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Prove that exist 50 bags which, when combined, contain at least half the apples and at least half the oranges. There are bags, each containing some number of apples, oranges and bananas. Prove that there exist 51 bags which, when combined, contain at least half the oranges, at least half the oranges, and at least half the bananas. When you roll two six-sided dice, their sum is a random number between 2 and Is there a different way to label two six sided dice with positive integers, so that their sum is a random number between 2 and 12 with the same frequencies?
Design nine playing cards, each labeled with a real number, so that when you randomly choose two of these cards without replacement , the probability of each sum is the same as the probability two dice would roll that sum. An island has 13 red, 15 blue, and 17 yellow chameleons. Whenever two chameleons of different colors meet, they change two the third color. Is it possible for all chameleons to become the same color? Each lockbox has a single key. A prankster shuffles all of the keys, then randomly places them in the boxes, one key per box.
He then closes the boxes, locking the keys inside.
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Though the steel boxes are impenetrable, the wooden ones can be broken easily. What is the probability that breaking open the wooden boxes will allow you to open the steel ones? Suppose that instead of placing each key in a different box, the prankster placed each key in a random box independently of the others, so that some boxes could contain multiple keys while others had none. How does this change the answer? Harder: What is the probability the number of heads will be a multiple of 3? There are 99 gunmen standing in a field.
At high noon, they all shoot the person closest to them. If several people are equally close, they shoot the tallest one. No two gunmen are the same height. Prove that there will be at least one gunmen who doesn't get shot. I have 40 dice, all of which are sided. Half are red, half are blue. I roll all of these dice.
Show that it will be possible to select a nonzero number of dice such that the red total equals the blue total. I've placed all 16 pawns on different squares of a chessboard. Prove that there exist four of them which form the vertices of a parallelogram. A chess knight is somewhere on an infinite chessboard. You are allowed to perform a series of swaps, where you switch any two checkers that are in the same row.
Is it possible to make it so that every column contains one checker of each color? There are six gopher holes. Every night, the gopher sleeps in one of these holes. Every night, you may examine one hole. What is the fewest number of nights it takes to catch the gopher? Prove that there will exist a row or column with at least 4 distinct digits. Find a filling where every row and column has at most 4 distinct digits. I have removed two opposite corners of a chessboard. Is it possible to tile the remained with dominoes, where each domino covers two squares? Can you tile a 10 by 10 chessboard board with dominoes so that half of the dominoes are horizontal?
A regular hexagon is divided into congruent equilateral triangles, and the resulting grid is covered with diamond shaped tiles which cover two such adjacent trinangles. The tiles can be in one of three orientations; prove that the number of tiles in each orientation will be the same. All US senators are about to have dinner at a circular table. Each place setting has a placard with the name the senator assigned that seat, but the senators ignore these and sit down randomly. Prove that it is possible to rotate the table so that no one is sitting at their placard.
A casino offers the following game. You roll a fair die, and can choose to either win the number of dollars rolled, or reroll. After your second roll, you get the same decision, but you must take whatever your third roll is. What is the expected value of this game under optimal play? The same casino offers a variation of the previous game: you get to roll a die as many times as you want, but each roll costs a dollar. When you decide you are done rolling, the casino pays you the value of your last roll in dollars. What strategy maximizes your expected winnings?
A hotel key card has two ways it can inserted it into the lock. However, it is so poorly labeled that as far you can tell, either way is equally likely to be the correct one. Trying the current side takes one seconds, and flipping takes half a second. What strategy minimizes the expected time it will take to open the lock? Your friend flips over the cards of a deck, one by one. Before each flip, you can assert that the next card will be red: you win if you are right, and lose otherwise.
What is your probability of winning when you play as well as possible? A store has a deal where the first person in line whose birthday is the same as someone ahead of them gets a free pizza. Since you know the owner, you can enter any place in the line. Assuming the current people in line have random birthdays uniformly distributed over a non leap year, where should you get in line? An emperor decided that his country didn't have enough males.
To remedy this, he enacted the following law: couples may have as many children as they want, until they have a girl, in which case they can't have any more children. How will this affect the gender distribution over time? Two gambling addicts, Helen and Taylor, flip a coin over and over. Once per minute, you randomly choose two differently colored balls from the urn, and paint the first one the color of the second one. On average, how long will it take for the balls to all be the same color?
