* You are guarding 100 murderers in a field, and you have a gun with a single bullet. If any one of the murderers has a non-zero probability of surviving, he will attempt to escape. If a murderer is certain of death, he will not attempt an escape. How do you stop them from escaping?
* One hundred people are in line to board a plane which has exactly 100 seats. Each passenger has a ticket assigning them to a specific seat, and the passengers board one at a time. The first person to board is drunk, picks a random seat, and sits in it. The remaining passengers board; if they find their assigned seat empty, they sit in it. If they find their seat taken, they pick a random seat to sit in. Everyone boards, and is seated. What is the probability that the final person who boards gets to sit in their assigned seat?
* A bag contains N socks, some of which are black, and some of which are red. If two random socks are picked, the probability that they are both red is 1/2. What is the smallest possible value of N for which this is possible?
* You shuffle a standard 52-card deck. What is the probability that the first ace appears exactly at the 20th card?
* For a 3 sets tennis game, would you bet on it finishing in 2 sets or 3 sets?
* I have a square, and place three dots along the 4 edges at random. What is the probability that the dots lie on distinct edges?
* A quantum coin exists in superposition: ǀψ⟩ = αǀH⟩ + βǀT⟩. You can measure it in the {H, T} basis or a rotated {Y, N} basis. Design a protocol to simulate a fair 50/50 coin toss, regardless of α and β. (Assume measurement collapses the state.)
* You have 10 people in a room. How many total handshakes if they all shake hands?
* Two decks of cards. One deck has 52 cards, the other has 104. You pick two cards separately from a same pack. If both of two cards are red, you win. Which pack will you choose?
* A group of people wants to determine their average salary on the condition that no individual would be able to find out anyone else's salary. Can they accomplish this, and, if so, how?
* At a fantasy auction, a dragon egg has a 60% chance to hatch a gold-producing dragon (yielding 100kg gold) and a 40% chance to explode (destroying all gold). The auction uses a "Vickrey" rule: you pay the average of your bid and the runner-up’s bid. How much should you bid if gold is worth $10,000/kg and you’re risk-neutral?
* A spaceship travelling at near relativistic speeds experiences time dilation. If the ship’s stock volatility scales with Lorentz factor γ, derive the adjusted Black-Scholes PDE for a call option expiring in "ship time."
* How many digits are in 99 to the 99th power?
* A line of 100 passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let's say that the nth passenger in line has a ticket for the seat number n.) Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?
* What is the sum of the numbers one to 100?
* Two entangled quantum prisoners choose Cooperate or Defect in superposition. If measured in a Bell basis, payoffs are superposed. Derive the Nash equilibrium when payoffs are \( \sqrt{2} \$ \) for mutual cooperation.
* You have a 3 gallon jug and 5 gallon jug. How do you measure out exactly 4 gallons? Is this possible?
* You have 17 coins and I have 16 coins. We both flip the all coins at the same time. If you have more heads then you win; if we have the same number of heads or if you have less then I win. What's your probability of winning?
* What is the probability you draw two cards of the same color from a standard 52-card deck? You are drawing without replacement.
* You're in a room with three light switches, each of which controls one of three light bulbs in the next room. You need to determine which switch controls which bulb. All lights are off to begin with, and you can't see into one room from the other. You can inspect the other room only once. How can you find out which switches are connected to which bulbs? Is this possible?
* Find the smallest multi-digit prime number that is a palindrome with an even number of digits.
* Two sentient dice adjust their faces after each roll. If you roll a 4, your die subtracts 1 from its faces; your opponent’s die adds 1. What’s the optimal strategy to maximize the expected sum over three rolls?
* In world series, what are the odds it goes 7 games if each team equal chance of winning?
* If the black hole information paradox resolves in favor of "information loss," you owe $1M; else, you gain $1M. Current physics consensus assigns 70% to "no loss." What’s your expected P&L? How would you hedge this?
* Given 100 coin flips, what is the probability that you get an even number of heads?
* There are 5 balls, 3 red, and 2 black. What is the probability that a random ordering of the 5 balls does not have the 2 black balls next to each other?
* A Dyson sphere captures energy from a star with luminosity \( L(t) = L_0 e^{-t/\tau} \). Energy is sold at $0.10/kWh. Calculate the NPV of the sphere if construction costs $10^{30} and \( \tau = 10^6 \) years.
* In a market where 30% of traders are ghosts who see true prices, and humans see noisy prices, derive the probability of an arbitrage opportunity existing. Assume human price errors are normally distributed.
* What is the least multiple of 15 whose digits consist only of 1's and 0's?
