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Every problem from the green book. Search, filter, find your next challenge.
The Problem Five pirates (A, B, C, D, E in descending seniority) must divide 100 gold coins. The most senior pirate proposes a split; if…
Brain Teasers · ~20 min
The Problem An island has 100 tigers and 1 sheep. Tigers can eat the sheep (turning the tiger into a sheep), or do nothing. Tigers are r…
Brain Teasers · ~15 min
The Problem A farmer needs to cross a river with a fox, a chicken, and a bag of grain. The boat holds the farmer plus one item. If left…
Brain Teasers · ~10 min
The Problem At a party with 23 people, what is the probability that at least two share a birthday? (Ignore leap years.)…
Brain Teasers · ~10 min
The Problem Three friends — Alice, Bob, Carol — play a card game. Each draws one card from a standard 52-card deck and holds it to their…
Brain Teasers · ~15 min
The Problem You have two ropes. Each burns end-to-end in exactly 60 minutes, but the burn rate is non-uniform (e.g., the first half…
Brain Teasers · ~15 min
The Problem You have 12 balls, one of which is either heavier or lighter than the other 11 (you don't know which). You have a balance sc…
Brain Teasers · ~15 min
The Problem How many trailing zeros does $100!$ have? Generalize: how many trailing zeros does $n!$ have?…
Brain Teasers · ~10 min
The Problem You have 25 horses and a 5-lane track. You can race 5 horses at a time and observe the order of finish (no timing). What i…
Brain Teasers · ~15 min
The Problem Find the next term in the sequence: $1, 11, 21, 1211, 111221, 312211, ?$…
Brain Teasers · ~15 min
The Problem You have a 6×6×6 box and bricks of size 1×2×4. Can you pack 27 such bricks into the box? (27 bricks × 8 = 216 = 6³, so the v…
Brain Teasers · ~15 min
The Problem You have two cubes. You can write a single digit (0-9) on each face of each cube. By arranging the two cubes side by side, y…
Brain Teasers · ~15 min
The Problem You're in a room with 3 doors. Behind one door is a job offer; behind the other two, nothing. You pick a door. The interview…
Brain Teasers · ~15 min
The Problem Alice wants to send a valuable item to Bob through a courier. The courier is dishonest and will steal anything sent unlocked…
Brain Teasers · ~15 min
The Problem You have a bag with 20 red balls and 20 blue balls. Each turn, you randomly draw 2 balls: - If they're the same color: disca…
Brain Teasers · ~15 min
The Problem You're in a room with 3 light switches, all off. Each controls one of 3 light bulbs in another room (you can't see the bulbs…
Brain Teasers · ~15 min
The Problem Three quants — Alice, Bob, Carol — want to find their average salary without revealing any individual salary to the others.…
Brain Teasers · ~10 min
The Problem There are 10 piles of coins. Each pile has 10 coins. One pile contains counterfeit coins that weigh 9 grams each; the other…
Brain Teasers · ~15 min
The Problem Three bags contain apples, oranges, and a mix (apple + orange). Each bag is labeled — "apples", "oranges", or "mixed" — but…
Brain Teasers · ~10 min
The Problem A king has 3 wise men. He places a hat on each one's head, drawn from a collection of 3 white hats and 2 black hats. Each wi…
Brain Teasers · ~15 min
The Problem If you break a clock face at two random points, what is the probability that the three resulting pieces can form a triangle?…
Brain Teasers · ~15 min
The Problem You have integers 1 through 100. Two distinct integers are removed. You can examine the remaining 98 integers (in any order,…
Brain Teasers · ~15 min
The Problem You have 10 machines producing coins. One machine produces counterfeit coins weighing 9 grams; the others produce genuine co…
Brain Teasers · ~15 min
The Problem A drawer has 10 red socks and 10 blue socks, all mixed up. You're in the dark and can only pull one sock at a time. What is…
Brain Teasers · ~5 min
The Problem At a party with $n$ people, each person shakes hands with some subset of the others (possibly none, possibly all). Prove tha…
Brain Teasers · ~15 min
The Problem At a party of $n$ people, each pair either has met or hasn't. Prove that there are always two people at the party with the s…
