151 Math Riddles With Answers
Maths riddles are the ones where the trick is arithmetic rather than vocabulary. Some are genuine number puzzles that need working out; others look like arithmetic and turn on a piece of wordplay hidden in the question.
Every riddle here has its answer and, where the answer needs it, the working. That matters more in this category than any other - "the answer is 5" is useless if you cannot see why.
These are a good fit for maths lessons: they get pupils reasoning about number relationships without it looking like a worksheet.
There are 500 coffins and 500 men who need them. The undertaker asks the first man to go to every coffin and open them. Then he tells the second man to go to every one and close it. The third one goes to every third coffin and so on. How many are open?
Only coffins with an odd number of divisors stay open in the end - the perfect squares - and there are exactly 22 perfect squares (1 squared through 22 squared) at or below 500.
How do you go from 98 to 720 using just one letter?
Spelling out "98" as "ninety eight" and inserting the letter x between the words creates "ninety x eight," which multiplies to 720.
Four years ago, Alex was twice as old as Jake. Four years from now, Jake will be 3/4 of Alex's age. How old is Alex?
Solving the two age equations together shows Jake is currently 8 and Alex is currently 12.
If it is two hours later, then it will take half as much time till it's midnight as it would be if it were an hour later. What time is it?
Working the algebra backward from midnight shows the current time must be 9 o'clock for the stated ratio to hold true.
Sally is 54 years old and her mother is 80, how many years ago was Sally's mother three times her age?
Solving 80-x = 3(54-x) gives x = 41, so 41 years ago Sally's mother was exactly three times her age.
If two hours ago, it was as long after one o'clock in the afternoon as it was before one o'clock in the morning, what time would it be now?
Working backward from "two hours ago," the elapsed time since 1 p.m. must equal the time remaining until 1 a.m., which only balances when the current time is 9 p.m.
In a bicycle race, the man who came two places in front of the last man finished one ahead of the man who came fifth. How many contestants were there?
If he's two places ahead of last (6th) he's in 4th, and finishing one ahead of 5th place also places him in 4th -- both conditions align only when there are 6 total contestants.
My daughter has many sisters. She has as many sisters as she has brothers. Each of her brothers has twice as many sisters as brothers. How many sons and daughters do I have?
With 4 daughters and 3 sons, each daughter has 3 sisters and 3 brothers (equal), and each son has 2 brothers and 4 sisters (twice as many sisters as brothers) -- satisfying both conditions.
One brick is one kilogram and half a brick heavy. What is the weight of one brick?
If one brick equals 1 kg plus half a brick, then half a brick must weigh 1 kg, making a whole brick weigh 2 kg.
Place three matches on a table. Tell a friend to add two more matches to make eight.
Three matches placed together can form the Roman numeral III; adding two more matches in a 'V' shape in front makes VIII, the Roman numeral for eight.
There are 25 red balls, 47 green balls and 3 blue balls in a basket. There is a blind man. What is the minimum number of balls that the blind man has to pick to make sure that there are at least 2 balls of different colors?
In the worst possible case, the blind man could draw all 47 green balls in a row before finally getting a red or blue one, so 48 draws guarantee at least two different colors.
Use the numbers 2, 3, 4 and 5 and the symbols + and = to make a true equation.
Both sides of the equation 2 + 5 = 3 + 4 equal 7, satisfying the challenge using each of the four given numbers exactly once.
Mr. Smith has 4 daughters. Each of his daughters has a brother. How many children does Mr. Smith have?
Since "each of his daughters has a brother" refers to the same single brother shared by all four daughters, Mr. Smith has 4 daughters plus 1 son, for 5 children total.
How can you add eight 8's to get the number 1000?
Try combining the 8's into different-length numbers (888, 88, 8) rather than adding eight separate 8's.
Three times what number is no larger than two times that same number?
Only when the number is zero does tripling it (still zero) stay no larger than doubling it (also zero); for any positive number, tripling it would exceed doubling it.
The combined age of Jack and John is 49. Jack is twice as old, as John was when Jack was as old as John is now. How old are the brothers?
Jack's age (28) is twice what John's age was seven years ago (14), which is exactly the gap between their ages now (28 - 21 = 7), and 28 + 21 = 49 as required.
How many days are there in 4 years?
