Alex, Bill, and Craig plan to start a new business.
They decided to try 4 different business models one by one.
An expert, Bill's dad, esteems that the probability of success is 1/4 for each of the models.
Estimate the probability that they succeed with at least one of them.
My local dice club is having having another open-evening to increase membership. Tonight's special is "squared dice".
You roll a single fair standard 6-sided die and receive the square of the number on the top face, expressed in £ (pounds, GBP).
What is the most you would be willing to pay to play this bizarre game?
Numbers from 1 to 9 are arranged in the cells of the 3x3 square.
What is the smallest number of different values that sums in 2x1 rectangles can take?
If you toss a fair coin there is a probability of one half (0.50) of getting a head.
If you toss three fair coins, all at the same time, what is the probability of tossing at least one head?
How many different ways are there to color the faces of a cube with six colors?
Two coloring ways are called different if the two cubes cannot be rotated so that the colors of their faces match.
The question of Moscow Mathematical Olympiad, 1935.
In a Gentleman’s club in London (England) all members must wear hats to enter, but must check them in at the entrance. Each member then receives a numbered ticket so they can retrieve their hats when they leave.
The hat-check clerk phones in sick before the start of his shift, so the manager, in a state of panic, grabs somebody off the street, dresses him in a suitable uniform, explains the job, hands him a bunch of cash, then retires to the sanctuary of his office.
It turns out that the random guy pulled in off the street, whilst able to smile pleasantly and nod his head intelligently, has no clue what he was supposed to do. He therefore stores and retrieves hats at random. 20 gentlemen stored and retrieved hats that evening.
What is the probability that exactly one gentleman received the wrong hat?
In a Gentleman’s club in London (England) all members must wear hats to enter, but must check them in at the entrance. Each member then receives a numbered ticket so they can retrieve their hats when they leave.
After a busy evening there are only 3 gentlemen remaining, so the hat-check clerk decides to leave early, entrusting the final 3 hats to the new guy.
It turns out that the new guy is a total loser (a waste of space) and when given a ticket just returns a hat picked at random! Up until that point, all hats had been returned to their correct owners.
What is the probability that at least one of the gentlemen receives their own hat?
What is the maximum number of wholly enclosed (bounded) regions created by drawing N congruent circles?
On Green island, 15 villages (green dots) are connected by straight roads.
Find a set of villages, no two of which are connected by a direct road.
How many villages are there in the largest set?
Guess the rule of writing numbers in the squares.
Replace dots by the numbers.
Which would be the largest number in the pyramid?
Jane is searching for mushrooms in a forest. She starts from a big oak tree and walks 1 meter (m) south, 2 m west, 3 m north, 4 m east, 5 m south, 6 m west, 7 m north, 8 m east, and so on.
After walking a total of 5,000 meters, her basket is full of mushrooms.
How far from the tree is she now?
Gerry throws a coin onto a chessboard. He only considers the results when the coin is completely inside the interior region of the board (the squares), with no part overlapping the border. The coin's diameter is half the side length of the board’s squares.
Estimate the probability that the coin covers a point with the vertexes of 4 squares.
I arrange these 9 cards in a line to form a 9-digit number.
What proportion (fraction) of the possible numbers are divisible by 4?
(Notice that the answer is equivalent to asking what is the probability of the randomly chosen 9-digit number being divisible by 4).
The probability of having accidents on a road in four decades (40 years) is 0.9375.
What is the probability of accidents in one decade?
I give you two sticks with the lengths of 20 cm and 40 cm.
If you randomly break each stick into two pieces, what’s the probability that they can form a quadrilateral (four-sided polygon)?
The probability of seeing a heavy truck on a country road in three hours is 0.65.
Estimate the probability of seeing a heavy truck in one hour on the road.
In a local lottery, the organizers draw 5 from 50 balls with different numbers.
What is the probability that the 5 numbers are in increasing order?
Ann, Ben, and Cenn visit Triangular city.
Ann starts at point A, and each minute, walks one block in the direction of red arrows randomly choosing one of the two directions.
Ben starts at point B, and each minute, walks one block in the directions of green arrows randomly choosing one of the two directions.
Cenn starts at point C, and each minute, walks one block in the directions of blue arrows randomly choosing one of the two directions.
What is the probability that all three meet at an intersection during their walks?
Alex, Bill and Carl want to know their average annual income without revealing their personal incomes. They do several steps:
1. Alex secretly chooses a random secret number 123 and writes it on a piece of paper. He adds his annual income, 230 thousand, to 123 (the secret number being interpreted as 123,000).
R1 = 123 + 230 = 353 and secretly communicates the result to Bill.
2. Bill adds his annual income 317 thousand to 353
R2 = 353 + 317 = 670 and secretly communicates the result to Carl.
3. Carl adds his annual income 353 thousand to 670
R3 = 670 + 353 = 1023 and secretly communicates the result to Alex.
4. Alex subtracts the secret number 123 from 1023 and divides the result by 3
(1023 - 123) / 3 = 900 / 3 = 300 thousand.
Alex tells the others that their average annual income is 300 thousand.
Which statement below is correct?
A point B is randomly chosen inside the square.
Estimate the probability that the triangle ABC is acute-angled.
A gigantic pizza is split amongst 10 mathematicians.
The first mathematician gets 1/10 of the pizza.
Then the second mathematician gets 2/10 of the remaining pizza.
Then the third mathematician gets 3/10 of the remaining pizza, and so on, until the last mathematician gets 100% of the remaining pizza.
Which mathematician gets the largest piece of pizza?
Square numbers are 1, 4, 9, 16, . . .
Rearrange the cards so that the sum or difference (left minus right) of the numbers on any two neighboring cards is a square number.
How many different arrangements are there?
You toss a fair coin 10 times and it always lands on heads.
What is the probability that it will land on heads with the next throw?
Ten lily pads form a circle in a pond. Jumpy sits on a lily. It always jumps either to the next lily, or hops over that and chooses the second lily. It never goes back, and it does not hop over the starting lily.
How many different ways are there to come back to the starting lily?
There are 800 students in a school and 50 classes, each of them has at least 15 students.
A class with more than 20 students is considered as overcrowded in the school.
What is largest possible number of overcrowded classes in the school?
Consider the following experiment.
STEP 1A: Two balls, labelled 1 and 2 respectively, are placed into an urn.
STEP 1B: One ball is randomly chosen from the urn and then discarded.
STEP 2A: Two balls, labelled 3 and 4 respectively, are placed into the same urn as in step 1A.
STEP 2B: One ball is randomly chosen from the urn and then discarded.
Define P(1) as the probability that ball 1 remains in the urn at the end of step 2B.
Evaluate P(1) + P(2) + P(3) + P(4).
Each child in a country has a 50-50 chance of being male or female. A mother gave birth to two babies, one of them is a boy.
Who is most likely the second baby?
I usually buy a healthy drink in the evening from one of the two drinks dispensers on my street.
The probability that my favorite drink is already sold out in the left dispenser is 0.3.
The probability that my favorite drink is already sold out in the right dispenser is also 0.3.
The probability that both of them do not have my favorite drink in the evening is 0.2.
What is the probability that both of them have my favorite drink in the evening?
Start at square S, and end at square X. Each step must be on the grid. Each step can be only one of UP or RIGHT.
To be clear, each step is only one grid square movement, and this cannot be a diagonal step.
Which square(s) most effectively block paths to X?
Using fair standard 6-sided dice, which event is most probable?
A: throwing a six on a single die.
B: throwing a total of 7 on two dice.
Three boys each sent a love letter to their unique and special girlfriends.
John wrote to Rosy, Pete wrote to Jade, and Steve wrote to Celeste.
The postman put the letters into three different boxes without looking at the name of the recipient.
What is the probability that each of the letters reached the correct recipient?
A farmer brought a bag of nuts to a market. The nuts weighed 100 kilograms (kg).
The first customer bought 1 nut, the second customer bought 2 nuts, the third customer bought 4 nuts, and so on: each next customer bought twice as many nuts as the previous one. All nuts have the same weight. At the end of the day the farmer had one nut left.
How many kilograms of nuts did the last customer buy?
We require the value of this ratio in the case that n tends towards infinity.
HINT: For each integer in the numerator (on top) there is exactly one n in the denominator (underneath).
by Leslie Green
The base-2 numeral system or binary numeral system uses only two symbols 0 and 1.