Brain Teasers · ~15 min
The Problem A square table has side length 1. Four ants start at the four corners. Each ant picks a direction (clockwise or counterclock…
Brain Teasers · ~15 min
The Problem You have 9 coins, one of which is counterfeit (lighter or heavier — you don't know which). You have a balance scale. Find t…
Brain Teasers · ~25 min
The Problem 100 prisoners are each assigned a number 1-100. A room has 100 boxes, each containing a slip with a number 1-100 (a random p…
Brain Teasers · ~25 min
The Problem Prove that a number is divisible by 9 if and only if the sum of its digits (in base 10) is divisible by 9. Equivalently: $n…
Brain Teasers · ~15 min
The Problem On an island, there are 13 red chameleons, 15 green, and 17 blue. When two chameleons of different colors meet, they both ch…
Brain Teasers · ~20 min
The Problem Prove by induction: if you split a pile of $n$ coins into two piles of $a$ and $b$ coins ($a + b = n$), and repeat this on e…
Brain Teasers · ~15 min
The Problem A chocolate bar has $m times n$ squares (in a grid). You can break it along any horizontal or vertical line, one break at a…
Brain Teasers · ~15 min
The Problem A race track has fuel at certain points along its length, totaling exactly enough fuel for one complete lap. Prove that th…
Brain Teasers · ~15 min
The Problem Prove by contradiction that $sqrt{2}$ is irrational.…
Brain Teasers · ~15 min
The Problem 100 prisoners wear hats, each hat a color from a set of 100 colors (each color used at least once, possibly with repeats). E…
Brain Teasers · ~25 min
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Brain Teasers · ~30 min
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Brain Teasers · ~30 min
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Brain Teasers · ~25 min
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Brain Teasers · ~25 min
Differentiation is the tool that turns slopes into numbers. Let's make sure the basic machinery is operational. Compute the derivative of $…
Calculus · ~10 min
Finding extrema is a recurring interview task — usually wrapped in some story about "optimising a payoff" or "minimising cost". Strip the st…
Calculus · ~15 min
Indeterminate forms are the gateway to asymptotic analysis. L'Hôpital's rule is the cleanest way through, but only when the hypotheses are a…
Calculus · ~15 min
Integration by parts is the product rule read backwards. It is the workhorse technique for integrals involving a product of "algebraic" and…
Calculus · ~15 min
Solids of revolution show up in volume calculations for everything from fuel tanks to option payoff regions. The disk method turns a 2D area…
Calculus · ~15 min
Continuous random variables live or die by their pdf. Computing expectation, variance, and moments is integration in a thin disguise. Let $…
Calculus · ~20 min
Multivariable calculus generalises the single-variable toolkit. Two operations matter most for quant work: partial derivatives (sensitivitie…
Calculus · ~20 min
Taylor series let you replace a nasty function with a polynomial — the foundation of almost every approximation in quant finance (Black-Scho…
Calculus · ~20 min
Newton's method is the iterative root-finder behind everything from implied volatility solvers to log-likelihood maximisation. The recipe is…
Calculus · ~15 min
Constrained optimisation is the bread and butter of portfolio theory and econometrics. Lagrange multipliers turn a "maximise $f$ subject to…
Calculus · ~25 min
Separable ODEs are the simplest non-trivial differential equation: when you can shuffle all the $y$'s to one side and all the $x$'s to the o…
Calculus · ~15 min
First-order linear ODEs cover an enormous range of quantitative models — interest accrual, cooling bodies, exponential mixing, and (with the…
Calculus · ~15 min
Second-order linear ODEs with constant coefficients model every damped oscillator, every resonant circuit, and (in finance) many short-rate…
Calculus · ~20 min
Nonhomogeneous second-order ODEs model forced oscillators — and resonance is the famous edge case every quant interview loves. Solve the in…
Calculus · ~25 min