Four years of 365 days would total 1,460 days, but one of the four is a leap year with an extra day, bringing the true total to 1,461.
What is the value of 1/2 of 2/3 of 3/4 of 4/5 of 5/6 of 6/7 of 7/8 of 8/9 of 9/10 of 1000?
Multiplying the fraction chain 1/2 x 2/3 x 3/4 ... x 9/10 causes every numerator and denominator to cancel except the very first and last, leaving 1/10 of 1000, which is 100.
If you toss a coin 10 times and it lands heads up every time, what are the chances it will land heads up if you toss it again?
A fair coin has no memory -- each flip is an independent event with exactly a 50% chance of heads, regardless of past results.
In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half the lake?
Since the patch's size doubles every single day, it must have covered exactly half the lake just one day before it covered the whole lake - day 47.
Dean Sam and Castiel are three brothers. Interestingly their current age is prime. What's more interesting that difference between their ages is also prime. How old are they?
Ages 2, 5, and 7 are all prime numbers, and the differences between them - 3, 2, and 5 - are also all prime, satisfying every condition in the riddle.
There are two fathers and two sons. They walk into a candy store and each buys a candy bar for 50 cents. The total for all of the candy bars was $1.50. How is that possible?
With a grandfather, father, and son, the middle person counts as both a father and a son, giving 'two fathers and two sons' among only three people who buy just three candy bars.
What number do you get when you multiply all of the numbers on a telephone's number pad?
Every phone keypad includes the digit 0, and multiplying any set of numbers that includes zero always results in zero.
How can you give someone $83 using exactly 7 bills, without using any one dollar bills?
A $50 bill, a $20 bill, a $5 bill, and four $2 bills together total $83 using exactly seven bills and no $1 bills.
If Mr. Flynn is 90 centimeters plus half his height, how tall is he?
If 90 cm equals half his height, his full height must be double that, or 180 centimeters.
Divide 20 by half and add 30, what do you get?
Dividing 20 by one-half means multiplying by 2, giving 40; adding 30 brings the total to 70.
There are 12 kids in a classroom. 6 kids are wearing socks and 4 are wearing shoes. 3 kids are wearing both. How many are bare feet?
Adding the sock-wearers and shoe-wearers together and subtracting those double-counted for wearing both (6 + 4 - 3 = 7) leaves 12 - 7 = 5 kids with bare feet.
When I take five and add six, I get eleven, but when I take six and add seven, I get one. What am I?
This is a clock-face riddle: five o'clock plus six hours lands on eleven, but six o'clock plus seven hours wraps back around to one on a 12-hour clock.
A bat and a ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?
If the ball costs 5 cents, the bat costs $1.05 (one dollar more), and together they total $1.10 -- the common wrong intuitive answer of 10 cents doesn't satisfy the 'one dollar more' condition.
What 3 positive numbers give the same result when multiplied and added together?
1 + 2 + 3 = 6 and 1 x 2 x 3 = 6, making them the only three positive numbers whose sum and product are equal.
When is 1500 plus 20 and 1600 minus 40 the same thing?
In 24-hour (military) time, 1500 plus 20 minutes is 1520, and 1600 minus 40 minutes is also 1520 -- both calculations land on the same time, 3:20 PM.
Payoff 2n (St. Petersburg). Charge to play?
The mathematical expectation is infinite, but most people wouldn't pay more than �20. This illustrates that humans value money based on 'utility' rather than just the raw expected amount.
Rule: Circle=Yellow. Cards: C, S, Y, R. Check?
You check the Circle (to see if back is Yellow) and the Red card (to see if front is a Circle, which would break the rule). Yellow and Square can't invalidate the rule.
Probability to be the last to try pitcher?
This is a property of a 'random walk' on a circle. Every person is equally likely to be the last one reached before the circle is completed.
Surprise test impossible logic?
If the test hasn't happened by Thursday, the students know it must be Friday (so it's not a surprise). This logic continues backward to Monday, 'proving' it can never be a surprise.
Sphere or Torus using rocks?
On a sphere, any two 'great circles' must intersect. On a torus, you can have a loop that goes around the center and another through the hole that never meet.
9 correct predictions in a row. Psychic?
The company sends 5,000 emails: 2,500 say Team A wins, 2,500 say Team B. They keep splitting the winners until one small group (like Kelly) has seen a perfect streak.