Consider 6-digit binary numbers. For one step you can change 5 symbols: all 0s to 1s and all 1s to 0s.
How many steps are needed to convert number 000000 into 111111?
Consider a 3 digit number abc, where a, b, and c represent unique digits.
Reverse the digits of abc to produce another 3 digit number.
Evaluate the sum abc + cba.
The sum is a palindromic 3 digit number, for example ded, where again d and e represent unique digits, not equal to any of a, b, or c.
What is the lowest possible digit sum of the sum abc + cba?
A prime number is a whole number greater than 1 whose only factors are 1 and itself.
The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, . . .
What could be the first number in a list of 100 consecutive integers without any prime numbers?
You have an infinite plane on which you place square tiles. One at a time, you add new tiles randomly such that each new tile shares at least one edge with a previously placed tile.
You choose randomly one of the existing edges with equal probability and attach the next tile.
What is the probability that you form a 2x2 square after 4 steps?
Inspired by Hairy Random Puzzles inspired by John Horton Conway, Quanta Magazine.
Assume that I can draw the squares and continue the pattern forever.
What fraction of the large square is green?
Place 1977 green points and 1 red point on a circle and draw all possible polygons with the vertexes in the points.
Which number of polygons is the largest?
Source: A. Savin's problem.
Alice, Bob, and Carly visit a gregarious billionaire who likes to reward clever schoolchildren. When the visit comes to its natural end, the butler shows the children to the special exit room. The children are to enter the room, one at a time, as directed by the butler, and in an order chosen by the butler. The children can discuss the situation and strategise before the first person is chosen, but after that, any communication is considered to be cheating, and the valuable prize is withheld.
Each child enters the room, turns over one of the 3 randomly but alphabetically labelled boxes, and views the concealed name tag. The under-butler then restores the boxes to their original state before ushering that child out of the room by a different door, and signalling for the next child. The box labelling is a single unique capital English alphabet character, so for example the boxes could be labelled D, K, and T.
If and only if all three children find their own name is the prize awarded: £1000 per person. Otherwise the children leave empty handed.
Assuming the children are intelligent, and follow a suitable strategy, what is the best probability for a successful outcome?
The director of a prison decides to give 4 prisoners a slim chance to leave. He arranges 4 numbered boxes [1 - 4] in a room. Under each box is a random unique ticket in the inclusive range 1 to 4. The 4 prisoners are also numbered 1 to 4, inclusive. The prisoners enter the room one at a time in a random order, then pick one box at random to see the hidden ticket number. The prisoners leave the room in exactly the same state as they entered it, and by another door, so they do not interact with the other prisoners.
If and only if all four prisoners pick their respective numbers do they get early release.
What is the probability of this outcome?
There are puppies in a pet shop. Three of them have already received all necessary vaccinations, the others have not.
Evguenia chooses two puppies without consulting their papers. The probability that the two of them have already been vaccinated is 1/5.
How many puppies are there in the shop?
You’re about to visit a gorgeous National Park. You want to know the chance of meeting a bear. You called three people you know. You estimate that each of them has a ⅔ chance of telling you the truth and a ⅓ chance of messing with you by lying. All three of them tell you that “There are no bears in the park”. To be clear, if all of them tell the truth, or all of them lie, their answers will agree with each other.
Based on this information, what is the probability that you will not meet a bear?
You have a laser pointer which needs two good batteries to work.
You have 4 good batteries and 4 bad (dead) batteries, but there is no visible difference between good and bad batteries. You randomly pick two batteries to go into the pointer.
What is the probability that this combination works, given that the pointer definitely works with good batteries?
There are 6 airports in a kingdom. Some of them are connected by direct flights. If two airports X and Y are connected by a direct flight, and airports Y and Z are also connected by another direct flight, then a direct flight between cities X and Z is considered to be decadent, and therefore not permitted.
The picture shows 9 direct flights between 6 airports.
The government decided to open 6 more airports.
What would be the largest possible number of direct flights in the kingdom after that?
Ten people are members of a parliamentary commission. We name them A, B, C, D, E, F, G, H, I, and J.
Everybody has less than three enemies within the commission:
A is an enemy of B and C,
B is an enemy of A and D,
C is an enemy of A and D,
D is an enemy of B and C,
E is an enemy of F,
F is an enemy of E,
G is an enemy of H and I,
I is an enemy of G and J,
J is an enemy of I.
Is it possible to divide them into two equal groups, so that there are no enemies in the groups?
Jack builds a long row of two hexagons from wooden sticks.
The picture shows the first 12 hexagons.
How many such hexagons would he build from 1000 wooden sticks?
If we roll the six fair dice, what is the probability that we obtain all the numbers 1, 2, 3, 4, 5, and 6?
You throw a fair 6-sided die. You will earn the face value of the die (in dollars) for either the first throw, or optionally the second throw.
This second option is the refusal of the money for the first throw, instead choosing to throw the die again and accept whatever score occurs.
What is the expected payoff of the game, assuming you use an optimum strategy?
We randomly pick two squares on the 3 x 3 grid.
What is the probability that the squares share at least one common vertex?
We randomly choose two squares on the chessboard.
What is the probability that they have a common side?
Two tokens are randomly taken from a bag containing only 2 red and 3 blue otherwise-identical tokens. Neither token is replaced.
What is the probability that both tokens are of the same colour?
The matchsticks make a spiral.
How many matchsticks do we need to make a spiral with 100 matchsticks on the last side?
Three soup bowls are placed upside down in a row going from left to right. Their positions are not changed.
On the soup bowls are placed two fair coins, with at most one coin per bowl.
How many arrangements (permutations) of coins on bowls are possible?
To be clear, we only care about the coins being heads up or heads down.
The exact position or rotation of the coins on any particular bowl is not considered.
The picture shows 13 squares whose side-lengths are 1, 2, 3, . . , 13 cm.
What fraction of the area of the outer square is green?
A square piece of paper is cut into N equal pieces by N - 1 parallel cuts. The pieces are fitted together to make a long rectangle.
The image shows the case of N = 2.
What is the ratio of perimeters of N of the original squares divided by the perimeter of the composite rectangle in the limit as N tends to infinity?
Three large triangles contain small blue and white triangles.
How many small white triangles are there in a large triangle that has 10 blue triangles in its base?
Continue the started series according to the pattern.
How many small squares does the eighth set contain?
by Sandor Roka
Roll a pair of standard six-sided dice, and calculate the following four products:
A. The product of the top numbers of the dice;
B. The product of the bottom numbers of the dice;
C. The product of the top number of the first die and the bottom number of the second die;
D. The product of the bottom number of the first die and the top number of the second die.
What is the sum of the four products?
A London dweller, John, estimates that during his life he will drink
as much water as two 35,000-litre water tanks contain.
It is the same volume as his dining room - about 18,000 gallons.
Estimate how much water he drinks per day.
1 litre is about 34 fluid ounces or a quarter of US liquid gallon.
An integer sequence is defined by terms of the form: n2 + 37.
The first term uses n=1 and has a value of 38.
What is the difference between the terms with n = 12345 and n = 12344?
HINT: Solve the problem analytically first.
The product of the ages of three kids, when expressed as whole numbers, is 216.
How many different sums of ages are possible?
Two standard, fair six-sided dice are rolled. The product of the two numbers rolled is calculated.
How many different results can we obtain?
An ordinary 1 m length of parcel string is randomly cut into two pieces.
What is the average length of the longer piece?
The road network in a county is symmetrical. The numbers show the lengths of the corresponding roads in miles.
A salesman travels from Aville to Biville and visits each city exactly once.
What is the shortest path?
There are four balls in a bag: 2 white, one black, and one red.
A student randomly draws two balls from the bag, and her teacher notices that at least one of the balls is white.
What is the probability that the other is white?
A young couple would like to have four children.
Which of the following is more likely?
The country's statistics shows that each birth has an equal chance of being a boy or a girl.
The infinite floor bears a uniform grid of squares, the side lengths of which are ten times the diameter of the coin in my hand.
If I roll the coin out onto the floor, what is the probability that it will come to rest entirely within a white square?
We wish to define some whole number V in such a way that it is guaranteed that having divided V by 2, the result is even.
Which is correct?
(m is a whole number)
Jim's team won 30% of the first 40% of 50 games of the Winnie Cup and 70% of the last 60% of the games.
