weighings: 10 + 13 + 5 = 28 kg. That counts every animal twice, so the three together weigh 14 kg. Take away each pair: the rabbit is 14 - 10 = 4 kg, the cat is 14 - 13 = 1 kg and the dog is 14 - 5 = 9 kg.
17
In 2 years, Noor will be twice as old as she was 5 years ago. How old is Noor now?

medium

Answer
12.
Call her age n. Then n + 2 = 2 x (n - 5) = 2n - 10, so n = 12. Check: in 2 years she will be 14, and 5 years ago she was 7.
18
A farmer counts 20 heads and 56 legs among her chickens and cows. How many chickens and how many cows are there?

medium

Answer
12 chickens and 8 cows.
If all 20 animals were chickens there would be 40 legs. There are 16 more, and each cow adds 2 extra legs, so there are 8 cows and 12 chickens. Check: 12 x 2 + 8 x 4 = 24 + 32 = 56.
19
A fish's head is 9 cm long. Its tail is as long as its head plus half its body. Its body is as long as its head and tail together. How long is the whole fish?

hard

Answer
72 cm.
Call the body B. The tail is 9 + B/2, and the body is 9 + tail, so B = 9 + 9 + B/2. Then B/2 = 18 and B = 36 cm. The tail is 9 + 18 = 27 cm. The fish is 9 + 36 + 27 = 72 cm.

Time, speed and rates

20
A boat travels at 12 km/h going downstream and 8 km/h going upstream. How fast is the current?

medium

Answer
2 km/h.
Going downstream the current adds to the boat's own speed, and going upstream it takes away the same amount. So the boat's own speed is halfway between: (12 + 8) / 2 = 10 km/h. The current makes up the difference: 12 - 10 = 2 km/h.
21
A clock chimes 6 times at 6 o'clock, and it takes 5 seconds from the first chime to the last. With the chimes spaced the same way, how long does it take from the first to the last of the 12 chimes at 12 o'clock?

medium

Answer
11 seconds.
Six chimes have 5 gaps between them, so each gap is 1 second. Twelve chimes have 11 gaps, which take 11 seconds, not 10.
22
One tap fills a tub in 3 hours and another fills it in 6 hours. How long does it take with both taps running?

medium

Answer
2 hours.
In one hour the first tap fills 1/3 of the tub and the second fills 1/6. Together that is 1/3 + 1/6 = 1/2 of the tub per hour, so the whole tub takes 2 hours.
23
Jo drives 120 km to her aunt's house at 40 km/h and drives the same 120 km home at 60 km/h. What is her average speed for the whole trip?

medium

Answer
48 km/h.
The trip there takes 120 / 40 = 3 hours and the trip home takes 120 / 60 = 2 hours. That is 240 km in 5 hours, or 48 km/h. It is not 50 km/h, because she spends more of the time at the slower speed.
24
Mia runs one lap of a track in 4 minutes and Leo runs one lap in 6 minutes. They start together from the same spot and run the same way. After how many minutes is Mia a full lap ahead of Leo for the first time?

medium

Answer
12 minutes.
Each minute Mia runs 1/4 of a lap and Leo 1/6, so Mia gains 1/4 - 1/6 = 1/12 of a lap per minute. She is a full lap ahead after 12 minutes, when she has run 3 laps and Leo 2.
25
What is the smaller angle between the hour hand and the minute hand of a clock at 3:15?

medium

Answer
7.5 degrees.
The minute hand points at the 3. The hour hand moves 30 degrees per hour, which is half a degree per minute, so at 3:15 it has moved 15 x 0.5 = 7.5 degrees past the 3. The angle between the hands is 7.5 degrees, not 0.
26
A car's odometer shows 15951 km, a number that reads the same backward. Exactly two hours later, it shows the next number that reads the same backward. What was the car's average speed?

medium

Answer
55 km/h.
The next number after 15951 that reads the same both ways is 16061. The car covered 16061 - 15951 = 110 km in 2 hours, which is 55 km/h.
27
Two trains start 100 km apart on the same track and head toward each other at 50 km/h each. A bird starts at the front of one train and flies at 75 km/h to the other, turns at once, and keeps flying back and forth until the trains meet. How far does the bird fly?

medium

Answer
75 km.
Skip the zigzags. The trains close the 100 km gap at 50 + 50 = 100 km/h, so they meet after 1 hour. The bird flies for that whole hour at 75 km/h, so it covers 75 km.
28
Two candles are the same height. One burns down completely in 4 hours and the other in 3 hours, each at a steady rate. You light them at the same time. When is the first candle exactly twice as tall as the second?

hard

Answer
After 2 hours 24 minutes.
After t hours the first candle has 1 - t/4 of its height left and the second has 1 - t/3. Twice the second equals the first when 2 - 2t/3 = 1 - t/4, which gives 5t/12 = 1, so t = 12/5 hours, or 2 hours 24 minutes. At that moment the first candle has 2/5 of its height and the second 1/5.
29
In a 100 m race, Ada beats Ben by 10 m. In another 100 m race, Ben beats Cy by 10 m. If they all run at the same speeds, by how many meters would Ada beat Cy in a 100 m race?

hard

Answer
19 m.
Cy covers 90 m while Ben covers 100 m, so Cy runs 90% of Ben's distance. When Ada finishes, Ben has run 90 m, and Cy has run 90% of 90 m, which is 81 m. Ada wins by 100 - 81 = 19 m, not 20 m.