Take the same setup as Painting Balls 1, except that each minute you randomly choose any two colored balls form the urn. A lemming is standing on the edge of a cliff, currently one step away from falling off. What is the probability it eventually falls off the cliff? An opera has attendees, each having assigned seats. The guests enter one by one to take their seats.
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However, the first guest is drunk, and simply sits in a random seat. Afterwards, every guest sits in their seat if it is unoccupied, and sits in a random seat otherwise. What is the probability that the th guest ends up in their seat? Warmup: Given a fair coin, how can you randomly choose among three options with equal probability? You have a bowl with noodles.
Over and over, you grab two noodle ends at random and tie them together, until there are no loose ends. On average, when there are no more loose ends, how many loops will there be? He misses his first, then sinks his second. Since confidence determines how well he plays, the probability he makes each subsequent throw is equal to the proportion of throws he has sank so far.
After throws including the first two , what is the probability that Shaq made exactly 50 baskets? Alice has 50 coins, and Bob has If they both throw all their coins in the air, what is the probability that Alice will end up with more heads than Bob? A frog is hopping down a long line of lily pads. Every hop he takes is either two pads forward or one pad backwards, with equal probability. Over time, what percentage of the lily pads does the frog land on?
A Hydra is a creature with many heads. They are born with heads, but they "shed" a random number of heads on each of their birthdays except for their actual day of birth. What is the life expectancy of a Hydra? The teacher then asks Alice, "Do you know what Bob's number is? Assuming Alice and Bob are perfect logicians, will this process necessarily stop?
There are two integers, each greater than one, but whose sum is less than Sam is told their sum, Polly is told their product. The two have the following conversation: Polly: I don't know what the numbers are. Sam: I already knew that. Polly: Now I know the numbers. Sam: Me too. What are the two numbers? All the logicians in the world are gathered on an island, and each given a tattoo of a word on their forehead.
There are no mirrors, and speaking is forbidden: in short, every logicians knows the words on everyone's head but their own. Still, every night at midnight, a boat comes to take away anyone who can deduce their forehead word.
Order of St. Sylvester - Wikipedia
Prove that all logicians will eventually deduce their forehead word, and leave the island. How long does it take? Every night at midnight, a boat comes to take away anyone who knows their own eye color. Since there are no reflective surfaces on this island, and talking about eye color is forbidden, no one knows their own eye color, though they know everyone else's.
One day, a green eyed oracle appears before the logicians. Loud enough for everyone to hear, she announces, "At least one of you has blue eyes.
Take the same setup as Green Eyed Oracle, with the following changes. Not everyone has blue eyes, but it is common knowledge that everyone has either blue or brown eyes. The oracle's announcement is changed to be any nontrivial statement about the number of people with blue eyes. The statement need not even be true! However, it is common knowledge that the logicians will believe anything the oracle says, unless their eyes tell them otherwise.
Will everyone necessarily leave the island? You come to a crossroads. There are two paths, one leading to a town of honest knights, the other leading to a town of knaves, who always lie. You want to get to the knight town. Sitting at the crossroads is an inhabitant of one of the towns, thought you don't know which. By asking a single yes or no question, determine which of the two paths leads to the knight town. Three identical looking men stand before you.
Sometimes Blue Knights Wear Black Hats
One of them is God, and always tells the truth. One is the Devil, and always lies. The last is an Idiot, who answers questions randomly. You enter a chamber with the three gods of the universe, Past, Present and Future, though you don't know which is which. You must determine their identities by asking them yes or no questions, each directed at a single god.
Present will truthfully answer the current question, Past will truthfully answer the previous question asked, and Future will answer the next question you plan to ask. If you ask Past resp. Future your first resp. To avoid paradoxes, you must plan what questions you plan to ask, and what order to ask them, ahead of time.
However, you may decide who to ask each question to based on previous answers. Furthermore, your questions may only refer to statements about the identities of the gods, combined with Boolean connectives like "and," "or", "not", etc. This means that questions about how a god would answer a question are forbidden, along with circular questions like "Will you answer no to this question? How many questions does it take to determine who is who, in the worst case? They are ranked so there is a captain, vice captain, vice-vice captain, etc.
They use this system to divide the gold: the current captain proposes a division of gold, and then all the pirates vote whether to accept it. If at least half vote yes, the gold is divided as proposed, otherwise the captain is killed and the process is repeated. The pirates are perfectly rational.