How many games did the team win?
You are given three rational values:
a/b, c/d, e/f,
where each letter represents a natural number.
No rational value is equal to any other of the rational values.
How many comparison operations are required to find the maximum value?
What is the least number n so that n! is evenly divisible by 2025?
n! = 1 x 2 x 3 x . . . x (n - 1) x n
Whilst the ordinary class-size in a school is 30 students, in a specialist class there are only 4 students. The front row of desks has 5 positions.
In how many distinct ways can the students occupy the front row?
We are allowed to use any of the letters { A, B, C, D, E }, and these letters in general can each be used any number of times. However, we are going to write a sequence of 4 such letters that must contain at least one B.
How many of these sequences can be written?
The union of two sets creates a new set which has elements which are in either or both of the two joined sets.
The set-difference operation, X - Y, takes all elements which are in X but not in Y.
Evaluate the specific expression shown in the image.
Eight children want to randomly but fairly choose one of them to go to a shop for ice cream. They have a single fair coin.
What is the least number of throws they need to make in order to choose the person?
Ninety-five percent of my gold coins and ninety-five percent of a coin cost as much as all my coins.
How many coins do I have?
Take a freshly shuffled pack of ordinary playing cards. Remove the top 14 cards and place them face-up on the table next to each other.
What is the probability that there is at least one match within the whole group of 14 cards?
To be clear, a match occurs when the same card in a different suit is within this group of 14 cards. There are 4 suits: hearts, clubs, diamonds, and spades. There are 13 distinct cards in each suit.
Agnetha and Benny play a simple game of cards. Agnetha picks the Jack, Queen, and King of Diamonds. Benny picks the Jack, Queen, and King of Spades.
Both players shuffle their three cards such that even they are unaware of the ordering. Both players simultaneously place their card next to the other, face-up on the table. If the cards match (such as Jack with Jack), Agnetha wins. This process is repeated twice more, so that all cards have been placed. At the end, if Agnetha has not won then Benny has.
What is the probability that Benny wins?
The image shows a 4 x 4 grid of coins. Instead imagine an N x N grid of sub-atomic particles. What is the minimum (non-zero) number of these particles which can be removed whilst still leaving the row and column parity unchanged at the end? (N > 16)
To clarify the question, suppose N is 20. Each row and each column of the grid has 20 particles, making the parity even for each of the rows and columns.
There are 16 students in a class: 10 boys and 6 girls.
The teacher randomly chooses a pair of students.
The probability of choosing which pair is the greatest?
A grandmother has two grandchildren, Alice and Brice. Alice calls her every second day and Brice calls her every third.
What is the fraction of days when she does not receive a call from the lovely grandchildren?
Seven children sit on a bench. The first girl tells a number to the second girl and she tells another number to the next kid.
If the first girl starts with 7 and each next girl either adds seven to the number she received or subtracts one from it, what is the most probable final result?
Three brothers received their father's inheritance.
Bardy received 20% more than Ardy and 20% less than Cardy.
How much more did Cardy receive than Ardy?
We put 12 marbles in a bag and blindly choose 6 of them.
What is the probability that one of the chosen marbles is red?
Three friends just found a ticket to the final of their favorite baseball team. They want to randomly (and fairly) define the lucky guy who will go to the final. They have only one bent coin.
How many times do they flip the coin to find the happy winner?
My pencils with hexagonal cross-section have the Swiss maker logo "Caran D'ache®" imprinted on exactly one of their faces.
If I roll all seven of my pencils on a glass table, what is the probability that at least one of the pencils stops with the logo facing up?
You had a set of five wooden sticks of lengths 1, 2, 3, 4, and 5 inches. Sadly, the three inch long stick has been lost.
If you pick three sticks from the remaining four, some combination(s) will allow you to form triangles, with the tips of the sticks touching each other.
What is the probability that a set of three randomly chosen sticks from the group of four will be able to form a triangle?
There are 10 identical links in the welded-link chain. Each link is painted in one of the three colours: red, green, and blue. Exactly 3 links are red, and exactly 3 links are green.
In how many different ways can the chain be painted?
We are designing flags. Each flag is divided into three horizontal parts just like the flag of the Netherlands shown at the right. We color the three stripes with red, blue, white, green or yellow. We cannot color two neighbor stripes in the same color.
Theoretically, how many different flags can we make?
Gerry has only one coin. Unfortunately it is slightly bent. He wants make a 50-50 decision using the coin.
How many times does he need to flip the coin to make the decision?
Ann hits a small target 80% of the time and Bob only 40% of the time.
After they shoot together, just one arrow each, only one of them hits the target.
What is the probability that it is Ann's arrow?
John is playing a game on a very long street which has lampposts at regular intervals. John is at lamppost number 1, the next being 2, and so on.
John misses zero lampposts then taps the next with his left hand, misses one lamppost then taps the next, and so on up to 100 tapped lampposts.
What is the number of the last tapped lamppost?
Seventy percent of the earth’s surface is covered in water. Only 3% of the earth’s water is freshwater. Forests cover 31% of the world’s land surface.
What part of the earth's surface is forest?
Mary has one more coin than John. All their coins are fair. They throw all their coins and count the number of tails.
What is the probability that she obtains more tails than John?
Gustaff and Gretchen wish to play a game using a standard 6-sided die. Whoever goes first gets to choose two numbers before throwing the die. If either of those two numbers comes up the first player wins. If the first player does not win then the second player gets to choose three numbers before throwing the die. The play continues in this fashion, where the original first player always chooses two numbers, whilst the original second player always gets to choose three numbers.
Which player has a higher chance of winning?
At a rural school in India there are 20 children in a particular class.
9 are wearing socks, 6 are wearing shoes, and 4 are wearing both socks and shoes.
How many children have nothing on their feet?
You have reached the final part of a game show. You are in the middle of a circle of 100 pedestals. On top of each pedestal is a cylindrical cover. Under just one of these covers is a set of keys to a $100,000 car (which you really, really, really want!).
You have to pick which cylinder to open. The game show host has no idea where the keys are, so it is no use looking at her. You make your choice by putting your hand on a particular cylinder. There is a whirring sound and the covers on 98 of the cylinders drop, revealing no keys anywhere. Your cylinder is still covered, and another cylinder is also covered.
What is the chance that your cylinder contains the keys to your dream car?
The image shows a quadrilateral, a planar 4-sided figure.
Supposing all four sides have different lengths,
and further supposing that we define the shape purely in terms of the side lengths,
and supposing that we don't consider symmetric duplicates,
how many variants of this shape are possible?
A teacher splits a group of six kids into two groups of three.
Every day the groups are different.
Over how many days can she do this?
I randomly choose a number from 1 to 500 inclusive.
The probability of choosing any of the 500 numbers is 1/500.
What is the probability that the chosen number starts with 1?
In this Venn diagram the outer rectangular box contains all integers.
The circle A contains all even counting numbers.
The circle B contains all odd integers.
What is in the grey region (which we can label as the set C) ?
Gerry says:
"Two nominally identical fair coins are tossed.
The probability of one head and one tail is 1/3.
Proof: There are only three possibilities:
(a) two heads
(b) two tails
(c) one head and one tail
The chance is therefore one in three, or 1/3."
Do you agree?
All the sides of an 5 × 5 × 5 cube are painted green.
The cube then is chopped into 125 small cubes.
The small cubes are put into a bag.
One small cube is randomly chosen and tossed across a table.
What is the probability that its top face is green?
All these rubber boots have the same size.
Jane picks a boot and asks Gerry to throw her another boot, which he randomly chooses.
What is the chance that it will be a good match for her?
There are 26 red and 26 black cards in a standard deck.
What is the probability that the top card is the same color as the bottom card?
The binary numeral system represents numeric values using two symbols: 0 and 1.
How many such symbols (binary digits 0 and 1) are necessary to write the decimal number one million (1,000,000) as a binary number?
Note: A bit is the smallest unit of measurement used to quantify computer data. It contains a single binary value of 0 or 1. Computers frequently use 64-bit integers.
Here we ask the difficult question,
"If x gets closer and closer to zero, what happens to the value of 1/x?"
Regular tetrahedral (triangular-based pyramid) dice have four faces, numbered 1 to 4 inclusive.
Having thrown two such dice, the ratio max/min of the two values is chosen as the score.
