- You are a prisoner in a foreign land. Your fate will be determined by a little game. There are two jars, one with 50 white marbles, and one with 50 black marbles. At this point, you are allowed to redistribute the marbles however you wish (e.g. swap a black marble with a white marble, etc.): the only requirement is that after you are done with the redistribution, every marble must be in one of the two jars. Afterwards, both jars will be shaken up, and you will be blindfolded and presented with one of the jars at random. Then you pick one marble out of the jar given to you. If the marble you pull out is white, you live; if black, you die. How should you redistribute the marbles to maximize the probability that you live; what is this maximum probability (roughly)?
- There are 3 white hats and 2 black hats in a box. Three men (we will call them A, B, & C) each reach into the box and place one of the hats on his own head. They cannot see what color hat they have chosen. The men are situated in a way that A can see the hats on B & C's heads, B can only see the hat on C's head and C cannot see any hats. When A is asked if he knows the color of the hat he is wearing, he says no. When B is asked if he knows the color of the hat he is wearing he says no. When C is asked if he knows the color of the hat he is wearing he says yes and he is correct. What color hat and how can this be?
- One train leaves Los Angeles at 15mph heading for New York. Another train leaves from New York at 20mph heading for Los Angeles on the same track. The distance between LA and NY is about 5000 miles. If a bird, flying at 25mph, leaves from Los Angeles at the same time as the train and flies back and forth between the two trains until they collide, how far will the bird have traveled?
- You have two ropes, each of which takes one hour to burn completely. Both of these ropes are non-homogeneous in thickness, meaning that some parts of the ropes are chunkier than other parts of the rope. Using these non-homogeneous ropes and a lighter, time 45 minutes.
- Willywutang is hanging out on a heavily forested island that's really narrow: it's a narrow strip of land that's ten miles long. let's label one end of the strip A, and the other end B. a fire has started at A, and the fire is moving toward B at the rate of 1 mph. at the same time, there's a 2 mph wind blowing in the direction from A toward B. what can willywu do to save himself from burning to death?! Assume that willywu can't swim and there are no boats, jet copters, teleportation devices, etc.. (if he does nothing, willywu will be toast after at most 10 hours, since 10 miles / 1 mph = 10 hours)
- You have 20 coin machines, each of which produce the same kind of coin. You know how much a coin is supposed to weigh. One of the machines is defective, in that every coin it produces weighs 1 ounce less than it is supposed to. You also have an electronic weighing machine. How can you determine which of the 20 machines is defective with only one weighing? (by one use, we mean you put a bunch of stuff on the machine and read a number, and that's it -- you not allowed to accumulate weight onto the machine and watch the numbers ascend, because that's just like multiple weightings). You are allowed to crank out as many coins from each machine as you like.
- You have two hourglasses: a 7 minute one and an 11 minute one. Using just these hourglasses, accurately time 15 minutes.
- You are an archaeologist that has just unearthed a long-sought pair of ancient treasure chests. One chest is plated with silver, and the other is plated with gold. According to legend, one of the two chests is filled with great treasure, whereas the other chest houses a man-eating python that can rip your head off. Faced with a dilemma, you then notice that there are inscriptions on the chests:
This chest contains the python.
One of these two inscriptions is true.
Based on these inscriptions, which chest should you open?
- You are an archaeologist that has just unearthed a long-sought triplet of ancient treasure chests. One chest is plated with silver, one with gold, and one with bronze. According to legend, one of the three chests is filled with great treasure, whereas the other two chests both house man-eating pythons that can rip your head off. Faced with a dilemma, you then notice that there are inscriptions on the chests:
Treasure is in this Chest.
Treasure is not in this Chest.
Treasure is not in the Gold Chest.
You know that at least one of the inscriptions is true, and at least one of the inscriptions is false. Which chest do you open?
- Speaker: "Brothers and Sisters, I have none. But this man's Father is my Father's son." Who is the speaker talking about?
- An analog clock reads 3:15. What is the angle between the minute hand and hour hand?
- Imagine an analog clock set to 12 o'clock. Note that the hour and minute hands overlap. How many times each day do both the hour and minute hands overlap? How would you determine the exact times of the day that this occurs?