Describe a chicken in computer programming.
In coding, a function is something that takes an input and provides an output. Grain is the parameter, and the Egg is the return value.
Hourglass weigh more falling or at rest?
When sand is in the air, its weight isn't recorded by the scale. However, the force of the sand hitting the bottom is greater than its static weight, resulting in a net increase during the flow.
Pays payout in Yen. Charge to play?
The expected value is the average of all outcomes: (1+2+3+4+5+6)/6 = 21/6 = 3.5. To make a profit, the 'house' must charge more than the expected payout.
5 black spins. Is red "due"?
The roulette wheel has no memory. Each spin is a fresh event with the same probability (ignoring the green zero), regardless of what happened before.
10 pennies in 3 cups, each has odd number.
Put 1 in the first, 1 in the second, and 8 in the third... but wait, the trick is nesting. Put 1 in cup A, 2 in cup B, and nest B inside cup C (which has 7). Now C has 9 (odd).
Spoon tea into coffee, mix, spoon back. More what?
Since you started and ended with the same volume in each cup, any 'tea' that stayed in the coffee must have been replaced by an exactly equal volume of 'coffee' in the tea.
Max slices from 4 straight cuts?
The formula is (n^2 + n + 2) / 2. For n=4, that is (16 + 4 + 2) / 2 = 11. This assumes cuts can overlap and pass through previous cuts.
Ways to arrange 8 books on a shelf?
The number of permutations is n factorial (n!). For 8 books, it is 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320.
1 poisoned bottle in 1000. Prisoners needed?
Give each prisoner a unique combination of bottles to sip from based on binary (0 and 1). The specific combination of prisoners who get sick creates a binary code for the poisoned bottle number.
P1 (4 of kind), P2 (3 of kind). Worst possible hand?
If Player 1 has four of a kind and Player 2 has three of a kind, there must be at least three cards of the same rank on the shared board. Your hand will use those same three cards.
Break stick at 2 random points. Triangle prob?
Using the 'triangle inequality' theorem, no piece can be longer than 0.5m. Visualizing this on a 2D plot of possible break points shows the 'success' area is 1/4 of the total area.
Monkey climbs rope with matching weight. Weight?
The force the monkey exerts to climb increases the tension in the rope equally on both sides. Thus, both the monkey and the counterweight accelerate upward together.
Reverses left/right but not up/down?
If you point your finger to the right, the mirror image points to the right. The reversal is actually in the 'depth' (front to back) dimension relative to the mirror's surface.
Camel 3000 bananas, 1k km. Max reach?
The camel must leave 'caches' of bananas. To minimize the 'cost' of travel, he moves the bananas in stages until the total supply matches his max capacity for the remaining distance.
Get fair toss from biased coin.
If you toss twice, the probabilities of HT and TH are both (p * (1-p)). Since these are equal, assigning one to 'Heads' and the other to 'Tails' creates a fair 50/50 result.
Spheres with 6cm holes. Which has more volume?
The volume of a spherical ring depends only on the height of the hole (h). The formula V = (pi * h^3) / 6 shows the radius (R) of the original sphere cancels out.
200 fish, 99% Gold. Remove how many for 98%?
In 200 fish, 198 are gold (99%). To make them 98%, the 2 non-goldfish must represent 2% of the total. 2 is 2% of 100. So you must have 100 fish total (remove 100 gold).
Brothers Chris (FL) and Connor (OR) same time. How?
Florida has a tiny portion in Central Time, and Oregon has a portion in Mountain Time. During the DST 'fall back' hour, the times can align perfectly across the country.
Irrational number ^ irrational power = rational?
Consider x = sqrt(2)^sqrt(2). If x is rational, we are done. If x is irrational, then x^sqrt(2) = (sqrt(2)^sqrt(2))^sqrt(2) = sqrt(2)^2 = 2, which is rational.
1st sits random. Last gets assigned seat prob?
This is a classic probability puzzle. Either the first person takes your seat, or they take their own, or they take someone else's which triggers a chain that eventually hits yours or theirs.
Does washer hole expand or shrink when heated?
Thermal expansion is like a photographic enlargement. Every part of the object, including the 'empty' hole, expands proportionally.
How many barbershops are in the USA?
Estimate based on one barber per ~2,500 people. With a US population of 330 million, 130,000 is a standard Fermi estimation result.