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Calculus · ~30 min
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Calculus · ~25 min
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Calculus · ~30 min
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Calculus · ~25 min
The Problem Let $ ec{u} = (1, 2, 3)$ and $ ec{v} = (4, -1, 0)$. 1. Compute the dot product $ ec{u} cdot ec{v}$. 2. Compute the cross p…
Linear Algebra · ~10 min
The Problem Let $$A = begin{pmatrix} 1 & 1 0 & 1 1 & 0 end{pmatrix}.$$ 1. Find a QR decomposition of $A$ using the Gram-Schmid…
Linear Algebra · ~25 min
The Problem Consider the matrix $$A = begin{pmatrix} 2 & 1 1 & 2 end{pmatrix}.$$ 1. Compute $det(A)$. 2. Find the eigenvalues $…
Linear Algebra · ~20 min
The Problem A real symmetric matrix $A in mathbb{R}^{n times n}$ is called positive semidefinite (PSD), written $A succeq 0$, if…
Linear Algebra · ~25 min
The Problem 1. LU decomposition. For $$A = begin{pmatrix} 2 & 1 & 1 4 & 3 & 3 8 & 7 & 9 end{pmatrix},$$ find a lower-trian…
Linear Algebra · ~25 min
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Linear Algebra · ~30 min
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Linear Algebra · ~30 min
The Problem A fair coin is tossed $n$ times. Let $X$ be the number of heads. 1. What is $mathbb{P}(X = k)$ for $0 leq k leq n$? 2.…
Probability · ~15 min
The Problem A standard 52-card deck is shuffled. You draw cards one at a time without replacement. 1. What is the probability that the…
Probability · ~15 min
The Problem 100 passengers board a plane with 100 seats. Each passenger has an assigned seat. The first passenger is drunk and picks a s…
Probability · ~20 min
The Problem $n$ points are placed independently and uniformly at random on the circumference of a unit circle. 1. What is the probabili…
Probability · ~25 min
The Problem A standard 52-card deck is dealt into 5-card poker hands. Compute the probability of each: 1. Royal flush (A-K-Q-J-10 all s…
Probability · ~15 min
The Problem A rabbit starts at position 0 on the integer lattice $mathbb{Z}$. At each step, it hops forward by an integer amount chosen…
Probability · ~20 min
The Problem A variant of the classic pirate puzzle. Five pirates (A, B, C, D, E in descending seniority) must divide 100 gold coins. The…
Probability · ~15 min
The Problem A chess tournament has $n$ players. Each pair plays exactly one game. A win is 1 point, a draw is 1/2 for each, a loss is 0.…
Probability · ~15 min
The Problem You have $n$ job applicants and $n$ positions. Each applicant has a strict ranking of positions (most to least preferred), a…
Probability · ~15 min
The Problem In a room of $n$ people, each person's birthday is independent and uniformly distributed over 365 days (ignore leap years).…
Probability · ~10 min
The Problem What is the 100th digit after the decimal point in the decimal expansion of $1/7$? Then, generalize: for any prime $p neq…
Probability · ~15 min
The Problem Prove that the cube of any integer has one of the following forms modulo 9: $$n^3 equiv 0, 1, text{ or } 8 pmod{9}.$$ E…
Probability · ~15 min
In a country where every family wants at least one boy, each family continues having children until they have a boy, then stops. Assume each…
Probability · ~10 min
Now flip the rule: in a different country, every family continues having children until they have a girl, then stops. Each birth is independ…
Probability · ~15 min
You have a biased coin that comes up heads with probability $p$ and tails with probability $1-p$. You flip it repeatedly until you see two c…
Probability · ~15 min
You have a biased coin with unknown probability $p in (0,1)$ of heads. You don't know $p$, but you can flip the coin as many times as you l…
Probability · ~20 min
Two players, A and B, take turns throwing darts at a target, with A going first. The first player to hit the bullseye wins. A hits the bulls…
Probability · ~15 min
People line up to play a game. Each person has an independent, uniformly random birthday (ignore leap year, so 365 days). Going through the…
Probability · ~25 min
Three fair six-sided dice are rolled. What is the probability that the three values appear in strictly increasing order (e.g., 2, 4, 5 — but…