- There are three closed and opaque cardboard boxes. One is labeled "APPLES", another is labeled "ORANGES", and the last is labeled "APPLES AND ORANGES". You know that the labels are currently misarranged, such that no box is correctly labeled. You would like to correctly rearrange these labels. To accomplish this, you may draw only one fruit from one of the boxes. Which box do you choose, and how do you then proceed to rearrange the labels?
- You are a contestant on the Monty Hall game show. Three closed doors are shown before you. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does.
The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors.
After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch?
- You are in an empty room and you have a transparent glass of water. The glass is a right cylinder, and it looks like it's half full, but you're not sure. How can you accurately figure out whether the glass is half full, more than half full, or less than half full? You have no rulers or writing utensils.
- You have a 6-foot long chain that is suspended at its ends, tacked to a wall. The tacks are parallel to the floor. Due to gravity, the middle part of the chain hangs down a little bit, forming a hump; the length of this hump in the vertical direction is 3 feet. Find the distance in between the tacks.
- A dragon and knight live on an island. This island has seven poisoned wells, numbered 1 to 7. If you drink from a well, you can only save yourself by drinking from a higher numbered well. Well 7 is located at the top of a high mountain, so only the dragon can reach it.
One day they decide that the island isn't big enough for the two of them, and they have a duel. Each of them brings a glass of water to the duel, they exchange glasses, and drink. After the duel, the knight lives and the dragon dies.
Why did the knight live? Why did the dragon die?
- How many places is there on the earth that one could walk one mile south, then one mile west, then one mile north and end up in the same spot? To be precise, let's assume the earth is a solid smooth sphere, so oceans and mountains and other such things do not exist. You can start at any point on the sphere. Also, the rotation of the earth has nothing to do with the solution; you can assume you're walking on a static sphere if that makes the problem less complicated to you.
- Three coworkers would like to know their average salary. However, they are self-conscious and don't want to tell each other their own salaries, for fear of either being ridiculed or getting their houses robbed. How can they find their average salary, without disclosing their own salaries?
- Arrange the numbers 1 to 8 in the grid below such that adjacent numbers are not in adjacent boxes (horizontally, vertically, or diagonally).
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The arrangement above, for example, is wrong because 3 and 4, 4 and 5, 6 and 7, and 7 and 8 are adjacent.
- You die and the devil says he'll let you go to heaven if you beat him in a game. The devil sits you down at a round table. He gives himself and you a huge pile of quarters. He says "ok, we'll take turns putting quarters down, no overlapping allowed, and the quarters must rest on the table surface. The first guy who can't put a quarter down loses." you guys are about to start playing, and the devil says that he'll go first. However, at this point you immediately interject, and ask if you can go first instead. You make this interjection because you are very smart, and you know that if you go first, you can guarantee victory. Explain how you can guarantee victory.
- Why are manholes round?
- Add punctuation to the following phrase to make something grammatically and logically coherent:
is is not not not is not is is is is not is not is it not
- In the city of Funky town, the following facts are true:
- No two inhabitants have exactly the same number of hairs.
- No inhabitant has exactly 483,207 hairs.
- There are more inhabitants than there are hairs on the head of any one inhabitant.
What is the largest possible number of inhabitants of Funky town?
25. You have two thermoses. The first contains a liter of milk, the second contains a liter of pure chocolate syrup. You pour one cup of milk out from the first thermos to the second one. Then, after mixing that, you take one cup of the mixture from the second thermos, and pour it back into the first thermos. After completing these two operations, which thermos is more pure?
26. There is an island of monks where everyone has either brown eyes or red eyes. Monks who have red eyes are cursed, and are supposed to commit suicide at midnight. However, no one ever talks about what color eyes they have, because the monks have a vow of silence. Also, there are no reflective surfaces on the whole island. Thus, no one knows their own eye color; they can only see the eye colors of other people, and not say anything about them. Life goes on, with brown-eyed monks and red-eyed monks living happily together in peace, and no one ever committing suicide. Then one day a tourist visits the island monastery, and, not knowing that he's not supposed to talk about eyes, he states the observation "At least one of you has red eyes." Having acquired this new information, something dramatic happens among the monks. What happens?