100 cheating husbands. Execution day?
If there was 1 cheater, his wife would know immediately. If 2, they wait to see if the other wife acts. By the 100th day, the lack of action by others proves your own husband is one.
Truth 3/4 times. Says 6 on a die. Prob it's a 6?
The probability he says 6 AND it is 6 is (1/6 * 3/4). The probability he says 6 but it isn't 6 is (5/6 * 1/4). Use Bayes to find the ratio.
5 bandits split 100 coins. Senior proposal?
The senior bandit knows he only needs two other people to agree. He offers 1 coin to those who would get nothing in the next-best scenario to secure their vote.
Families stop childbearing after 1st boy. Sex ratio?
Every individual birth has a 50/50 chance. No matter the 'stopping rule,' the total expected number of boys and girls in the population will remain equal.
How many squares on an 8x8 chessboard?
Don't just count the 1x1 squares. You must sum all possible sizes: 1x1 (64), 2x2 (49), 3x3 (36), 4x4 (25), 5x5 (16), 6x6 (9), 7x7 (4), and 8x8 (1).
How many ping pong balls fill a 747?
Estimate the volume of a 747 (approx 1000 m^3) and the volume of a ping pong ball, then account for the 'packing fraction' of empty air space between balls.
C beats T by 10m in 100m. C starts 10m back. Winner?
Since Chandler runs 100m in the same time Toni runs 90m, when Chandler is 10m back, they will meet at the 90m mark. Chandler is faster and will pull ahead in the last 10m.
Does level change when ice in water melts?
According to Archimedes, the floating ice displaces a volume of water equal to its weight. When it melts, it becomes water of that exact weight and volume.
4 people: 1, 2, 5, 10 mins. Bridge (max 2). Min time?
Fastest pair (1&2) cross (2m), 1 returns (1m), slowest pair (5&10) cross (10m), 2 returns (2m), fastest pair (1&2) cross again (2m). 2+1+10+2+2 = 17m? Wait, logic says 15m is possible.
31 games in elimination tournament. Players?
In any knockout tournament, every player except the winner must lose exactly one game to be eliminated. If there were 31 games, 31 people lost.
3 doors: Heaven, 1-day Hell, 2-day Hell. Avg days?
E = 1/3(0) + 1/3(1+E) + 1/3(2+E). Solving for E: E = 1 + 2/3E -> 1/3E = 1 -> E = 3.
110m rope. 1ft slack added/pulled up. Truck fits?
Adding 1ft of slack to 110m (360ft) creates a triangle with base 180 and hypotenuse 180.5. sqrt(180.5^2 - 180^2) = 13.4 ft, plenty for a truck.
Which is greater: e? or ?e?
Take the e-th root of both. e^(1/e) is the maximum value for the function x^(1/x). Since e is the base of the maximum, e^pi > pi^e.
Surgeon with 2 gloves must operate on 3 patients. How?
Patient 1: Gloves A and B. Patient 2: Only Glove B (top one removed). Patient 3: Turn Glove A inside out and put it over B. A's clean side is now outside.
Duck in pond. Cat runs 4x faster on edge. Escape?
The duck swims in a circle of radius R/4. The cat has to run halfway around the pond (pi * R). The duck's distance is shorter than the cat's travel time allows for.
1/2 of 2/3 of 3/4... of 9/10 of 1,000?
The series is (1/2)*(2/3)*(3/4)*...*(9/10). The numerators and denominators cancel out diagonally, leaving 1/10. 1/10 * 1,000 = 100.
Pick 1 of 3. Empty removed. Do you switch?
Your initial pick had a 1/3 chance. The remaining 2/3 chance is now concentrated in the one unopened envelope the interviewer didn't discard.
Explain a database to an 8-year-old.
Use a metaphor of an organized filing system or library where everything has a specific label and 'shelf' to make it easy to find later.
Standard deviation of {1, 2, 3, 4, 5}?
The mean is 3. Squares of deviations: (1-3)^2=4, (2-3)^2=1, (3-3)^2=0, (4-3)^2=1, (5-3)^2=4. Average is 10/5 = 2. SD is sqrt(2) ? 1.414.
1/4 into tunnel. Meet train at start or end. Speed ratio?
If she meets the train at the end after running 3/4 of the way, and she would have met it at the start after running 1/4 back... set up the time/distance equations.