Probability · ~15 min
You're on a game show with three doors. Behind one door is a car; behind the other two are goats. You pick door 1. Monty, who knows what's b…
Probability · ~10 min
A population starts with a single amoeba. At each step, every amoeba independently either splits into 2 amoebas with probability $frac{3}{4…
Probability · ~20 min
A jar contains 10 candies: 6 red and 4 blue. You draw 2 candies at random without replacement. Given that at least one of the drawn candies…
Probability · ~15 min
A bag contains 3 coins: 2 are fair ($P(H) = frac{1}{2}$) and 1 is double-headed ($P(H) = 1$). You pick a coin uniformly at random and flip…
Probability · ~15 min
A 6-chamber revolver contains 1 bullet. Before every trigger pull, the cylinder is spun so the bullet is equally likely to be in any chamber…
Probability · ~15 min
You are dealt a 13-card hand from a standard 52-card deck. Compare two conditional expectations: (a) The expected number of aces in your ha…
Probability · ~15 min
The Problem Two friends agree to meet at a cafe between 12:00 and 1:00 PM. Each arrives independently at a uniformly random time in this…
Probability · ~10 min
The Problem A stick of length 1 is broken at a uniformly random point. The longer piece is then broken at a uniformly random point along…
Probability · ~15 min
The Problem Customers arrive at a store according to a Poisson process with rate $lambda$ per hour. 1. Given that exactly $n$ customer…
Probability · ~25 min
The Problem Let $Z sim mathcal{N}(0, 1)$ be a standard normal random variable. 1. Compute $mathbb{E}[Z^n]$ for $n = 0, 1, 2, 3, 4$.…
Probability · ~20 min
The Problem You have $n$ noodles in a bowl. You reach in, pick two free ends uniformly at random, and tie them together. Repeat until no…
Probability · ~15 min
The Problem A portfolio manager holds $N_S$ shares of stock $S$ and wants to hedge using $N_F$ futures contracts on a related asset $F$.…
Probability · ~25 min
The Problem You roll a fair six-sided die once. You can either keep the value shown (in dollars) or pay $$1$ to reroll. After the secon…
Probability · ~15 min
The Problem A deck has cards numbered 1 through $n$, shuffled randomly. You turn over cards one at a time. At any point you may say "sto…
Probability · ~15 min
The Problem Let $X$ and $Y$ be independent random variables with $X sim mathcal{N}(mu_X, sigma_X^2)$ and $Y sim mathcal{N}(mu_Y,…
Probability · ~15 min
The Problem There are $n$ distinct coupon types. Each time you buy a coupon, you get a uniformly random type (independently). What is th…
Probability · ~20 min
The Problem Two bonds default independently with probabilities $p_1$ and $p_2$ over a 1-year horizon. 1. What is the probability that b…
Probability · ~25 min
The Problem Let $X_1, X_2, ldots, X_n$ be i.i.d. $text{Uniform}(0, 1)$ random variables. Define $M_n = max(X_1, ldots, X_n)$ and $m_…
Probability · ~25 min
The Problem Let $X_1, X_2, ldots, X_n$ be i.i.d. $text{Uniform}(0, 1)$ random variables. Define $M_n = max(X_1, ldots, X_n)$ and $m_…
Probability · ~25 min
The Problem Three ants are placed at the three vertices of an equilateral triangle. Each ant picks a direction (clockwise or countercloc…
Probability · ~20 min
The Problem A gambler starts with $$i$ and plays a game where, at each step, they win $$1$ with probability $p$ and lose $$1$ with pr…
Probability · ~25 min
The Problem You roll a fair six-sided die repeatedly. Let $X_n$ be the cumulative sum after $n$ rolls. What is the probability that the…
Probability · ~20 min
The Problem Two players, A and B, watch an infinite sequence of fair coin flips. Player A picks a triplet of heads/tails (e.g. HHH); pla…
Probability · ~25 min
The Problem An urn contains 5 red, 3 green, and 2 blue balls. You draw balls one at a time without replacement until the urn is empt…
Probability · ~15 min
The Problem A drunk man stands at position 0 on the integer lattice $mathbb{Z}$. At each step he moves left or right with equal probabi…
Probability · ~20 min
The Problem You repeatedly roll a fair six-sided die. After each roll, you may either stop (collect the value of the most recent rol…
Probability · ~25 min