Their priorites are, in this order, survival, then wealth, then bloodthirst meaning if all else is equal, they would prefer to see more pirates killed. How will the gold be divided? There is a pride of perfectly rational lions. Suddenly, the alpha lion dies, leaving the 99 remaining lions to decide the next leader. Here's how their "election" process works. The oldest living lion is the initial candidate. At any point, a lion can eat the curent candidate, which makes them become the candidate. If multiple lions want to eat someone, the older lion gets priority.
The process ends when no one wants to eat the leader. The lions all want to be leader, and if that is not possible, would rather be alive than dead. Which lion will be the leader? They are silently sitting in a line, so each logician can see the hats in front of them, but not behind them. There is also a bell which rings every minute.
If a monk ever deduces their hat color, they must silently leave the line the next time the bell rings so people behind them will notice them leaving, but not those in front. A gamemaster then installs a lightbulb which every monk can see. He explains that the lightbulb is wired to stay lit if and only if at least one monk in the line is wearing a white hat. Prove that all the monks will eventually leave. When all of the hats are white, how long does it take?
You have an unlimited number of torpedoes. At the top of every hour, you can fire a torpedo to a place on the number line, sinking the sub if it is there. How can you guarantee eventually sinking the sub? In this case, the torpedo will destroy a region of length one centered around the place it is fired. Alice, Bob, Charlie and Diana have to cross a bridge at night. It takes them respectively 1,2,5 and 10 minutes to cross this bridge. At most two people can be on the bridge at a time, moving at the slower person's rate.
The Knights Templar Rulebook Included No Pointy Shoes and No Kissing Mom
Furthermore, you need a flashlight to safely cross the bridge, and the team only has one flashlight though two people crossing at the same time can share it. What is the fastest that the team can cross the bridge? Yusef has bananas, and a hungry camel. The camel can carry bananas, but has to eat a banana for every mile it walks.
How many bananas can Yusef deliver to the neighboring city miles away? He is allowed to leave bananas on the side of the road to pick them up later. Airplanes hold enough fuel to fly exactly halfway around the world. There is an airport at the north pole, which is the only place from which planes can take off and land. If two planes meet in midair, one can instantly transfer any amount of fuel to the other, and a plane can instantly refuel at the airport.
All planes fly at the same speed. How many planes do you need for one to circumnavigate the globe? There are 17 rabbit holes in a row, with a rabbit sleeping in one of them. Every day, a fox checks one of the holes, catching the rabbit if it is there. Every night, the rabbit moves to an adjacent hole. Can the fox catch the rabbit?
A magician and her assistant perform the following "magic trick. An audience member is invited onstage, and chooses any five cards he pleases from a standard 52 card deck. The assistant chooses one of these cards for the volunteer to put in his pocket, and then arranges the other cards in a neat, face down pile. The magician returns, examines the pile, and successfully guesses the card in the volunteer's pocket. How do they perform this trick? There is no trickery used: the only way the assistant sends any information to the assistant is by choosing the order the cards are stacked in.
What is the largest value of N for which the Fitch-Cheney card trick can performed with an N card deck, instead of a 52 card deck? The magician leaves unable to see or hear what transpires onstage , while the assistant invites a volunteer onstage and gives him nine dice. The volunteer is asked to roll the dice, arrange them in a horizontal line, then return to his seat. The assistant then covers two of the dice. Finally, the magician returns, and successfully guesses the values of both the hidden dice. How is this trick be done? You have an ice cream cake, with white icing on top, and the black chocolate interior showing when looked at from beneath.
The position of each slice is clockwise adjacent to the previous. There are several fuel stations around a circular racetrack. Combined, they have precisely enough fuel for one car to make it all the way around. Prove there is some fuel station a car can start at and be able to drive clockwise around the entire track. You have two equal volume cans of red and blue paint, which have the same density. You take a cupful of red and pour it into the blue, and then a same volume cupful of this mixture and pour it back in the red.
Which is greater: the percentage of blue paint in the red can, or the percentage of red paint in the blue can? Their commander yells at them to form a line going East to West, with everyone facing East. To remedy the situation, any soldier who sees the face of their neighbor assumes they were facing the wrong direction, and turns around. This takes one second. The same process happens every second until no two soldiers face each other. What's the maximum length of time it will take for the soldiers to stop turning? The paper currency in Decistan comes in four denominations: 1, 10, and 1, dollars.