How many distinct scores are there?
A tetrahedron (triangular-based pyramid) has four faces. Dice can be made from regular tetrahedra, with the faces numbered 1 to 4 inclusive.
Having thrown two such dice, the maximum value from one of them is chosen as the score.
What is the average (mean) value of the score?
We wish to form as many distinct (unique) sequences of the letters A, B, C, and D as possible.
How many are there?
A fair coin is flipped five times and the sequence of heads (H) and tails (T) noted.
Which sequence is more likely?
There are 4 standard fair coins on the table. Sam picks up and flips two of them. Pat picks up and flips the other two.
What is the probability that Sam gets more heads than Pat?
Ben and Ann each throw a fair 6-sided normal die.
What is the probability that Ben gets a higher value than Ann?
Three students share a house. All have similar cars, with similar keys, and these keys are placed in a bowl in the hallway. It is nighttime, and the hall light has failed. They therefore take the keys from the bowl randomly.
What is the probability that at most one of them takes the wrong key?
Al, Bill and Cal love Mary and each of them is going to give her a present. She can accept Al's present with probability 4 out of 5, Bill's present with probability 3 out of 5, and Cal's present with probability 2 out of 5.
Estimate the probability that the girl accepts at least one of the presents.
Define the set O as the set of all counting numbers which are odd multiples of 2.
Define the set L as all elements in set O except those which are larger than 2000.
What is the sum of all elements in set L?
One hundred distinct smallest positive integer multiples of 4 form set A,
A = {4, 8, 12, 16, ... }.
One hundred distinct smallest positive integer multiples of 6 form set B,
B = {6, 12, 18, ... }
How many elements are common in A and B?
The team captains of the two five-a-side football teams are chosen at the beginning of the year. The captains are of course also players. The captains pick their players, alternately, but the order in which players are picked affects their self-worth, and therefore is important.
How many ways are there to choose the teams, given that the same 10 students always form the two teams each week?
Three pool balls, numbered 1, 2, and 3, are placed in a line from left to right in such a way that not more than one adjacent pair are in increasing order from left to right.
The sequence: 1, 2, 3 is not allowed because there are two adjacent pairs in increasing order ([1 - 2] and [2 - 3]).
How many of these sequences exist?
A ballot sorting machine has two possible output bins. Any ballot paper entering the machine must end up in one or other of these two bins, even under fault conditions.
If three ballot papers are fed into the machine, how many different sets of bin counts are possible in the case of a fault condition?
Gerry sent three letters to three girlfriends.
The postman put the letters into three different boxes without looking at the name of the recipient.
What is the probability that each girl received a wrong letter?
Six playing cards, consisting of three pairs, are randomly placed face down on a table in a grid pattern.
Another person, completely unaware of the card positions, and without any trickery or assistance, picks two cards to turn face up.
What is the probability that the two cards are a pair?
In a country of 100,000,000 people, 50% are male. 25% are children. 10% are retired. 5% can't ever work for health reasons, and are not counted as part of the workforce.
1% of the working population are professional scientists, with no gender bias.
How many female professional scientists are there in that country?
The four counters, marked '1', '2', '3', and '4', are randomly place in a line.
What is the probability that they form a string of at least 3 digits, each increasing by 1 from left to right?
You have counters labelled 0 to 9 as shown, and one blank yellow counter.
Given that the labelled counters always have to be placed in increasing order to the right (as shown), how many different arrangements are possible?
You start on the green square at the left. You move to the red square at the right. Each move is either up, down, left, or right.
There are several paths, all of the same minimal length.
How many of these minimal length paths are there?
It is your turn on guard duty, rifle in hand. Whilst you hate guns, you have come to realise that it is even less desirable to have your brains chewed on by one of the wandering zombies you are defending against. There are 4 range markers out in the field to assist you with your gun sights. At the outer marker you have a 1/5 chance of putting the zombie down. The other markers have probabilities of 1/4, 1/3, and 1/2 because as the zombie gets closer, your shot is more on-target.
A zombie has appeared and is heading your way. You only fire at the markers, and only get off one shot at each as a maximum.
What is the chance of your surviving this attack?
When we were very young, my sister was 33% older than me.
But the next year she was only 25% older than me.
And the year after that she was only 20% older than me.
Then finally she was only 10% older than me.
How old was I at the start of this progression?
In a particular game, each turn consists of throwing a fair 6-sided die, and depending on the outcome of that throw, tossing a fair coin.
If the die shows an even number, or a number evenly divisible by 3, the coin is tossed.
What is the probability of a head on each turn?
Five numbers have an average (arithmetic mean) that is 10.
The smallest number is 5 and the largest number is 30.
What is the largest of the three other numbers?
This is a typical ACT question.
The playing cards have been arranged in ascending order, with the ace considered as higher than the king.
From a freshly shuffled complete deck of cards the first card drawn is a 9 of diamonds, which remains face up on the table.
What is the probability that the next card drawn is higher?
(For the purposes of this question, we do not consider the suits, hearts, clubs, diamonds, and spades, as being different in value.)
A thumb war tournament had ten players: five girls and five boys. They form five teams of two players.
If the players are assigned so that all possible arrangements are equally likely, which of the following arrangements is more likely?
Boys and girls go on a field trip.
More than 45% and less than half are boys.
What is the least number of children there can be in the group?
The 5 x 5 grid shown consists of 25 little squares, and a blue border which we will neglect. Having drawn a single square, we wish to create the full grid by a series of copy and paste operations. Each copy is one operation, and each paste is one operation. It is permitted to do one copy, and multiple pastes of that copied pattern, in order to reduce the operation count.
What is the minimum operation count required to create the whole grid, given that squares cannot be deleted once pasted?
Gerry holds 2 ropes. Jane randomly picks two ends and ties them together. She does the same with the two other ends.
What is the probability that there are two loops when Gerry opens his hand?
We wish to form a 5-digit number using the digits 0, 1, 2, 3, 4, 5 not more than once in any particular number.
Given that leading-zero suppression will be used, how many of these numbers are available?
You have to choose 11 different positive whole numbers whose average value is 11.
What is the largest possible number you could choose which fits this requirement?
In a class, each student has a brother/sister and/or a pet.
¼ of students with a brother/sister have a pet.
⅓ of students with a pet have a brother/sister.
What fraction of the class in total have pets?
We have 3 shapes to be placed along a line.
The order is which they are placed is important.
Each shape only occurs once in the line.
Each shape can be one of four colors.
Each color only occurs once in the line.
How many different arrangements are there?
James took 3 tests and scored 66%, 72%, and 80%.
The second test had twice as many questions as the first one.
The third test had three times as many questions as the first one.
What percent of all the questions of all the tests did he answer correctly?
A bag contains 1 green and 3 red tokens. Mary and John take turns blindly drawing a token out of the bag without replacement.
Whoever draws the green token wins. Jane draws first.
Who has the greater chance to win?
Four friends randomly sit down at a round table. Ann notices that to her right are Ben, then Carol, then Derek.
Ben is appalled to find out that these four friends are sitting alphabetically, but in an anti-clockwise direction as seen from above. Being scientifically minded they all set about working out how many interchanges of adjacent people are necessary to correct their unnatural rotational orientation.
In simple terms, only people sitting right next to each other can swap places, and obviously they need to orient themselves alphabetically, in a clockwise direction, as seen from above.
How many swaps are needed to correct this distressing situation?
by Leslie Green
The maharaja, in his splendid palace, is bored, sooooooo very bored. He has to make everything a game to add interest to his otherwise meaningless life of inaction.
To choose the color of his clothes for the day he gets a servant to randomly pick from an urn containing red, white, blue, and green gemstones. Each is picked with equal probability, but that alone is inadequate.
If red is chosen he discards that choice with probability 2/3.
If white is picked he discards that choice with probability 3/4.
If blue is picked he discards that choice with probability 4/5.
If green is picked he discards that choice with probability 5/6.
A discarded choice simply means the process starts again from the beginning with equal probabilities for all outcomes.
Which color of clothing does he wear more frequently?
by Leslie Green
With equal probability we randomly pick a red, white, blue, or green sweet from a huge tub. If the sweet is anything other than green the operation is complete.
If the sweet is green we eat it and select another sweet randomly from the tub.