The Problem A queue has $n$ people, each holding either a $$5$ bill or a $$10$ bill. Tickets cost $$5$. The cashier starts with no ch…
Probability · ~25 min
The Problem You flip a fair coin until the pattern HHT appears. Let $tau$ be the stopping time (the number of flips needed). 1. What i…
Probability · ~25 min
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Probability · ~30 min
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Probability · ~30 min
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Probability · ~25 min
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Probability · ~35 min
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Probability · ~30 min
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Probability · ~30 min
The Problem Dynamic programming is the workhorse of quant interviews u2014 it powers American-option pricing, optimal-stopping puzzles,…
Dynamic Programming · ~15 min
The Problem You roll a fair six-sided die repeatedly. After each roll you may either stop and walk away with your accumulated total, or…
Dynamic Programming · ~20 min
The Problem The Red Sox and the Yankees are playing a best-of-7 series: the first team to 4 wins takes the trophy. The Red Sox win each…
Dynamic Programming · ~20 min
The Problem You may roll a fair six-sided die up to $n$ times. After each roll you either accept the value shown (the game ends and you…
Dynamic Programming · ~25 min
The Problem A deck has 26 red and 26 black cards. Cards are drawn one at a time without replacement. After each draw you may stop. If yo…
Dynamic Programming · ~25 min
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Dynamic Programming · ~30 min
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Dynamic Programming · ~35 min
Brownian motion is the cornerstone process of quantitative finance. Once you internalize three properties — start at zero, independent Gauss…
Brownian Motion and Stochastic Calculus · ~20 min
First passage times have a brutal punchline: Brownian motion hits every level almost surely, yet the expected time to do so is infinite. Int…
Brownian Motion and Stochastic Calculus · ~25 min
Itô's lemma is the chain rule with a twist. The twist is $(dW_t)^2 = dt$, and the entire edifice of stochastic calculus exists to make that…
Brownian Motion and Stochastic Calculus · ~30 min
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Brownian Motion and Stochastic Calculus · ~35 min
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Brownian Motion and Stochastic Calculus · ~30 min
You're pricing a European option on a non-dividend-paying stock. The trader next to you shouts a series of market moves and wants to know, i…
Option Pricing · ~10 min
A European call and a European put are written on the same non-dividend-paying stock $S$, with identical strike $K$ and maturity $T$. The co…
Option Pricing · ~15 min
An American option gives the holder the right to exercise at any time up to maturity $T$; a European option only at $T$. The extra right sho…
Option Pricing · ~15 min
Assume the stock price follows geometric Brownian motion under the physical measure: $$dS = mu S,dt + sigma S,dW.$$ Let $f(S, t)$ be t…
Option Pricing · ~30 min
For a non-dividend-paying stock with current price $S$, volatility $sigma$, and continuously compounded risk-free rate $r$, derive and stat…
Option Pricing · ~20 min
Using the Black-Scholes formula, derive closed-form expressions for the Greeks of a European call on a non-dividend-paying stock: - Delta $…
Option Pricing · ~25 min
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Option Pricing · ~35 min
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Option Pricing · ~35 min
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Option Pricing · ~30 min
A trader constructs a bull spread on a stock currently trading at $S_0 = $100$ by: - Long 1 call with strike $K_1 = $95$, premium $C_1 =…
Option Portfolios and Exotic Options · ~15 min
An investor expects a major move in a stock currently trading at $S_0 = K = $100$ but has no view on direction. They buy an at-the-money st…
Option Portfolios and Exotic Options · ~15 min