- "There are 10 Brown Eyed Monks"
- "There are at least two Red Eyed Monks"
- "There is an odd number of Red Eyed Monks"
- "There is an even number of Red Eyed Monks"
- "There is more than one Red Eyed Monk"
27. Compare the numbers 0.99999... (Infinitely many 9s) and 1. Which of the following statements is true? Why?
- 0.99999 ... < 1
- 0.99999 ... = 1
- 0.99999 ... > 1
28. Imagine a disk spinning like a record player turn table. Half of the disk is black and the other is white. Assume you have an unlimited number of color sensors. How many sensors would you have to place around the disk to determine the direction the disk is spinning? Where would they be placed?
29. A man has a gold chain with 7 links. He needs the service of a laborer for 7 days at a fee of one gold link per day. However, each day of work needs to be paid for separately. In other words, the worker must be paid each day after working and if the laborer is ever overpaid he will quit with the extra money. Also he will never allow himself to be owed a link. What is the fewest number of cuts to the chain to facilitate this arrangement and how does that guarantee payment?
30. This is the logic game of Mastermind. If you haven't played it before, here's how it works. There is a board that is sectioned off into many rows, each row having four slots in which pegs can be inserted. There are six different colors of pegs: green, red, yellow, brown, dark-blue, and light-blue. There are two players, A and B. First, A makes up some arrangement of four pegs along a row, the colors and ordering of which are his or her choice. Then B spends the rest of the game trying to guess what A's arrangement is. For every guess that B makes, A will respond by putting some black and/or white pegs right next to A's guess; the black and white pegs are interpreted as follows
- Black keypeg = one of B's pegs is the correct color and in the correct position
- White keypeg = one of the B's pegs is the correct color but in the wrong position
So if B manages to guess all four colors and positions correctly, A will respond with four black keypegs, and the game is over. The goal is to determine A's secret arrangement in the minimum number of guesses. Below, we see a completed game of Mastermind. Apparently the player was able to determine A's arrangement by using only four guesses. What’s A's arrangement?
31. In a two-child family, one child is a boy. What is the probability that the other child is a girl?
In a two-child family, the older child is a boy. What is the probability that the other child is a girl?
32. Four people, A, B, C, and D, are on one side of a bridge, and they all want to cross the bridge. However, it's late at night, so you can't cross without a flashlight. They only have one flashlight. Also, the bridge is only strong enough to support the weight of two people at once. The four people all walk at different speeds: A takes 1 minute to cross the bridge, B takes 2 minutes, C takes 5 minutes, and D takes 10 minutes. When two people cross together, sharing the flashlight, they walk at the slower person's rate. How quickly can the four cross the bridge?
33. You have two cubes. Mark there faces such they can represent all the days of a month, means can represent numbers 01, 02, 03, 04, …..31.
34. There is 100 floor building and you have two eggs. You need to find out the floor, from where if the egg is dropped it will break. Below that floor the egg won’t break. You nee to find this out in minimum iteration.
35. Ten people land on a deserted island. There they find lots of coconuts and a monkey. During their first day they gather coconuts and put them all in a community pile. After working all day they decide to sleep and divide them into ten equal piles the next morning. That night one castaway wakes up hungry and decides to take his share early. After dividing up the coconuts he finds he is one coconut short of ten equal piles. He also notices the monkey holding one more coconut. So he tries to take the monkey's coconut to have a total evenly divisible by 10. However when he tries to take it the monkey conks him on the head with it and kills him. Later another castaway wakes up hungry and decides to take his share early. On the way to the coconuts he finds the body of the first castaway, which pleases him because he will now be entitled to 1/9 of the total pile. After dividing them up into nine piles he is again one coconut short and tries to take the monkey's coconut. Again, the monkey conks the man on the head and kills him. One by one each of the remaining castaways goes through the same process, until the 10th person to wake up gets the entire pile for himself. What is the smallest number of possible coconuts in the pile, not counting the monkeys?
The question can be like:
36. My fancy new digital alarm clock is broken! The time 'jumps' around. When I reset it, it reads 12:00:00. Then it runs as it should, but after 12:04:15 it resets back to 12:00:00. It counts up to 12:04:15 again and then it jumps to 12:08:32 ! Weird stuff. Do you know what's wrong with my alarm clock?