* Is 1027 a prime number?
* Does the price of a call option increase when volatility increases?
* What is the singles digit for 2^230?
* In a poker game, players read opponents’ hands with 50% accuracy. Model the equilibrium bluffing frequency when raising all-in on a flush draw. Assume pot odds are 2:1.
* Why might two bonds issued by the same company with the same coupon/maturity be trading at different prices?
* The probability you'll see a falling star in the sky over the course of one hour is 0.44. What's the probability you'll see one over half an hour?
* We are playing Russian roulette, with a standard 6-chamber revolver. I put two bullets in adjacent chambers, spin, point the gun at my head, and pull the trigger. Click. I'm still alive. It's now your turn, and I hand the gun to you, and give you two choices. Would you rather, assuming you want to live, a) Re-spin, aim at your own head and pull the trigger. b) Do not spin, aim at your own head, and pull the trigger.
* You are given two ropes that when lit burn in one hour. Which one of the following time periods CANNOT be measured with your ropes? a) 50 min b) 30 min c) 25 min d) 35 min.
* You have three cards, each labeled n, n+1, n+2, and you don't know n. All cards start facing down so you can see them. You flip one card. If you choose to "stay", you get that card's value. If you don't "stay", then you flip another card. Again, choose to "stay" (and keep the 2nd card's value) or flip the final card and keep the final card's value. Design the optimal strategy to maximize the value of the card that you choose and find the expectation of that value.
* Calculate 119^2 no pen and no paper, in 1 minute.
* How much are you willing to pay to play the following game? You start with $1. You flip a fair coin. If it lands on heads, you double your winnings and flip again. If it lands on tails, the game is over and you collect the money you've won. You continue playing until you land on tails.
* If two people buy 48 oranges. Pat buys 5 times more than Charles. How many oranges did Pat buy?
* Is it ethical to consume meat?
* There is a fair coin (one side heads, one side tails) and an unfair coin (both sides tails). You pick one at random, flip it 5 times, and observe that it comes up as tails all five times. What is the chance that you are flipping the unfair coin?
* You and your friend are playing a game. The two of you will continue to toss a coin until the sequence HH or TH shows up. If HH shows up first, you win. If TH shows up first, your friend wins. What is the probability of you winning?
* What is the probability that a seven-game series goes to 7 games?
* Facebook has a content team that labels pieces of content on the platform as spam or not spam. 90% of them are diligent raters and will label 20% of the content as spam and 80% as non-spam. The remaining 10% are non-diligent raters and will label 0% of the content as spam and 100% as non-spam. Assume the pieces of content are labelled independently from one another, for every rater. Given that a rater has labelled 4 pieces of content as good, what is the probability that they are a diligent rater?
* Say you draw a circle and choose two chords at random. What is the probability that those chords will intersect?
* 1/1000 people have a particular disease, and there is a test that will say you have the disease 98% of the time if you do have. If you don't have the disease, there is a 1% false positive rate. If someone tests positive, what are the odds they have the disease?
* There are 50 cards of 5 different colors. Each color has cards numbered between 1 to 10. You pick 2 cards at random. What is the probability that they are not of same color and also not of same number?
* A fair six-sided die is rolled twice. What is the probability of getting 1 on the first roll and not getting 6 on the second roll?
* What is the expected number of rolls needed to see all 6 sides of a fair die?
* Three friends in Seattle each told you it's rainy, and each person has a 1/3 probability of lying. What is the probability that Seattle is rainy? Assume the probability of rain on any given day in Seattle is 0.25.
* Say you roll three dice, one by one. What is the probability that you obtain 3 numbers in a strictly increasing order?
* Three ants are sitting at the corners of an equilateral triangle. Each ant randomly picks a direction and starts moving along the edge of the triangle. What is the probability that none of the ants collide? Now, what if it is k ants on all k corners of an equilateral polygon?
* What is the expected number of coin flips needed to get two consecutive heads?
* How many cards would you expect to draw from a standard deck before seeing the first ace?
* A and B are playing a game where A has n+1 coins, B has n coins, and they each flip all of their coins. What is the probability that A will have more heads than B?
* Say you are given an unfair coin, with an unknown bias towards heads or tails. How can you generate fair odds using this coin?
* Say you have N i.i.d. draws of a normal distribution with parameters μ and σ. What is the probability that k of those draws are larger than some value Y?
* A fair die is rolled n times. What is the probability that the largest number rolled is r, for each r in 1..6?
* What is the singles digit for 2²³⁰?
* f A, B, and C are integers between 1 and 10 (inclusive). How many different combinations of A, B, and C exist such that A<B<C?
