Interview prep · Brain Teasers
Brain Teasers interview questions
Brainteasers and mental math test how you think when you cannot rely on preparation — which is precisely why some interviewers still love them. Nobody expects genius; they want to watch you stay calm, structure an unfamiliar problem, and do clean arithmetic under observation.
The gentlest version is pure mental math: quick multiplication, percentages, and estimation, sometimes rapid-fire. The next tier brings probability and expected value — coin and dice setups where the test is defining outcomes carefully before computing. The hardest versions are market-sizing and logic puzzles, where there is no single right number and the interviewer grades the decomposition: stating assumptions, breaking the problem into estimable pieces, and sanity-checking the result.
The winning habit is thinking out loud. A structured attempt that lands near the answer scores better than a silent pause followed by a guess, because the process is the product. Drill arithmetic until it is fast, learn the handful of classic puzzle structures, and practice narrating your reasoning as you go.
Brain Teasers: what candidates ask
Are brainteasers still common in investment banking interviews?+
It varies a great deal by firm, region, and individual interviewer. Pure puzzle questions are generally more associated with trading and quant seats than with banking, but mental math and estimation questions still appear in IB processes often enough that skipping them is a gamble. The math practice pays off regardless — paper LBOs and accretion questions use the same muscle.
Can I ask for a moment to think or use paper?+
Usually, yes — a brief, composed pause to structure your approach reads far better than an instant wrong answer. Whether paper is available depends on the format, so practice both ways. Whatever you do, narrate your reasoning: interviewers ask these questions to watch you think, and silent calculation wastes the one thing being graded.
How do I get faster at mental math?+
Short daily practice beats cramming. Drill multiplication tables beyond ten, percentage shortcuts, and decomposition tricks — breaking awkward numbers into round pieces — then rebuild speed under mild time pressure. Estimating first and refining afterward also helps: an interviewer would rather hear a fast approximation with a stated error direction than watch you grind out exact digits.
Every Brain Teasers question in the bank
All 118 published questions from Brain Teasers & Mental Math — each links to its own page. Free ones show the full model answer.
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- You roll one fair six-sided die. What is the probability you roll a number greater than 4?
- Flip a fair coin twice. What is the probability of getting at least one head?
- Draw one card from a standard 52-card deck. What is the probability it is a face card (J, Q, or K)?
- Draw one card from a 52-card deck. What is the probability it is a heart OR a king?
- What is the goal of a market-sizing question in an IB/consulting interview, and how should you open your answer?
- Why should you always 'round to nice numbers' in a market-sizing question, and what's the risk?
- An interviewer says 'just give me a number, fast' for a market size. How do you handle the time pressure without losing rigor?
- Mental math: what is 17 × 24?
- A stock falls 50%, then rises 50%. Where does it end up versus the start?
- Rule of 72: roughly how long does it take money to double at 8% annual compounding? At 12%?
- Two fair six-sided dice are rolled. What is the probability the sum equals 7?
- You flip a fair coin 3 times. What is the probability of getting exactly two heads?
- Expected value: a game costs $10 to play. You roll one fair die and win $3 times the number shown. Should you play?
- Fraction/ratio estimation: a company has revenue of $4.2bn and you estimate net margin around 11%. Roughly what is net income?
- Probability / EV: a coin is biased so P(heads) = 0.6. You win $10 on heads and lose $5 on tails. What is the expected value per flip?
- Mental math: what is 1/8 as a percentage, and 7/8?
- Mental math: what is 25 × 32?
- Quick percentage trick: what is 4% of 75?
- A perpetuity pays $50 per year forever. At a 5% discount rate, what is it worth today?
- A stock rises 25%, then falls 20%. Where does it end up versus the start?
- Quick sum: what is 1 + 2 + 3 + … + 100?
- You roll a fair die and flip a fair coin. What is the probability of getting an even number AND heads?
- Interest rates rise. Which bond's price falls more: a 2-year or a 10-year bond (same coupon and credit)?
- What is 2^10, and why is it a useful anchor for compounding math?
- A retailer buys a product for $60 and sells it for $80. What is the gross margin %? The markup %? Why do they differ?
- Ten people at a meeting each shake hands exactly once with every other person. How many handshakes?
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- Roll two fair dice. What is the probability the two dice show the same number (a double)?