When the operation is complete, what is the probability of having picked a red sweet?
by Leslie Green
If you flip a fair coin two times, the probability of obtaining 1 head and 1 tail (in any order) is 50%.
If you flip the coin four times, what is the probability of obtaining 2 heads and 2 tails (in any order)?
Lewis Carroll's puzzle
A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white.
What is now the chance of drawing a white counter?
Alice wants to visit 4 cities. She starts from her hometown, flies to every city exactly once, and flies back to her town.
How many different ways are there to do so?
Each of the four cards has a number on one side and a colored patch on the other side.
Which two cards must you turn over in order to test the truth of the statement that if a card shows an even number on one face, then its opposite face is red?
Source: Wason Selection Task
100% of 100 adults, 90% of 90 girls and 80% of 80 boys have shown up in the concert hall on time.
How many people are late?
Farmer John has to pick out two piglets from a group of four to send off to market.
How many different ways are there for him to do this?
Gerry is sure that he knows the answers to half of 44 multiple-choice math questions on the SAT. There are five answer choices for each multiple-choice question. He answered all the questions. He answered half of the questions correctly. He answered 11 questions by guessing between one of two answers, one of which was certainly correct. He had no clue about the remaining questions, so he just guessed them as well.
Calculate his expected score.
You throw a pair of fair 6-sided dice, one of which rolls under the table and is not immediately visible. The other shows a 6.
What is the probability that not more than one of the pair is a 6?
In which range is the percentage of prime numbers the largest?
A prime number is a positive integer that has no positive divisors other than 1 and itself.
A graph is a network of points and lines (edges).
A tree is a graph in which any two vertexes are connected by exactly one path (edge).
Each edge has a vertex at each end.
A tree is a connected graph with no closed circuits (or loops).
Which statement is true?
Father promised John 1 cent for 1 solved problem, 2 cents for 2 solved problems, 4 cents for 3 solved problems, 8 cents for 4 solved problems, . . .
John finished the weekend with an extra $20 and several cents in his pocket.
How many problems did he solve?
Five friends decided to give a rare coin to one them. They want to randomly choose the person. They want that Jane who is sick today has 4 times more chances than another student.
What is the minimum number of times they flip the coin to define the winner?
The managing director of a company decides to dismiss 60% of the company employees and to increase the wages of the others by 60%.
What is the salary reduction for the company?
Eleven cities are connected by a circular road. The distance between two neighboring cities is 10 miles.
What is the average length of a journey from one city to another one that is randomly chosen?
You have 240 gold coins, exactly one of which is fake, and therefore significantly lighter than the rest. You have a pair of balance scales which will balance for any equal number of true gold coins on both sides.
What is the least number of balance operations required in order to guarantee finding the fake coin?
by Leslie Green
The expression shown in the image is probably unfamiliar to you.
In English you could interpret this as, "If you increase n indefinitely, what becomes of the expression in brackets in this limiting case?"
by Leslie Green
We want to decide the winner among 4 students with an equal probability by tossing a fair coin.
What is the necessary number of tosses?
A coin collector has 7 large gold coins, 5 large silver coins, and 8 large bronze coins. His assistant has been told to remove them from the locked security cabinet and dust them, before returning them to their drawers. There are exactly 10 drawers, and the assistant successfully puts two coins in each drawer, albeit in some haphazard way.
If the collector opens a drawer and finds a gold coin, what is the least* probability that the other coin is also a gold coin?
*Consider the case when the assistant places the coins in a way that is not favorable for the selection.
by Leslie Green
The jeweller's assistant is playing with 3 real diamonds and 3 fake diamonds. He cannot distinguish between them. Suddenly he sneezes, mixing them all up.
Trying to cover up his mistake, he decides to randomly put the jewels and fakes back in their respective bottles.
He has several options: Write trying to pick a Real diamond as R, and trying to pick a Fake diamond as F.
Which method gives him the greatest chance of success?
by Leslie Green
As this is a mental exercise, you are not allowed to point at or touch the screen.
You are not allowed to write anything down.
You are not allowed to mumble, move your lips, or frown.
Sum the digits on the complete diagonal starting from 3 and running down to 8. Give the result mod 5.
Tom and Mary can't decided whether to go left or right at a road junction. Mary wants to go left; Tom wants to go right. They decide to settle the issue by tossing a fair coin. Whoever first gets a 'head' chooses the direction.
Mary goes first.
What is the probability that they go left?
by Leslie Green
Estimate the approximate value of the expression given in the image.
Although you are free to use a calculator, if you wish, it is not necessary to do so.
We cut a corner of a square piece of paper, so that only 75% of the area is left. All sizes are in cm.
What is the area of the triangle that was cut off?
A password is a four-digit odd number.
Repetition of digits is allowed.
The number can be started with any digit, except 0.
Determine the number of different passwords that can be created.
There are two boxes, each containing 3 balls: { green, red, yellow }.
You pick a ball from each box at random.
If they both are yellow you get $3.
If they both are green you get $6.
You get nothing if the colors are different or red.
The game costs $1 to play.
Would you expect to gain or lose money on average, given that the boxes are returned to their original states at the end of each round?
Calculate the expected score of a student who guesses randomly each of the 44 multiple-choice math questions on the SAT.
There are five answer choices for each multiple-choice question.
Gerry either walks (when it snows), drives (when it rains) or cycles to his college.
He walks about 10% of the time and drives five times more frequently.
Estimate the number of sunny days (no rain, no snow) in the country per year.
The picture show a Venn diagram showing the relationships between even numbers, odd numbers, multiples of 7, and factors of 77.
Which one represents the even numbers?
NOTE: Any integer that can be divided exactly by 2 is an even number. All other integer numbers are odd.
The entire 50 man captain and crew of a pirate ship sit evenly-spaced around a huge round table. The cabin boy is diametrically opposite to the captain, and being the most lowly crew member gets the least share of the profits of the latest mission.
Starting from the cabin boy, and moving around in either direction, each crew member gets one extra gold doubloon than the previous crew member.
The cabin boy gets one gold doubloon. What was the total haul of gold doubloons?
by Leslie Green
James shares a secret with his three classmates.
Only one of two children in the class shares secrets with another student.
What is the probability that the number of students who know the secret is smaller than 5?
You throw two fair dice: one red, and one green.
What is the probability that the value on the green die is at least 2 more than the value on the red die?
Alan, Bryan, and Dylan all have different weights.
Assume that each young man is equally likely to have any particular weight between 65kg and 70kg.
If Alan is heavier than Dylan, what is the probability that Bryan is also heavier than Dylan?
Only Mary, Gerry and Jane correctly solved a math puzzle.
If the teacher said that 85% failed to solve the puzzle, how many students tried to solve it?
Grandma is not very strong, but she is very patient. Every week she washes the thingamajig, taking off 50% of the remaining dirt. No extra dirt is accumulated during the week.
How many times does she need to wash the thingamajig in order to reduce the dirt to below 0.1% of its original value?
(Don't use a calculator.)
Statistics shows that doctor A determines the correct diagnosis of a disease with the probability 90%, doctor B with the probability 95%.
Estimate the probability of wrong diagnosis if a patient is diagnosed by both doctors.
Estimate how many minutes per year on average a web server is unavailable, if its availability is declared as 99.999%.
The picture shows a subway map.
A team of inspectors verifies passengers' tickets at a station on a line, or all lines through it if there are many lines.
Then they randomly choose the next neighboring station and move there to make their inspection.
At what station does Gerry have more chance of being inspected?
Gerry gave $100 to his dad.
His dad says that the money grows at a rate of 10% every year and can be redeemed at any time without penalty.
Gerry asks for his money after 1 year and 9 months.
How much will he receive?
Once a year a dragon drinks some of the Elixir of Life from the Magic Lake.
The Elixir is never replaced, and never evaporates or leaks from the Magic Lake.
At the end of the first year the dragon drinks 1/2 of the Lake, which was full beforehand.
At the end of the second year the dragon drinks 1/3 of the original full volume, and so on.
In which year does the Magic Lake run dry?
Percent (or percentage) just means how many per hundred, on the basis that there are 100 cents in a dollar.
Per-unit just means how many per one, and as such is typically fractional.
A 10% increase is a factor of 1.100 as a multiplier, a per-unit amount.
A 5% decrease is a factor of 0.950 as a multiplier.
What multiplier represents a 13% increase?