* What is 89²?
* I roll a die up to three times. You can decide to stop and choose the number on the die (where the number is your payoff) during each roll. What's your strategy?
* If you look at a clock and the time is 12:15, what is the angle between the hour and the minute hand?
* A car travels a distance of 60 miles at an average speed of 30 mph. How fast would the car have to travel for the same 60 mile distance home to average 60 mph over the entire trip?
* How many Hershey's chocolate bars were sold in the US last year?
* You have a five-gallon jug and a three-gallon jug. You must obtain exactly four gallons of water. How will you do it?
* You are faced with two doors. One door leads to your job offer (that's the one you want!), and the other leads to the exit. In front of each door is a guard. One guard always tells the truth. The other always lies. You can ask one question to decide which door is the correct one. What will you ask?
* You are playing a card game against one opponent. The game starts with 21 cards on a table. You and your opponent alternate turns, and during each turn, a player may pick up 1, 2 or 3 cards. The winner is the person that picks up the last card. You go first. What is your first move, and what is the optimal strategy to win this game?
* You are given 9 marbles that look the same, but 1 of them weighs slightly less than the other 8. You are also given a balance scale. What is the least number of times you could use the balance to determine which of the marbles is the lighter one? (Also explain the different balance weightings you would perform).
* Four people are on one side of a bridge at night and would like to cross it. Only two can cross at a time. In order to cross, a flashlight must be used. There is only one flashlight. The four people each take different amounts of time to cross the bridge, so if two people cross together, they will cross at the speed of the slower person (since they need to be together and use the flashlight). Describe how all four people can reach the other side of the bridge in 17 minutes, with the following times for each person to cross as follows: Person A: 1 minute; Person B: 2 minutes; Person C: 5 minutes; Person D: 10 minutes.
* A company has ten machines that produce gold coins. One of the machines is producing coins that are a gram light each. How do you tell which machine is making the defective coins with only one weighing?
* You have 18 blue socks and 14 black ones in a drawer. It is very dark. How many do you have to pull out before you have matching pair?
* Say you are driving on a one-mile track. You do one lap at 30 miles an hour. How fast do you have to go on the second lap to average 60 miles an hour?
* You and your friend go out to dinner together, and the bill is $25. You and your friend each pay $15 in cash which your waiter gives to the cashier. The cashier hands back $5 to the waiter. The waiter keeps $3 as a tip and hands back $1 to each of you. So, you and your friend paid $14 each for the meal, for a total of $28. The waiter has $3, and that makes $31. Where did the extra dollar come from?
* A bear walks north one mile west one mile and south one mile and ends up in the same place. What color is the bear?
* There are 3,182 players in a head-to-head knockout tennis tournament. How many matches must be played to crown a winner?
* How many times a day do a clock's hands overlap?
* There are three boxes of eggs. In each box is either big eggs, small eggs or big and small eggs. The boxes are labelled "big," "small," and "mixed," but every box is mislabelled. What is the least number of boxes you can open to know which eggs are in which box?
* You are blindfolded and sit in front of a table. On the table is a large number of coins, 10 of which have heads facing up. How do you split the group of coins into two groups such that the same number of coins are heads-up in each group? Note: You don't know how many coins there are and you can't tell which side is facing up in any way.
* It's 3:30pm. What is the angle formed by the hour hand and the minute hand?
* McNuggets come in boxes of 6, 9, and 20. What is the largest number of nuggets that it is not possible to obtain by purchasing some combination of these boxes?
* You own a pet store. If you put in one canary per cage, you have one canary too many. If you put in two canaries per cage, you have one cage too many. How many canaries and cages do you have?
* A man leaves home for a mountain at 1pm and reaches the top at 3pm. The following day he departs from the top at 1pm and gets home at 3pm, by following the same path as the day before. Was he necessarily ever at the same point on the path at the same time on both days?
* You drive to the store at 20 mph and return by the same route at 30 mph. Discounting the time spent at the store, what was your average speed?
* You have 100 balls (50 black balls and 50 white balls) and 2 buckets. How do you divide the balls into the two buckets so as to maximize the probability of selecting a black ball if 1 ball is chosen from 1 of the buckets at random?
* A car travels a distance of 60 miles at an average speed of 30 mph. How fast would the car have to travel the same 60 mile distance home to average 60 mph over the entire trip?
* You are given 12 balls and a scale. Of the 12 balls, 11 are identical and 1 weighs EITHER slightly more or less. How do you find the ball that is different using the scale only three times AND tell if it is heavier or lighter than the others?