Counting and chance

30
How many different ways can you arrange the four letters of MATH?

easy

Answer
24.
There are 4 choices for the first letter, then 3 for the second, 2 for the third and 1 for the last. 4 x 3 x 2 x 1 = 24.
31
You roll two ordinary dice. What is the chance that they add up to 7?

easy

Answer
1/6.
Two dice give 6 x 6 = 36 equally likely results. Six of them add to 7: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. 6 out of 36 is 1/6. No other total is as likely.
32
A prize is behind one of three closed doors. You pick door 1. The host, who knows where the prize is, always opens one of the other two doors with nothing behind it, then lets you switch to the door that is still closed. Should you switch?

hard

Answer
Yes. Switching wins 2 times out of 3.
Your first pick is right 1 time in 3, and staying wins only then. The other 2 times in 3, the prize is behind one of the doors you did not pick, and the host must open the empty one, so the door he leaves closed has the prize. Switching wins in all of those cases.
33
One card is red on both sides, one is white on both sides, and one is red on one side and white on the other. You draw a card at random and lay it on the table. The side facing up is red. What is the chance that the other side is red too?

hard

Answer
2/3.
Count sides, not cards. Three sides are red: two on the red-red card and one on the red-white card, and each is equally likely to be the one facing up. In 2 of those 3 cases the card is red-red, so the chance is 2/3, not 1/2.
34
How many people must be in a room before there is a better than even chance that two of them share a birthday? Ignore February 29 and assume every birthday is equally likely.

hard

Answer
23 people.
It is easier to work out the chance that all the birthdays are different: 365/365 x 364/365 x 363/365 and so on, one fraction per person. For 22 people that chance is about 0.524. For 23 it falls to about 0.493, so the chance of a shared birthday is about 0.507, just over half.
35
Ten bags hold coins. In nine bags every coin weighs 10 g, and in one bag every coin weighs 11 g. You may use a scale that shows exact grams, but only once. How can you find the heavy bag?

hard

Answer
Take 1 coin from bag 1, 2 coins from bag 2 and so on, up to 10 coins from bag 10, and weigh them together. The number of grams over 550 is the number of the heavy bag.
That is 1 + 2 + ... + 10 = 55 coins, which would weigh 550 g if every coin weighed 10 g. Each heavy coin adds 1 g, and bag number k gave k coins, so a reading of 553 g means bag 3 is the heavy bag.

Shapes and space

36
A 13 m ladder leans against a wall with its top 12 m above the ground. How far is the foot of the ladder from the wall?

medium

Answer
5 m.
The ladder, the wall and the ground make a right triangle, with the ladder as the longest side. 13 x 13 = 169 and 12 x 12 = 144. 169 - 144 = 25, which is 5 x 5, so the foot is 5 m from the wall.
37
Each side of a square grows by 10%. By what percentage does its area grow?

medium

Answer
21%.
A square with 10 cm sides has an area of 100 square cm. With 11 cm sides the area is 121 square cm, which is 21% more. In general the area grows 1.1 x 1.1 = 1.21 times.
38
How can you cut a round cake into 8 equal pieces with only 3 straight cuts, without moving any pieces between cuts?

medium

Answer
Make two cuts straight down through the center at right angles to each other, then one flat cut through the cake halfway up.
The two downward cuts make 4 equal wedges. The flat cut splits every wedge into a top half and a bottom half, giving 4 x 2 = 8 equal pieces.
39
What is the largest number of pieces you can cut a round pizza into with 4 straight cuts? The pieces do not need to be the same size, and you may not stack or move them.

hard

Answer
11 pieces.
Each new cut adds one piece for every earlier cut it crosses, plus one. The first cut makes 2 pieces, the second adds 2, the third adds 3 and the fourth adds 4: 2 + 2 + 3 + 4 = 11. For this, every pair of cuts must cross inside the pizza, with no three cuts meeting at one point.
40
A rope is tied tight around the equator of a perfectly round Earth. It is then made 1 m longer and lifted evenly all the way around. Roughly how high above the ground is it now?

hard

Answer
About 16 cm.
The distance around a circle is 2 x pi x radius. Adding 1 m to that distance adds 1 / (2 x pi) m to the radius, which is about 0.16 m. The size of the Earth does not matter: a rope around a ball, made 1 m longer, rises by the same 16 cm.

Timer and team scores

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Read a prompt, allow thinking time, then invite an answer or a pass. For a timed round, choose a turn length before starting. A host can keep the printed answers out of view until the round ends.

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