Is it possible to have half a million bills which are worth a million dollars? The owner sorts each row alphabetically by author, and then does the same with the columns. Prove that the rows will still be sorted. A lecture had five students in attendance. They all started awake, and all ended awake, but they each fell asleep twice. Furthermore, for every pair of students, there was a time when they were both asleep.
Prove that, at some point, there were three students asleep. In Los Angeles, there are seven gangs, each of whom claims half of Los Angeles as territory. Call a place in Los Angeles "hot" if a majority of the gangs claim it. What is the smallest that the hot region can be? Information Theory. Number Hats 10 people will stand in a circle, and each have a hat placed on their head.
Red and Blue Hats, Vegas Style Take the same setup as Red and Blue Hats in a Circle, except that when each player announces a guess, they also announce a bet that their guess is correct. Alternating Hat Colors A prison warden offers his inmates a game for their freedom. Red and Blue Hats in a Line 10 gnomes have been captured by an evil wizard, and forced to play a game for their lives. Red and Blue Hats: Coordination Challenge people will stand in a circle and have a red or blue hat placed on their head, so they know everyone's hat color except their own.
Two Sheriffs Two sheriffs are working on a case that started with 8 suspects. Spy Behind Enemy Lines You are a spy about be sent behind enemy lines. Locks and Thieves Warmup : Three thieves have stolen a treasure chest. Overhead Projector Before large computer screens and digital projectors, professors would use overhead projectors.
Butcher and Scale A butcher measures out quantities of meet using a two pan balance, and several reference weights. The Poisoned Wine Barrel A king is preparing a feast, and has barrels of wine. Three Dial Lock A certain combination lock has three dials, each number 1 to 8. Mostly Truthful I am thinking of a number between 1 and Conway's Soldiers On an infinite checkerboard, a certain row and all the rows below it have checkers on their squares. Zeroise Me Satan challenges you to the following game. Robot in a Maze 1 A chessboard maze consists of an 8 by 8 chessboard, surrounded by walls, where Between every pair of walls, there either is a wall, or there isn't, One square is marked start, and a different one is marked finish, It is possible to get from start to finish.
Robot in a Maze 2 For this question, a chessboard maze is just like it was defined in Robot in a Maze 1, except that the start square is always at the NW corner, and the finish is always in the SE corner. Matrix Sums There is a matrix rectangular array of real numbers. Magic Money Machine Scientists have designed a magic money machine. Numbers on a Blackboard You have an empty blackboard. Switch the Knights There are four knights at the corners of a 3 by 3 chessboard, the bottom two white, the top two black.
Switch the Knights 2 On a 4 by 3 chessboard, the bottom row is filled with white knights, the top row with black knights. Measuring Water You have two jugs, whose volumes are 3 and 5 gallons. Five Fencing Friends There are five fencers. Integral Rectangles A rectangle is partitioned into smaller rectangles.
Four Points, Two Distances The vertices of a square have the following property: there are only two distinct values among the distances between each pair of points. Strips in the Plane Let's say that a strip is the region of the plane between two parallel lines. Color the Plane Prove it is possible to color each point of ah infinite plane with 7 colors so that any two points that are at distance one are different colors.
Eighty Percent Shaq is shooting free throws. Elephant Balancing Act A circus owns 13 elephants.
Names for the Templars
Cross, Dot and Scalar Products In multivariable calculus, there are vectors and scalars. Knight Game Alice and Bob play a game with a standard chessboard and a knight. Subtraction Games Two players play a game with a pile of 50 candies. Quarters on a Table Two players take turns placing quarters on a rectangular table. Cart 0. Item s Added To cart Qty. If you are a new user Register login. Help Center. Exchange offer not applicable. New product price is lower than exchange product price. Exchange offer is not applicable with this product. Exchange Offer cannot be clubbed with Bajaj Finserv for this product.
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View Order. Hover to zoom. Sold Out! Be the first to review. A work pant or an event unto itself? Why insist on utilitarian descriptions? We should wear the looks we love, all day every day, and forget the rest. Let the world catch up to our style. And to make this point even more clearly, we decided to catch up with Musgraves, who was on tour and in Denver as New York Fashion Week drew to a close. The director Bardia Zeinali and fashion editor Jorden Bickham headed west, rounding up some local talent along the way, and had a little fun of a David Lynch stripe with the new reigning queen of country.
The results are fantastical and fantastic, much like Fall Kacey Musgraves. A shimmery catsuit for cool, confident cats, much like Musgraves. A dress to buy now, wear forever. Musgraves pairs her with a cowboy hat, just because.