Preparing for Fresher Quant Roles

Interview arithmetic isn't a memory contest, it's a small set of restructuring moves applied under a ticking clock. Learn the moves, and the clock stops mattering.

You don't need a calculator to know what 8% for ten years does to your money, a single approximation, added instead of multiplied, gets you within a rounding error every time.

"How many piano tuners are there in Chicago?" isn't a trivia question, it's a test of whether you can break an impossible number into a chain of guessable ones.

A repeatable five-move routine for puzzles you have never seen before, pin the rules, shrink the problem, hunt for the quantity that does not change, guess the shape of the answer, then try to break it.

Twelve coins, one fake of unknown weight, three weighings. The puzzle looks like luck but it is pure information budgeting, every weighing has three outcomes, so aim to split the live possibilities into three equal thirds.

A room of people can see everyone's hat but their own. With no communication except silence and timing, they can still pass information, and that trick is the seed of every hat puzzle an interviewer will throw at you.

Two of the most reusable tools in puzzle solving. An invariant is a quantity no legal move can change, which proves things impossible; a monovariant only moves one way, which proves things must end.

Most "expected number of flips/steps until X" puzzles look intractable until you ask one question: what happens after the very first move, and can I write the answer in terms of itself?

The expected value of a sum always equals the sum of the expected values, even when the pieces are wildly dependent on each other. That one fact quietly solves half the hard-looking counting puzzles in an interview.

Candidate A beats candidate B in the final count, what's the chance A was strictly ahead for the entire tally, from the very first vote counted? A clever mirror-image trick turns a hard counting problem into simple subtraction.

HTH and HHT are both three-flip patterns with 1/8 probability each, so why does one take longer, on average, to see than the other? The pattern's own internal structure is the hidden variable.

Instead of building a state machine, imagine an army of gamblers betting on every flip, the optional stopping theorem turns their fair-game payouts into the waiting time itself, with almost no algebra.

Given a biased coin and permission to bet any fraction of your bankroll on repeated flips, how much should you actually bet? Too little leaves growth on the table; too much and you eventually blow up. The Kelly fraction is the exact answer.

You roll a die and can either keep the number shown or pay a cost to reroll, once or repeatedly. The optimal rule is a threshold: keep any number above a cutoff, reroll anything below it, and the cutoff is just an expected-value comparison.

A shuffled deck is turned over one card at a time. You can stop whenever you want and bet the next card is red. Is there a strategy that beats 50/50, and if so, what edge does it give you?

A puzzle gives you a number's remainder against several small divisors and asks you to find it, or to prove it's unique. The Chinese Remainder Theorem is the reusable machinery for turning a stack of remainder clues into one answer.

When a direct count gets messy, find a one-to-one correspondence to something you already know how to count. If every object in the hard pile matches exactly one object in the easy pile, the two piles are the same size.

When events overlap, you can't just add their sizes, you'll double-count the overlap. Inclusion-exclusion is the systematic fix: add the singles, subtract the pairs, add back the triples, and so on.

Sometimes the most useful sentence in a puzzle isn't a stated fact, it's someone announcing they *can't* figure something out. That single admission of ignorance rules out entire branches of possibility for everyone listening.

If you stuff more pigeons than pigeonholes, some hole gets at least two, an observation so obvious it feels useless, until it's the entire proof that a solution must exist, with no construction required.

One of twelve coins is fake, lighter or heavier, you don't know which, and a balance scale gives you exactly three weighings to find it and say which way it's off. The trick is designing weighings whose outcomes are informative even when they balance.

A Fermi estimate isn't one guess, it's a product of several smaller guesses you're more confident about. A decomposition tree is the discipline of choosing which factors to multiply, and it's the difference between a defensible number and a shrug.

The number your decomposition tree spits out is a draft, not an answer. A sanity check runs it through a handful of independent tests, bounds, benchmarks, a second route, before you say it out loud.

'What's the next number in 2, 6, 12, 20, 30, ...?' is really asking whether you have a checklist for sequence puzzles. Differences, ratios, and known sequence families cover almost everything you'll be handed.

Most 'count the ways to...' interview puzzles are recurrences wearing a costume. The skill is translating a story into an equation that relates the answer for n to the answer for smaller n, after that, it's mechanical.

A generating function packs an entire infinite sequence into one polynomial-like expression, where the coefficient of x^n is the n-th answer. Counting problems that look hard as combinatorics often become easy algebra once repackaged this way.

Binary search is the purest example of a whole family of interview puzzles: any question you can answer with a single yes/no query should be attacked by cutting the space of possibilities in half, not by guessing.