A cash-or-nothing binary call pays a fixed cash amount $Q = $100$ if $S_T > K$ at expiry, and $0$ otherwise. Given: $$S_0 = $100,; K =…
Option Portfolios and Exotic Options · ~20 min
An exchange option (Margrabe option) gives the holder the right, at expiry $T$, to exchange one unit of asset 2 for one unit of asset 1. Pay…
Option Portfolios and Exotic Options · ~25 min
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Option Portfolios and Exotic Options · ~30 min
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Option Portfolios and Exotic Options · ~30 min
You're rebalancing a portfolio with two risky assets. Asset 1 has expected return $mu_1 = 8%$ and volatility $sigma_1 = 15%$; Asset 2 ha…
Other Finance Questions · ~25 min
Your desk runs a $10M equity portfolio. Daily log-returns are assumed normal with mean $mu = 0.0005$ and standard deviation $sigma = 0.02$…
Other Finance Questions · ~20 min
A 5-year annual-coupon bond pays $6%$ on a $100 face value. Its yield to maturity is $y = 5%$. Part A Compute the bond's price, Macaul…
Other Finance Questions · ~20 min
A stock trades at $S_0 = $100$. The continuously compounded risk-free rate is $r = 4%$. The stock will pay a $$2$ cash dividend in 3 mont…
Other Finance Questions · ~15 min
A quant is choosing between the Vasicek and Cox–Ingersoll–Ross (CIR) short-rate models to calibrate to a curve where the policy rate is curr…
Other Finance Questions · ~25 min
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Other Finance Questions · ~35 min
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Other Finance Questions · ~35 min
Swap the values of two integer variables a and b without using a temporary variable. You may use arithmetic or bitwise operations, but y…
Algorithms · ~5 min
Given an array of $n$ integers, remove the duplicates and return the unique elements. Variants - A. You may use extra memory. What…
Algorithms · ~10 min
You are given coefficients $(a_0, a_1, dots, a_n)$ of a polynomial $$p(x) = a_n x^n + a_{n-1} x^{n-1} + cdots + a_1 x + a_0,$$ and a val…
Algorithms · ~15 min
Design a data structure that supports a stream of numbers and, after each new number $x_t$ arrives, returns the average of the most recent $…
Algorithms · ~10 min
Implement a sorting algorithm and discuss its complexity. A. Quicksort Implement quicksort in place. What is its average-case and worst…
Algorithms · ~20 min
Given an array of $n$ distinct elements, generate a uniformly random permutation — every one of the $n!$ orderings must be equally likely. Y…
Algorithms · ~15 min
Given a sorted array of $n$ distinct integers and a target $t$, find the index of $t$ (or report absence) in $mathcal{O}(log n)$ time.…
Algorithms · ~15 min
The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n ge 2$. Implement a function to compute $F_…
Algorithms · ~10 min
Given an array $a_1, a_2, dots, a_n$ of integers (possibly negative), find the maximum sum of any contiguous subarray $a_i + a_{i+1} + cdo…
Algorithms · ~20 min
Given a positive integer $n$, determine whether $n$ is a power of $2$ (i.e. $n = 2^k$ for some integer $k ge 0$) using a single expression.…
Algorithms · ~10 min
Multiply an integer $n$ by $7$ using only bit shifts and a single subtraction (no operator, no loops). Generalise: how would you multip…
Algorithms · ~10 min
You have a fair coin. You want to simulate a biased "coin" that comes up heads with probability $p$ for any $p in [0, 1]$ whose binary expa…
Algorithms · ~20 min
You have $1000$ bottles of wine. Exactly one is poisoned. The poison takes $24$ hours to kill anyone who drinks even a drop, and you have $2…
Algorithms · ~30 min
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Algorithms · ~25 min
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Algorithms · ~25 min
The Problem You want to estimate $mathbb{E}[g(X)]$ where $X$ has a known distribution and $g$ is a payoff function (for instance, $g(x)…
Numerical Methods · ~20 min
The Problem The Black-Scholes-Merton PDE for a derivative with value $V(S, t)$ is $$frac{partial V}{partial t} + rS frac{partial V…
Numerical Methods · ~25 min
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Numerical Methods · ~30 min