- Two cards are drawn without replacement from a 52-card deck. Given the first card is a king, what is the probability the second card is also a king?
- You flip a fair coin 10 times. What is the probability of getting exactly 5 heads?
- A jar has 3 red and 2 blue marbles. You draw 2 without replacement. What is the probability of one red and one blue (in any order)?
- Roll three fair dice. What is the probability that all three show different numbers?
- Flip a fair coin 4 times. What is the probability you get more heads than tails?
- You roll a die. If even, you win that many dollars; if odd, you lose that many dollars. What is the expected value of the game?
- Roll two dice. What is the probability the sum is a prime number?
- A coin is biased with P(heads) = 0.7. You flip it 3 times. What is the probability of getting exactly one head?
- Explain the difference between a top-down and a bottom-up market-sizing approach.
- Estimate the annual market size (revenue) for coffee sold in the United States.
- Estimate the number of smartphones sold in the US per year.
- Estimate how many piano tuners work in Chicago. (The classic Fermi problem.)
- Estimate the annual market size for diapers in the United States.
- An interviewer asks you to size a market and you realize one assumption could be off by 3x. What do you do?
- Estimate the number of Starbucks stores in the United States.
- Estimate the annual revenue of a single mid-volume McDonald's restaurant in the US.
- Estimate the number of pizzas eaten in the US per day.
- What is a good closing 'sanity check' technique for any market-sizing answer?
- Estimate the number of windows in New York City.
- Estimate the annual US market for pet food.
- Estimate how much money is spent on Halloween candy in the US each year.
- Estimate the number of dentists in the United States.
- Estimate the US market size for gym memberships (annual revenue).
- Estimate the annual US market for bottled water (revenue).
- Estimate the number of ATMs in the United States.
- Estimate the annual US market for movie theater tickets (box office revenue).
- Mental math: what is 88 × 88?
- Mental math shortcut: square any two-digit number ending in 5, e.g. 35² and 65².
- Mental math: what is 13 × 17? (multiply two numbers both in the teens)
- A stock drops 20%. What percentage gain is needed to break even?
- An investment compounds at 6% per year. Approximately how many times does it multiply over 36 years?
- Roll two fair dice. What is the probability of at least one 6?
- I flip a fair coin until the first head appears. What is the expected number of flips?
- Two cards are drawn without replacement from a standard 52-card deck. What is the probability both are aces?
- Clock angle: what is the angle between the hour and minute hands at 3:15?
- Clock angle: how many degrees does the hour hand move in one minute, and how many times per day are the hands exactly overlapping?
- The 8-ball weighing puzzle: you have 8 identical-looking balls; one is slightly heavier. Using a balance scale, find the heavy one in the minimum number of weighings.
- River crossing: a farmer must take a wolf, a goat, and a cabbage across a river. The boat holds only the farmer plus one item. Left alone, the wolf eats the goat, and the goat eats the cabbage. How does he get all three across?
- Mental math / valuation: a company trades at 14x EV/EBITDA with EBITDA of $250m. What is enterprise value, and if net debt is $400m, what is equity value?
- Estimate (Fermi): how many ping-pong balls fit in a typical office conference room?
- Probability: you roll a fair die repeatedly. What is the expected number of rolls to see your first 6?
- Rule of 72 covers doubling. Roughly how long does money take to TRIPLE and to QUADRUPLE at 9% a year?
- Reverse rule of 72: an investment doubled in 6 years. Roughly what annual compound return did it earn?
- A cash flow of $100 arrives next year and grows 3% forever. At an 8% discount rate, what is it worth today?
- A perpetuity pays $60 per year and trades at a price of $800. What discount rate is the market implying?
- Two fair dice are rolled. What is the probability the sum is 10 or higher?
- Draw two cards from a standard 52-card deck without replacement. What is the probability they are the same suit?
- Market sizing: estimate the number of gas stations in the US. What structure is the interviewer looking for?
- A company grows revenue 10% and its operating margin expands from 20% to 25%. What happens to operating profit?
- Breakeven math: fixed costs are $200k per year, price is $50 per unit, variable cost is $30 per unit. How many units to break even?
- A company with a 30% gross margin cuts prices by 10%. How much must volume rise to keep gross profit dollars flat?
- Mental math: compute 96² using the base-100 method.