Sum of digits from 1 to 1,000,000?
The sum for 0-999,999 is (sum of digits 0-9) * (number of positions) * (frequency). 45 * 6 * 100,000 = 27,000,000. Add 1 for the '1' in 1,000,000.
Helium balloon in car accelerating forward. Direction?
In an accelerating car, heavier air is pushed to the back. This creates a pressure gradient that 'buoys' the lighter helium balloon toward the front.
HH = Jamie takes 2. Any other = Ashley takes 1. Winner?
There are 4 outcomes: HH, HT, TH, TT. HH (Jamie) happens 1/4. HT, TH, TT (Ashley) happen 3/4. Ashley wins 3 times for every 1 time Jamie wins (actually 3 coins to Jamie's 2).
100 ants on 1m stick at 1cm/s. Max time to clear?
Because all ants are identical, a 'bounce' is mathematically the same as two ants passing through each other. The max time is just the time for the furthest ant to walk to the end.
Shortest distance from room corner to opposite corner?
Imagine the room as a flat 2D net. The shortest path is a straight line across two adjacent 'unfolded' faces of the room.
Measure 4L using 3L and 5L jugs.
Fill 3, pour into 5. Fill 3 again, pour into 5 until full. 1L remains in the 3L jug. Empty 5, put that 1L in. Fill 3 and add to the 5L jug to get 4L.
Angle between clock hands at 3:15?
At 3:15, the minute hand is at 15m (90�). The hour hand has moved 1/4 of the way between 3 and 4. 30� per hour * 0.25 = 7.5�.
3-way. Accuracy 33/50/100. You (33) go first. Action?
If you miss on purpose, the 50% shooter will try to kill the 100% shooter (his biggest threat). This leaves you a 1-on-1 duel against a surviving, possibly weaker, opponent.
20% men/40% women attend. 35% employees men. Total %?
Assume 100 people: 35 men, 65 women. Attending: (20% of 35 = 7) + (40% of 65 = 26). 7 + 26 = 33 people total out of 100.
How is world population measured?
It is a combination of national censuses (counting) and statistical sampling/modeling for areas with less reliable data.
Alex (RR/BB wins). Sam (BR/RB wins). Better position?
Variance measures risk. Sam's payouts are always �2 for mixed colors, while Alex fluctuates between �3 and �1. Lower variance is 'safer' for the same return.
7-part bar. Pay 1 part/day for 7 days. Snap twice?
Binary counting: Pieces of 1, 2, and 4 units can be combined to form any total from 1 to 7. You swap pieces with the gardener daily like giving change.
Weigh an elephant without a massive scale?
Archimedes' Principle: Put the elephant on a boat, mark the water line. Remove elephant and add stones until the boat hits the same line. Weigh the stones.
3R, 2B hats. 1st/2nd can't guess. What is 3rd's color?
If the 3rd professor saw two blue hats, he would know his is red (only 2 blue exist). Since he doesn't know, the others must have at least one red hat... follow the deduction chain.
How do you move Mount Fuji?
This tests lateral thinking. You can 'move' it by moving yourself relative to it, or by waiting for continental drift over millions of years.
Double every 5m. 1 fills in 12h. Time for 2 to fill?
Starting with 2 is exactly where the 1-bacterium pond would be after 5 minutes (first doubling). So it saves 5 minutes off the 12-hour total.
Guess 1-1000 with higher/lower. Max guesses?
Binary search halves the search space each time. 2^10 = 1024, which covers 1-1000 in exactly 10 steps.
Can the mean of two consecutive primes be a prime?
The average of two numbers is the number exactly between them. All numbers between consecutive primes are composite by definition.
Can you swim faster in syrup or water?
The increased friction in syrup is balanced by the increased resistance the water provides for your stroke to push against. Studies (and Mythbusters) confirm this.
6 chambers, 2 consecutive bullets. Fired empty. Spin or stay?
Two bullets in 6 chambers leaves 4 empty slots. Since you just fired an empty one, there are 5 slots left; only one of the empty slots is followed by a bullet. 1/4 chance vs 1/3 if you spin.
Times digit "3" is printed between 3,000 and 4,000?
The digit 3 appears 1,000 times in the thousands place (3000-3999) and 100 times in each of the other three places (hundreds, tens, units) within that range.
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