Because her birthday falls in the school summer holidays, the only people who come to Octavia's party are her three cousins. The round children's table has 4 places set and Octavia (who is seated first) gets to sit wherever she likes. As her cousins enter the room they always randomly sit next to somebody who is seated, this being the social norm. Her smelly cousin always arrives last.
What is the probability that she has to sit next to her smelly cousin?
by Leslie Green
An apple juice producer has a problem with its latest batch of 800 liters of apple juice.
The specification is <10% by weight of natural sugar,
but this batch is 12.5%.
How much water do they need to add to bring the batch within the specification?
(Neglect measurement uncertainties.)
by Leslie Green
Twenty-six students are randomly assigned to two equal-size teams.
What is the probability that Mary and John are in the same team?
In a class of 50 students, 24 speak French and 34 speak Spanish.
What is the probability that at least 6 speak both French and Spanish?
Eight cards are taken from an ordinary pack of playing cards. There are 3 red cards and 5 black cards. These 8 cards are well shuffled, and two are taken out at random.
What is the probability that the cards are of the same color (colour)?
by Leslie Green
The average of 10 tests is 74.
The lowest score is 0 and the highest is 100.
What is the average score if these two test results are removed?
In adult relationships, by the time a couple has gone out more than a few times we suppose that the probability of the woman being happy with the pairing is 1/4 and the probability of the man being happy is 1/3. A long-term success is only likely if both are happy with the arrangement.
What are the chances of that happening?
by Leslie Green
In the design of multi-stage rockets, electrically-triggered explosive bolts can be used to separate the stages. If the bolt fails to break, the previous stage is still held in place -- and that is typically catastrophic. If the bolt breaks too early then that again can be catastrophic. Having made the part as reliable as possible, further improvement can be made by using series and parallel combinations.
Consider a series connection of explosive bolts. If any bolt fails to break on command, the breaking of any of the other bolts allows the mission to continue. The series connection is less likely to fail to separate.
If the probability of separation failure of one explosive bolt is p, what is the most accurate formula for separation failure of two bolts in series (assuming random failures)?
by Leslie Green
Jim said:
"I always give 200% to my job.
80% on Monday, 50% on Tuesday, 40% on Wednesday, 20% on Thursday, and 10% on Friday."
What is the real percent of Jim’s engagement?
I collected 75 cherries. If all of them except 3 are in a small basket, what is the percentage of cherries that are in the basket?
What is the fewest number of people that could have visited a movie show in the open-air cinema, if exactly 99.2% of the people watched the film until the end?
A cube has a green face, two yellow faces, and three red faces.
Gerry throws it three times.
What is the probability of getting a red first, then yellow, and then green at the end?
"Mathematics all looks like Greek to me", you say. And much of it is, so you are right!
On the left of the equation we have what looks like the letter U with a tail at the start. It is the lower case Greek letter mu. To the right of the equals sign we have a huge symbol that is a bit like a capital E, but different. It is the capital letter sigma, also Greek. What we have here is sigma notation which you should think of as summation.
It is inconvenient to write a + b + c + d + ... for hundreds of values. Instead we might call all the values x and use a subscript to identify the different values. This is very convenient for experimental results as reading 1 has subscript 1, reading 2 has subscript 2, and so on.
Remember that "sub-" means below like submarine, sub-optimum, sub-standard; the index is just below the main symbol.
What is mu as given by this equation?
by Leslie Green
A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
What is the tenth term of the geometric sequence below?
3, 6, 12, . . .
You spin a wheel and it randomly lands on $1, $2, $3, or END. If you land on $1, $2, or $3 you get that money and spin the wheel again. You keep receiving money until you land on END.
What is the probability that the game ENDs with you having won exactly $2?
Four girls dance in a circle.
Every girl has a handkerchief, that she gives either to the left neighbor or to the right neighbor.
The choice is random.
What is the expected number of girls with at least one handkerchief?
What is the largest possible remainder that can be obtained when the product of digits of a two-digit number is divided by its sum?
How should you split a 90-dollar check (bill) with tax in a restaurant among 4 friends?
Remember to tip 20%.
The estimation of the number of humans who have ever been born on the Earth is about 109 billion.
The world population was estimated to have reached 7.6 billion as of October 2017.
What is the percentage of the living people to the number of those who have ever been born in 2017?
Students, their parents, and teachers of Green-Planet school planted trees along a pathway. There are 120 trees in total. Every second tree is planted by students, and every third tree is either by teachers or by students.
The remaining trees are planted by parents.
How many trees did the parents plant?
A wristwatch has had a chaotic price history, which consists of adding $10, a $10 reduction, doubling in cost, and being at a 50% discount.
If these operations were applied in one order the final price would give a maximum, and in another order would give a minimum.
What is the difference between these two costs?
Author: Leslie Green
In a state, the car license plate number has three digits from 0 to 9. The smallest number is 000 and the largest 999.
What is the probability that a license number has the sum of digits smaller than 3?
Five identical balls are numbered 1 to 5 and then placed in a bag. The bag is shaken so the balls are mixed up. Jane draws three of the balls from the bag. Jane wins if the sum of the numbers on the chosen balls is greater than the sum of the balls left in the bag.
What is the probability that Jane wins the game?
Gerry drinks Swiss Mountain whole milk (3.9% fat), while Jane drinks fat-reduced milk (2.5% fat).
If Gerry drinks 1000 ml, how much does Jane need to drink to get the same quantity of fat as Gerry?
In a strange and far-off land, supermarkets typically sell date-limited food at a discount at a particular time of day. Successful hunter-gatherers can then benefit from a product which was at its full price until only a few minutes beforehand.
The image depicts the price in units of pounds, a decimal currency such that 29p = £0.29.
What discount has been applied?
Author: Leslie Green
I put three dice into a cup, shake them up, and roll them out onto a table. A camera looks at the table from above, and computer software correctly finds and displays the die which is closest to the center (centre) of mass of the collection of dice.
What is the probability that the value displayed is 1, 2, or 5?
Author: Leslie Green
A googol is 10 to the 100th power (which is 1 followed by 100 zeros).
A googol is larger than the number of elementary particles in the universe, which amount to only 10 to the 80th power.
How many zeros does the sum of three hundred googols have?
How many different ways are there to compose the word KAYAK, linking the circled letters by following the line segments in the picture?
Logic Puzzle
There are only X types of people:
those who understand binary and those who don't.
What number can replace X?
Mathematicians love using symbols to avoid writing. If you don’t understand what the symbols mean, the resulting expressions are completely meaningless.
Can you crack the code? Can you guess what the upside-down U is supposed to mean?
Author: Leslie Green
I have unlimited stock of all digits, except the digit "5" . I have only 165 fives.
I write all counting numbers starting from 1.
How many numbers can I compose?
Jane has just been learning about the binary number system. She counts using just the fingers and thumbs on her hands, with a curled finger representing a binary 0 and an outstretched finger representing a binary 1. She is counting up from zero (from left to right for you) in whole numbers.
First finger: 0 or 1
Second finger: 0 or 2
Third finger: 0 or 4
Fourth finger: 0 or 8
What number does she show?
In each move, you can only swap two adjacent cards.
What is the minimum number of moves required for the cards to be arranged backwards, that is
9, 8, 7, 6, 5, 4, 3, 2, 1?
In a country, a ticket is considered to have a "lucky" number if the sum of the first 3 digits is equal to the sum of the last 3 digits for a six digit number.
For example, the number 376079 marked green is lucky.
What is the most frequent sum of three digits?
A secret service of a country has 8 spies on payroll in a neighbour country. It is predicted that the network will increase the number of its spies by 50% each year.
How many spies will the secret service have in three years?
Jim is going on a tour around the world. He has 5 tops, 4 bottoms, and 3 pairs of footwear. He wants to post a selfie everyday to show different places and his different outfits to his girl-friend Mary, who stays at home. He does not want to wear the same outfit twice.
For how many days does he have enough clothing?
Since the year 2058, 11 year olds have been required to get high exam marks in one of four elective subjects in order to graduate; harder subjects can earn more points.
The maximum possible scores are
100 for Set Theory,
200 for Vector Calculus,
300 for Orbital Mechanics, and
500 for Quantum Cryptography.
Typically, students of Set Theory get 95 out of 100 questions correct in the exam, whereas the figures for the other subjects are 25 out of 50 for Vector Calculus, 11 out of 30 for Orbital Mechanics, and 21 out of 100 for Quantum Cryptography.