- Fast trick: multiply by 11 in your head — what are 45 × 11 and 78 × 11?
- Pattern: what comes next in the sequence 2, 6, 12, 20, 30, … ?
- Pipe A fills a tank in 3 hours; pipe B fills it in 6 hours. Running together, how long to fill the tank?
- Two trains 100 miles apart drive toward each other, each at 25 mph. A fly leaves one train at 50 mph, flying back and forth between them until they collide. How far does the fly travel?
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- A test for a disease is 99% accurate (99% sensitivity and 99% specificity). The disease affects 1 in 10,000 people. You test positive. What is the probability you actually have it?
- Roll a fair die repeatedly. What is the expected number of rolls to see all six faces at least once (the coupon collector problem)?
- Roll two fair dice. Given the sum is even, what is the probability both dice are even?
- Roll a single die. If it lands on 6, you roll again and add; you keep rolling and adding as long as you roll a 6. What is the expected total?
- Two players alternately flip a fair coin; the first to flip heads wins. What is the probability the player who goes first wins?
- Roll two dice. Given that at least one die shows a 4, what is the probability the sum is 7?
- You draw 5 cards from a 52-card deck. What is the probability all five are spades (a spade flush from a fresh deal)?
- A bag has 2 fair coins and 1 two-headed coin. You pick one at random and flip a head. What is the probability you picked the two-headed coin?
- In a best-of-3 series, each game is independent and you win each with probability p = 0.6. What is the probability you win the series?
- You draw cards one at a time from a shuffled 52-card deck. What is the expected number of cards drawn to get the first ace?
- How many gas stations are there in the United States? Walk through your estimate.
- Estimate how many golf balls fit inside a standard school bus.
- Estimate the total annual revenue of the US airline industry (domestic passenger).
- Estimate the weight of a fully loaded Boeing 747 at takeoff (order of magnitude).
- Estimate how many cars are sold (new) in the United States each year.
- Estimate the total addressable market (TAM) for ride-sharing in a major US city like New York.
- Estimate how many barbers/hair salons are needed to serve the US population.
- Estimate the global market size for toothpaste (annual revenue).
- Estimate how many commercial flights take off worldwide each day.
- Estimate the number of streetlights in a city of 1 million people.
- Estimate the global revenue of the smartphone app store ecosystem (consumer spend).
- The Birthday Problem: in a room of 23 people, what's roughly the probability that at least two share a birthday — and why is it surprisingly high?
- The Monty Hall problem: 3 doors, a car behind one and goats behind the other two. You pick a door; the host (who knows what's behind them) opens a different door revealing a goat, then offers a switch. Should you switch?
- The 100-floor, 2-egg puzzle: find the highest floor from which an egg can be dropped without breaking, minimizing the worst-case number of drops. You have exactly 2 eggs. What's the optimal strategy and worst-case count?
- The burning-rope puzzle: you have two ropes, each burns in exactly 60 minutes but unevenly (rate varies along the rope). Using only these ropes and matches, how do you measure exactly 45 minutes?
- Probability: a family has two children, and you're told at least one is a boy. What is the probability both are boys? (The classic two-child puzzle.)
- You roll a fair die and may either keep the result or reroll once (you must keep the second roll). Playing optimally, what is your expected value?
- 1% of a population has a disease. A test is 90% accurate both ways (90% sensitivity, 90% specificity). You test positive. What is the probability you actually have the disease?
- You have 25 horses and a track that races 5 at a time, with no stopwatch. What is the minimum number of races to find the 3 fastest horses?
- 100 lockers, all closed. Student 1 toggles every locker, student 2 every 2nd, student 3 every 3rd, … student 100 toggles locker 100. After all 100 students pass, which lockers are open?
- Roll two fair dice. What is the expected value of the LARGER of the two numbers?
- What is the expected number of fair-coin flips to get two heads IN A ROW?
- 100 passengers board a full 100-seat plane. Passenger 1 lost his ticket and sits in a random seat. Each later passenger takes their own seat if free, otherwise a random free seat. What is the probability passenger 100 gets their own seat?
- You flip 3 fair coins. Given that at least one is a head, what is the probability all three are heads?
- At a 10% discount rate, roughly what fraction of a perpetuity's value is captured by just the first 30 years of cash flows?
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