With two eggs and a 100-floor building, find the highest safe floor using as few drops as possible, the puzzle that teaches you to balance the cost of a bad early guess against the cost of testing too cautiously.

Making the locally best choice at every step works for some problems and quietly produces the wrong answer for others, the puzzle is learning to tell which is which before you commit to an approach out loud.

The interviewer says "make me a market" and waits. Here is what the game actually is, how to pick your two numbers, and what happens on every turn after your first quote.

A famous puzzle where a visitor announces something everybody already knew, and a hundred people leave the island anyway. The trick is the difference between everyone knowing a fact and everyone knowing that everyone knows it.

You interview candidates one at a time and must hire or reject on the spot. Look at 37 percent of them, then take the next one who beats everybody so far, and you land the very best more than a third of the time.

A market-making game where you can't just quote the 'fair' price forever, once your inventory gets close to its limit, your quotes have to lean the other way, and the puzzle is working out exactly how much.

In an auction where the item's true value is the same for everyone but unknown, winning the auction is itself bad news, it means you were the most overoptimistic bidder in the room.

What actually happens between submitting a quant application and getting an offer, the stages, what each one is really testing, and where most candidates lose ground.

Quant interviewers grade the reasoning, not just the answer, which means silence, even silence that ends in a correct number, is usually a worse outcome than narrating a wrong turn out loud.

A hedge fund trades other people's money under a fee arrangement designed to align the manager's incentives with investor returns, understanding that arrangement explains most of how these firms actually behave.

Base salary, bonus, and P&L-linked pay work very differently across prop firms, hedge funds, and banks, the mix tells you as much about a firm's incentives as the headline number does.

Prop trading firms, hedge funds, banks, and asset managers all hire quants, but they pay for different skills, take different risks, and feel very different to work inside, a map of who's who before you specialize your prep.

Prop firms trade only their own capital, mostly in liquid exchange-traded products, and their whole hiring and culture model, fast, meritocratic, high-variance, follows from that one fact.

A public research portfolio is a small set of well-documented projects that let a hiring manager watch you think, not just read a claim that you can. Three careful pieces beat ten rushed ones.

No single skill makes someone a quant, it's a specific combination of math, programming, statistics, and market intuition, and knowing the shape of that stack tells you where to actually spend study time.

Bit manipulation uses a number's binary representation directly, AND, OR, XOR, and shifts, to pack data tighter and do certain checks in a single CPU instruction instead of a loop.

A recursive function solves a problem by calling itself on a smaller version of the same problem. The call stack is the bookkeeping that makes it work, and understanding those stacked frames is what turns recursion from magic into a tool you can reason about and cost.

A binary search tree keeps data sorted while still allowing fast inserts and deletes, which arrays and linked lists cannot do at the same time. Everything costs O(height), so the whole subject is really about keeping the tree short.

Dijkstra finds the cheapest route from one node to every other node when edge costs are non-negative. It is BFS with a priority queue instead of a plain queue, and it is what sits underneath any routing decision where hops have different prices.

A greedy algorithm takes the best-looking option at every step and never reconsiders. Writing one is easy; the hard part is proving it is optimal, and the exchange argument is the standard tool for that.

A segment tree answers "what is the sum, min or max over positions i to j?" in O(log n) while still letting you change individual values in O(log n). It is the standard answer when prefix sums break because the data keeps updating.

Merge sort, quicksort, heapsort and insertion sort all put things in order, but they differ in worst case, memory, stability and cache behaviour. Knowing which trade-off you are buying is the interview question, and it decides real things like how a book of orders is ranked.

Union-find answers "are these two things in the same group?" and "merge these two groups" in effectively constant time. It is the right structure whenever connections arrive one at a time and you need to track what has become connected to what.

A strikingly simple relationship, the average number of items in a system equals the average rate items arrive times the average time each one spends there, that holds for almost any queue or system, from an order book to a coffee shop line, regardless of the details of how arrivals or service times are distributed.

A model of an insurer's (or a trading desk's) capital cushion steadily growing from premiums or carry but taking random hits from claims or drawdowns, and the mathematics of how likely that cushion is to ever hit zero.

A single-server queue with random arrivals and random service times has a wait time that doesn't grow smoothly as the server gets busier, it blows up, and the M/M/1 model is the cleanest way to see exactly how.

A result about processes that repeat in cycles, each cycle earning some reward and taking some random length of time, showing that the long-run average reward per unit time simply equals the average reward per cycle divided by the average cycle length.

The mathematical object that tells you, at every moment, exactly how much a decision you could walk away from right now is worth, computed by working backward from the last possible moment.