37. The UC Berkeley bus had a minimal number of passengers. When it arrived at Telegraph Avenue, 3/4 of the passengers got out, and 7 people got on. At the next two stops, Shattuck and Hearst, the same thing happened. How many got off at Hearst?
38. What is the beginning of eternity, the end of time and space, the start of every end, and the end of every race?
39. A snail is at the bottom of a well that is 20 meters in depth. Every day the snail climbs 5 meters upwards, but at night it slides 4 meters back downwards. How many days must elapse till the snail reaches the top of the well?
40. Pairs of primes separated by a single number are called prime pairs. Examples are 17 and 19. Prove that the number between a prime pair is always divisible by 6 (assuming both numbers in the pair are greater than 6).
41. You are a landscape specialist, and have been asked to design a garden for a math professor. He wants four trees that are all equidistant from each other. How do you place the trees?
42. A boat of mass M1 is floating in a lake of water. The volume of the lake is V. The water surface is initially at height h, as measured relative to the lake's floor. There is an anchor of mass M2 sitting on the boat's deck. A person standing on deck picks up the anchor and throws it overboard. The anchor then sinks to the bottom of the lake, and the water surface height becomes h'.
Which of the following qualitative relationships is correct? What assumptions are you making about the values of M1, M2, h, and V?
a. h' < h
b. h' = h
c. h' > h
- Consider a list of 2000 statements:
1) Exactly one statement on this list is false.
2) Exactly two statements on this list are false.
3) Exactly three statements on this list are false.
. . .
2000) Exactly 2000 statements on this list are false.
· Which statements are true and which are false?
· What happens if you replace "exactly" with "at least"?
· What happens if you replace "exactly" with "at most"?
· What happens in all three cases if you replace "false" with "true"?
- A patient has fallen very ill and has been advised to take exactly one pill of medicine X and exactly one pill of medicine Y each day, lest he die from either illness or over dosage. These pills must be taken together. The patient has bottles of X pills and Y pills. He puts one of the X pills in his hand. Then while tilting the bottle of Y pills, two Y pills accidentally fall out. Now there are three pills in his hand. Because both types of pill look identical, he cannot tell which two pills are type Y and which is type X. Since the pills are extremely expensive, the patient does not wish to throw away the ones in his hand. How can he save the pills in his hand and still maintain a proper daily dosage?
- A bunch of nifty match configuration problems. Starting with the following configuration:
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A. Remove two matches to get two squares -- one larger than the other.
B. Move 3 matched to get 3 identical squares.
C. Move 4 matched to get 3 identical squares.
D. Move 2 matches to get 7 (non-identical) squares. hint: you may place one match over another
Without calculating 21!, what are the digits marked x and y?
- Say you have some bendable wires (any number, any length). What is the minimum number of solder connections needed to make a cube? Prove it. Also, what is the minimum number of wires necessary? Prove it.
- The master of a college and his wife has decided to throw a party and invited N guest and their spouses. On the night of the party, all guests turned up with their spouse, and they all had a great time. When the party was concluding, the master requested all his guests (including his wife, but not himself) to write down the number of persons they shook hands with, and to put the numbers in a box. When the box was opened, he was surprised to find all integers from 0 to 2N inclusive.
Assuming that a person never shakes hands with their own spouse and that no one lied, how many hands did the master shake?
- Consider an answering machine with remote inquiry facility, where you can call the answering machine and enter a 4 digit passcode into your telephone keypad, so you can listen to the messages from anywhere you like. Many of these machines will let you in if you enter the correct consecutive sequence of digits, regardless of what preceded that sequence.
Example: Passcode is 1234.
if you feed the machine 1234, you're in
To brute-force hack the machine, you could try all numbers from 0000 to 9999, sending 40000 sounds across the wire. But since you are a smart hacker, you see that there's room for optimization. What is the shortest series of digits you have to send to the answering machine in order to break the code in any case?
- You are presented with a ladder. At each stage, you may choose to advance either one rung or two rungs. How many different paths are there to climb to any particular rung; i.e. how many unique ways can you climb to rung "n"?
After you've solved that, generalize. At each stage, you can advance any number of rungs from 1 to K. How many ways are there to climb to rung "n"?