Which subject gives a typical student the highest score?
Author: Leslie Green
The average height of an American is about 170 cm, which is 67 inches, which is approximately 5 foot – 7 inches tall.
The entire population of a town consisted of just 20 adults.
If two babies were recently born what is the average height of the town population?
Kitty has 1000 beads.
100 of them are marble, and 200 are red.
If Kitty offers you one bead at random, what is the probability that it will be a red marble?
Three cantons are being allocated 20 seats in a Parliament. Canton A has 27% of the population, canton B has 33%, and canton C has 40%. The seats are counted by rounding to the nearest integer.
How many seats does canton B get?
A teacher explains to his students about taxes by eating 30% of their ice cream.
If each of 24 students has a cone of ice cream, estimate the amount of ice cream the teacher ate?
Jane is choosing a passcode for her computer.
If her passcode needs to be 6 characters in length and has to contain 2 capital English letters and 4 digits in any order, how many possible passcodes can she make?
In a village, 80% of houses contain two or more people.
Of those homes containing only one person, 20% contain a female. Forty men live separately.
How many houses are there in the village?
John refused to learn how to cook, despite the best efforts of his parents and teachers.
Now he is at college, any meals he prepares can contain only some combination of boiled eggs, baked beans, and pizza.
How many different meals can he prepare?
In an office the ratio of men to women is 7 : 3
The ratio of right handed men to left handed men is 7 : 3
The ratio of right handed women to left handed women is 3 : 7
What is the percent of right handed employees in the office?
I wrote all the integers from 11 to 99 inclusive.
Which digit appears less than the others mentioned below?
For every birthday, Jane's parents put as many dollars in a piggy bank as her age in years.
There is a total of $136 in the piggy bank.
How old is she?
In an election, Hillary received 60% of the votes and Donald received all the rest.
If Hillary won by 12 votes, how many people voted?
When the ball hits the ground, it bounces back up to half its original height. It stops bouncing if the height is smaller than one tenth of a foot.
How many times would it bounce if it is dropped from 30 feet?
If Alex gets 90% on her math test, Beatrice gets 90% of Alex's results and Craig gets 90% of Beatrice's results, what is the approximate average score of the team?
John gets a wage increase of 4% plus an extra $9 per week.
His present wage is $750 per week.
What will his new wage be?
Scientists and engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 0.000000000023 is impractical. Scientific notation uses a number between 1 and just less than 10, multiplied by a power of 10.
What is the scientific form for 0.000123?
Jane and Gerry work at a call center selling worthless rubbish to unsuspecting customers. They are in competition with each other to be the best seller today. In the morning they work from the easy customer list and in the afternoon they are forced to work from the difficult customer list.
In the morning Jane sells to 90 out of her 100 calls, whereas Gerry sells to 85 out of his 100 calls. 90% to 85% means Jane wins.
In the afternoon Jane sells to 30 out of her 100 calls, whereas Gerry, who is sad that Jane is winning, only makes 20 calls, and only makes 5 sales. With 30% to 25% sales figures, Jane again wins.
At the end of the day the boss totals the sales, totals the calls, and computes the aggregated percentage successful sales figure for each seller.
Who wins the competition?
The problem was suggested by Leslie Green
There is a 6% sales tax in a state.
How much tax did I pay for a car, if the check (cheque) was $265,000 in total?
A survey shows that 85% of the population of a country speak English, 75% of the population speak French.
How many per cent of the population can speak both these languages if 5% of the population speak neither of them?
If you toss eight dimes and three of them land heads up, what are the chances the ninth dime will land heads up?
Gerry wants to buy Jane a $100 gift in a pet shop.
If the sales tax is 10% and Benny is on 10% sale today, how much would he have to pay to buy Benny?
If the ratio of men to women in a company is 5 : 7, which of the following could not be the number of employees in the company?
According to Wikipedia Mrs. Vassilyeva holds the record for most children: she gave birth to a total of 69 children: sixteen pairs of twins, seven sets of triplets and four sets of quadruplets between 1725 and 1765, in a total of 27 births.
What is the average number of children per birth?
Jane picks a random integer between 1 and 5 (inclusive) and Gerry does the same.
They then compared their numbers.
What is the probability that Jane's number is greater than Gerry's number?
In a country, there is a baby born every 20 seconds, 1000 deaths every day and 6,000 people immigrate to the country every month.
To the nearest hundred, how many people are added to the population of the country each day?
John and Mary borrow $3,000 to pay for new furniture.
They will pay back the loan by making 12 monthly payments of $333.
How much does the loan cost?
Compare the loan with the original price of the furniture.
The average IQ of 10 students is 101.
If nine of these students each have an IQ of 100, what is the IQ of the tenth student?
This is a typical SAT question.
Seven soldiers have lined up in order of height, tallest to shortest.
They have to be lined up from small to tall.
This will be achieved by a number of interchanges of two soldiers next to each other.
How many interchanges are needed?
The amount of water flowing into a tank doubles every minute.
The tank is full in an hour.
When was the tank a quarter full?
What is the average (arithmetic mean) of all the multiples of ten from 10 to 1000 inclusive?
This is typical SAT question.
Four bugs are at the four corners of an equilateral tetrahedron.
Each bug randomly picks a direction and moves along the edge of the tetrahedron until it reaches the next corner.
What is the probability that none of the bugs will meet one another?
Three apples were weighed in pairs and the weights were 200, 212, and 224 grams.
What is the weight of the lightest apple?
There are six traffic lights on Speedup Avenue.
In how many ways can the lights be set so that no neighboring lights have the same color?
(We assume a traffic light sequence where only one colour is lit at a time. This would not be the case for traffic lights in the United Kingdom and some other countries.)
John answered 100 4-option multiple-choice questions.
He is sure that he correctly answered 60% of the questions.
In 30% of the questions, he chose the answer among two options, and he answered all other questions by randomly guessing among the four options.
What score does John expect to receive on the exam?
A chip is placed at the bottom left corner square of a 5 x 5 grid.
The chip is moved one space upwards or to the right. One of the directions is randomly chosen, for example, by flipping a fair coin.
What is the probability that the chip does not reach the center square of the grid in 4 moves?
I have ten dimes; nine are real, and one is fake.
Whenever a real dime is flipped, it comes up heads with a probability of 0.5.
A fake dime comes up heads up with a probability of 9/10.
What is the probability that a randomly chosen coin will come up heads?
Eugenia earned 60, 65, 71, 79, 85 and 90 on her mathematics examinations.
If she receives a score of 68 on the seventh exam, then the average will . . .
The average of a set of 12 numbers is 1.
If 18, 35 and -20 are added to the set, what is the new average?
99 numbers have an average of 100.
Ninety of these numbers have an average of 99.
What is the average of the other nine numbers?
John arrives at crossroads A from the North.
He makes moves each time randomly choosing one of three directions.
For example, he could be at point B after making four moves.
What is the probability that he finds himself back at point A after 5 moves?
One person is chosen at random from the list.
What is the probability that his/her name contains the letter "n"?
As each of five eggs is weighed, the average weight increases by two grams each time.
If the first egg weighs 50 grams, what is the weight of the last egg?
Statistics show that for every 100 babies born in Funny Town, there are 10 more boys than girls.
What is the probability that Mr. Smith's newborn twins are girls?
Alex received a 70 on his essay and an 80 on his final.
He got a 90 on class participation.
The essay counts as 30% of his grade.
Class participation counts as 20% of his grade.
What is his grade?
A box contains at least five of each of four different types of apples.
I select apples from the box without looking.
How many apples must I draw to be sure of getting at least two of one kind?
The first person is 100 cm tall.
Each next person is 10% taller than the person before.
Who will be taller than 2 meters first?
Which number could be added to the set without changing the average (arithmetic mean of all the numbers)?
If two dice are rolled 72 times, how many times is the sum of the two top numbers expected to be 10?
A bag contains 6 green, 5 gray, and 4 violet disks.
If a disk is drawn at random from the bag, what is the probability that the disk drawn is not green?
Five percent of the marbles in a jar are black.
One fourth of the marbles are green.
One half of the marbles are yellow.
The rest are white.
If there are 12 white marbles, how many marbles are in the jar?
The slice that has been eaten had 16 mushrooms on it.