* Design a voting system for three candidates that respects unanimous preferences and avoids dictatorships. What’s the minimum number of voters required?
* You've got a 10 x 10 x 10 cube made up of 1 x 1 x 1 smaller cubes. The outside of the larger cube is completely painted red. On how many of the smaller cubes is there any red paint?
* There are 10 black socks and 10 white socks (no left-right distinction) in the wardrobe. Your task is to draw the minimum number of socks at random to be sure you have a pair of a single color. How many socks should you draw?
* Aliens offer you a perpetuity paying $1/year, but their time dilation field makes your discount rate follow a hyperbola 1/(1 + t) instead of exponential. What’s the present value?
* Assuming that temperature varies continuously, prove that there are always two opposite points on the Earth's surface that have the same temperature.
* A rabbit sits at the bottom of a staircase with n stairs. The rabbit can hop up only one or two stairs at a time. What kind of sequence is formed by the different ways possible for the rabbit to ascend to the top of the stairs of length n=1,2,3...?
* You pick one of two envelopes containing $X and $2X respectively for some value X. After seeing $10 inside, should you switch? Calculate the expected value of switching.
* A. B & C live together and share everything equally. One day A brings home 5 logs of wood, B brings 3 logs and C brings none. Then they use the wood to cook together and share the food. Since C did not bring any wood, he gives $8 instead. How much to A and how much to B?
* A group has 70 members. For any two members X and Y there is a language that X speaks but Y does not, and there is a language that Y speaks but X does not. At least how many different languages are spoken by the members of this group?
* An 8x8 chessboard can be entirely covered by 32 dominoes of size 2x1. Suppose we cut off two opposite corners of chess (i.e. two white blocks or two black blocks). Prove that now it is impossible to cover the remaining chessboard with 31 dominoes.
* There are 51 ants sitting on top of a square table with side length of 1. If you have a square card with side 1/5, can you put your card at a position on the table to guarantee that the card encompasses at least 3 ants?
* What is the minimum number of colors needed to paint the plane so no two points 1 unit apart share the color?
* A bond prospectus states: “This bond will default if and only if its default cannot be proven in ZFC set theory.” What is the bond’s credit rating?
* Assume 100 zombies are walking on a straight line, all moving with the same speed. Some are moving towards left, and some towards right. If a collision occurs between two zombies, they both reverse their direction. Initially all zombies are standing at 1 unit intervals. For every zombie, you can see whether it moves left or right. Can you predict the number of collisions?
* Two bullets are put into a gun's round barrel consecutively, which has capacity of 6. The gun is shot once, but no bullet is fired. Does rolling the barrel (shuffling) before next shot increase the probability of firing a bullet?
* A horn is formed by rotating 1/x around the x-axis from 1 to infinity. What's the probability that a random point inside the horn has x-coordinate less than 2?
* Estimate the number of piano tuners in Chicago if pianos can tune themselves using AI. Provide a 95% confidence interval.
* We have a weighted coin which shows a head with probability p, (0.5<p<1). How do we get a fair toss from this? That is, how do we toss this coin in such a way that we can have probability of winning = losing = 50%?
* Calculate the VaR of a portfolio whose returns follow a nowhere-differentiable function (e.g., Weierstrass curve) with Hausdorff dimension 1.5.
* You’re on a game show with countably infinite doors. Behind one is a car; the rest have goats. After you pick a door, the host (who knows what’s behind all doors) opens all but one other door, revealing goats. Should you switch?
* There is a regular die and a special invisible die. You know that regular die has integers 1 to 6, but don't know what's on the invisible dice. After tossing, I speak the sum of outcome of both die. It so happens that the outcome is an integer between 1 to 12, with equal probability (1/12 each). Can you guess what are the numbers printed on special invisible dice?
* Two witches make a nightly visit to an all-night coffee shop. Each arrives at a random time between 0:00 and 1:00. Each one of them stays for exactly 30 minutes. On any one given night, what is the probability that the witches will meet at the coffee shop?
* In a repeated Prisoner’s Dilemma, what is the Nash equilibrium?
* A stick is broken into 3 parts, by choosing 2 points randomly along its length. With what probability can it form a triangle?
* p and q are two points chosen at random between 0 & 1. What is the probability that the ratio p/q lies between 1 & 2?
* A very innocent monkey throws a fair die. The monkey will eat as many bananas as are shown on the die, from 1 to 5. But if the die shows '6', the monkey will eat 5 bananas and throw the die again. This may continue indefinitely. What is the expected number of bananas the monkey will eat?
* What is the expected number of coin tosses required to get n consecutive heads?