What is the estimated number of mushrooms on the entire pizza?
A two-digit integer is 4.5 times the value of the same number read from right to left.
Find the number.
I participate in a street-legal car race (on a race track).
One second before the finish I am the first who overtakes the last.
In what position do I finish?
Given that 80% of IT projects fail, and that 20% of IT managers are female, what percentage of IT projects are both successful and have male managers?
The national debt of a country is $16.5 trillions.
It has a population of 300 million.
If each person paid back $100 per month, how long would it take to pay off the national debt, assuming there was no requirement to pay interest?
[ NOTE: Use the definition: 1 trillion = 1 million million ]
There are 10 green and 10 red jellybeans in a jar.
If I randomly remove 2 jellybeans, what is the probability that both are the same color?
(Note: the jelly beans are not replaced in the jar after having been removed.)
Bob shaves himself 4 minutes per day.
If he lives 70 years, what is the total amount of time spent shaving?
The binary numeral system represents numeric values using two symbols: 0 and 1.
How is the number 999 written in binary code?
There are 9 children in a class and each knows a different piece of a short story. They are allowed to exchange what they know via email.
What is the minimum number of emails required to enable each child to know the whole story?
In a bakery store, 400 customers participated in a survey on Tuesday.
On Wednesday, 500 customers were asked at random, of which 40 confirmed that they were surveyed the day before.
Estimate the number of customers per day in the store.
I have four boxes, each containing identically shaped balls: one of them is black.
A ball is randomly drawn out of each box.
What is the probability that at least one of the four balls is black?
I arranged eight wooden sticks in the shape of a fish.
What is the minimum number of sticks that must be moved to make the fish face another direction?
There is a legend that a stork brings babies to a village.
All parents continue to have children until they have a boy.
If they have a girl, then they try to have another child.
They stop if they have a boy.
The probability of giving birth to a boy is 50%.
What is the proportion of boys to girls in the vilage?
The photograph courtesy of Roland Sauter
There are four referees on a rectangular pitch, with one at each corner.
They all set off for a different corner along a side at random at 19:00.
What is the probability that none of the referees meet another referee?
John sent an email to his three partners.
Everybody answered and copied the answer to all others.
A total of 12 emails were sent.
How many emails are sent if there are 40 people communicating in the same manner?
The Venn diagram shows the result of a survey of 80 clients.
It indicates the number of people using 4 different service providers.
There are 8 clients who use all four providers.
If the proportion is the same in a region of 10,000 clients, how many are expected to use at least 2 providers?
Place the number tiles in the squares so that no two consecutive numbers are next to each other horizontally, vertically, or diagonally.
What is the sum of the two numbers in the red squares?
This is a standard set of dominoes.
One domino has no spots and the greatest number of spots is 12.
If I randomly choose a domino, which number of spots has the greatest probability of appearing?
I put 100 pennies, 20 nickels, 10 dimes, and 4 quarters in a bag.
I take 4 coins from the bag without looking.
What is the expected value of these four coins?
Penny = 1 cent.
Nickel = 5 cents.
Dime = 10 cents.
Quarter = 25 cents.
The Venn diagram shows the result of a survey of 80 students.
It indicates the number of people listening to 4 popular music genres.
If the proportion is the same in a school of 500 students, how many are expected to listen to only one music genre?
Henry has 100 roses.
He gives them to his dancing partners in such a way that each girl gets at least one flower and no two girls get the same number of flowers.
What is the largest number of girls with whom he could dance?
A frog jumps one meter 3 times, and every time he randomly chooses one of four directions (either north, south, east, or west).
What is the probability that the frog is 3 meters away from where he started?
Bob the plumber has 10 pockets and 60 nuts.
He wants to put the nuts in all of his pockets so that no two pockets have the same number of nuts.
What is the largest number of nuts that a pocket can contain?
Two cards are randomly chosen.
The two numbers are added.
What sum of numbers is most likely to occur?
One person is chosen at random from the list.
What is the probability that his/her name contains the letter 'a'?
How many times in a 12-hour period does the sum of the digits on a digital clock equal 5?
(NOTE: This clock uses AM / PM rather than 24-hour format.)
There are 4 male and 3 female players on a national tennis team.
How many sets of mixed doubles teams can be formed?
A pass code must comprise four different numbers from 0 to 9.
How many different pass codes can you use?
The Smiths and Petersons went out to dinner.
The two couples chose a round table and randomly picked their places, uniformly spaced around the table.
What is the probability that Mr Smith is sitting opposite to his beloved wife (facing her)?
Statistics at statemaster.com shows the numbers of elementary and secondary school pupils per teacher in different states of the USA.
If the quality of education depends only on the ratio of teachers to students, which state has the best education system?
By drawing two marks on a wooden ruler, we can measure 1, 2, 3, 4, 5 and 6 units. We can measure ALL units from 1 to 6.
What is the maximum number of units we can measure in such manner if we draw three marks on a longer ruler?
John earned an average of 80% on his five exams.
What is the lowest possible percentage score he could have received in any one of the exams?
Everyone in tenth grade classes voted on a motion.
Sixty percent of the girls voted YES.
Forty percent of the boys voted YES.
The motion passes if over 50% of the votes are YES.
In what class did the motion pass?
Three boxes contain two coins each.
One contains two nickels, one contains two dimes, and one contains a dime and a nickel.
All three boxes are mislabeled.
If you are permitted to remove only one coin at a time, how many must you remove in order to be able to label all three boxes correctly?
Source: Fixx, James F Solve It!, 1978
In a class, 90% of the students earned A-grades on the first test, 80% earned A-grades on the second test, 70% on the third test, and 60% on the fourth test.
What is the smallest possible percentage of students who earned A-grades on all the tests?
The average (arithmetic mean) of a set of 10 different numbers is 100.
If the numbers 48 and 160 are removed from the set, what is the average of the remaining numbers?
The average (arithmetic mean) of the 9 numbers is 60.
Which pair of numbers could be removed from the list without changing the average?
What is the probability that the x-coordinate of a point randomly selected within the triangle is less than the y-coordinate?
Fifteen numbers have an average of 15.
Five of these numbers have an average of 5, four other numbers have an average of 4, three an average of 3, and two an average of 2.
What is the remaining number?
The average of eleven numbers is 9 and the average of another set of nine numbers is 11.
What is the average of all these numbers?
Bob is rolling two dice and adding the numbers on the top faces.
What sum is Bob most likely to roll?
The population of Funnytown increased from 9,999 people in 1999 to 11,111 in 2001.
What was the percentage increase?
The slice that has been eaten had 11 mushrooms on it.
What is the estimated number of mushrooms on the entire pizza?
You throw a pair of standard fair 6-sided dice. 
How many different triangles have their vertexes in the vertexes of an irregular pentagon?
What is 2 plus 2 (base 3)?
What is the value of this infinite series?
What are the possible numbers of intersections of 7 lines?
How many times is 30% of 20% of 10 dollars greater than 10% of 20% of 30 dollars?
What is the percentage suggested by the poster?
Find x.
Find the number of different pathways that spell out the word MATH.
Infinite exponential: 
How many different ways are there to tile a 2x7 rectangle with unmarked 1x2 dominoes?
In a mathematical context, which of these objects is a sequence?
How many different ways can these nine balls be put into two identical bags?
What is the probability that there are 5 Tuesdays in February of a randomly picked year?
Estimate the mean of these numbers without using a calculator and without using pen & paper.
If one elf can make two elves after an hour of work, how many elves are there after 5 hours?
In how many ways can you pick all six apples from the tree?
How many squares are there in such a pyramid with 20 rows?
Divide 115 into two parts so that one will be 130 percent of the other. 
What day of the week will it be 44 years after Sunday, January 1st?
How many days are there in 21th century?
Determine the mean (the average) number of legs of the group.
Find two consecutive integers between which the result of the square root of 777 is. 
Which of the following is closest to 
The letters A, E, I, O, and U are vowels. 
How many different bracelets can you make from four beads?
There are 21 cubes in the 6-cube tower. 
Order the shape by weight from lightest to heaviest.
How many coins must you move to form two infinite lines, each with exactly five coins?
What is the logic behind these calculations? 
I plan to roll a die 10 times. 
How many different lines can you draw through N points that are evenly spaced on a circle?
What is the difference between the sums of numbers in the sixth and third columns?