Brain Teaser Questions for Interviews: 10 Classic Puzzles With Answers

The puzzle. A farmer must take a fox, a chicken and a bag of grain across a river. The boat holds the farmer and one item. The fox cannot be left alone with the chicken, and the chicken cannot be left alone with the grain.

The answer. Seven crossings, and the step people miss is bringing the chicken back.

Taking the grain on trip 3 and the fox on trip 5 works just as well. Seven is the minimum.

In the interview. Start by naming the item that causes all the trouble. The chicken is in both conflicts, and the fox and the grain are safe together. That tells you the first trip must be the chicken, and that the chicken can never be left on a bank with either of the others. Then say the state of both banks after every trip, as in the table. If you get stuck after trip 3, ask aloud whether any rule stops an item from coming back.

The puzzle. Two doors, one to freedom and one to a trap. Each has a guard. One guard always tells the truth and the other always lies, and you do not know which is which. You may ask one guard one question.

The answer. Ask either guard: "Which door would the other guard tell me leads to freedom?" Then take the opposite door.

Both guards point to the trap door, so you never need to know who you asked.

In the interview. Explain the idea, not only the question. One question to one guard gives an answer you cannot trust. Routing the question through both guards puts exactly one lie into the answer every time, and an answer that is always wrong is as useful as one that is always right. If asked for a second solution, point at a door and ask, "If I asked you whether this door leads to freedom, would you say yes?" The liar lies about his own lie, so both guards say yes only for the freedom door.

The puzzle. You have 12 coins that look identical. One is fake and has a different weight, and you do not know whether it is heavier or lighter. Using a balance scale three times, find the fake coin and say whether it is heavy or light.

The answer. Number the coins 1 to 12. First weigh 1, 2, 3, 4 against 5, 6, 7, 8. There are three outcomes.

If the first weighing balances, coins 1 to 8 are genuine and the fake is among 9 to 12. Second weighing: 9, 10, 11 against 1, 2, 3.

If 1, 2, 3, 4 is the heavier side, the fake is either a heavy coin among 1 to 4 or a light coin among 5 to 8, and coins 9 to 12 are genuine. Second weighing: 1, 2, 5 against 3, 6, 9.

If 5, 6, 7, 8 is the heavier side, do the same thing with the two groups swapped. Second weighing: 5, 6, 1 against 7, 2, 9.

That covers all 24 possibilities (12 coins, each heavy or light), and each one ends at a different result.

In the interview. Nobody expects this tree from memory. Show the counting argument that shapes it. A weighing has three outcomes, so three weighings can tell apart at most 3 x 3 x 3 = 27 situations, and you have 24. That means every weighing must split the remaining cases into three roughly equal groups, which is why you start with 4 against 4 and leave 4 aside. Splitting 6 against 6 wastes the "balances" outcome. The second idea is in the second weighing: move some suspects to the other pan, take some off, and add a known genuine coin. How the scale changes tells you which group holds the fake. Ask for paper and write down the suspects after each weighing.

The puzzle. How many people do you need in a room before it is more likely than not that two of them share a birthday?

The answer. 23. With 23 people the chance of at least one shared birthday is about 50.7%.

Work out the chance that nobody shares a birthday and subtract it from 1. Assume 365 equally likely birthdays.

1. The first person can have any birthday: 365/365.
2. The second must avoid one day: 364/365.
3. The third must avoid two days: 363/365.
4. Keep going to the 23rd person: 343/365.
5. Multiply all 23 fractions: about 0.493.
6. Chance of at least one match: 1 - 0.493 = 0.507.

In the interview. Say the trick before the arithmetic: "It is easier to find the chance that there is no match and subtract from 1." Then explain why the answer feels too low. You are not asking whether someone matches your birthday. You are asking whether any of the 253 pairs match. Nobody multiplies 23 fractions in their head, so set the product up and state the result.

The puzzle. You have two fuses and a lighter. Each fuse takes exactly 60 minutes to burn from end to end, but it burns unevenly, so half the length is not half the time. Measure exactly 45 minutes.

The answer.

Total: 30 + 15 = 45 minutes.

In the interview. Say why cutting or folding a fuse in half does not work: the burn is uneven, so length tells you nothing about time. Then state the one fact that does hold: a fuse lit at both ends is gone in half its remaining time, however unevenly it burns, because the two flames together must consume the whole fuse. Build up from there. One fuse at both ends gives 30 minutes. A fuse with 30 minutes left, lit at both ends, gives 15.

The puzzle. A lily pad on a pond doubles in size every day. It covers the whole pond on day 30. On which day did it cover half the pond?

The answer. Day 29. If it doubles every day, then the day before it was full it was half full.

In the interview. The wrong answer is day 15, which treats the growth as a straight line. Say "I will work backward from the end" and give the answer in one sentence. Expect the follow-ups in the table. A good closing remark: on day 25 the pond is only about 3% covered (1/32), so something that doubles looks small until just before the end.

The puzzle. Four people must cross a bridge at night with one torch. The bridge holds two people at a time, and anyone crossing needs the torch. They take 1, 2, 5 and 10 minutes to cross, and a pair moves at the slower person's pace. What is the shortest time to get everyone across?

The answer. 17 minutes. The insight is to send the two slowest people across together, so that the 5 minutes is hidden inside the 10, and to use the two fastest as couriers for the torch.

The obvious plan has the fastest person walk everyone over one at a time. It takes 2 + 1 + 5 + 1 + 10 = 19 minutes.

In the interview. Give the 19-minute plan first as a baseline, then ask where the time goes. Its return trips are already as cheap as they can be (1 minute each). The waste is that the 5 and the 10 each cross separately, costing 5 + 10 = 15 minutes. Putting them on the bridge together costs 10. To do that, a fast person must already be waiting on the far side to bring the torch back, which is why 1 and 2 go first. Overall the trips forward fall from 17 minutes (2 + 5 + 10) to 14 (2 + 10 + 2) and the trips back rise from 2 minutes (1 + 1) to 3 (1 + 2), a net saving of 2 minutes.

The puzzle. Three people sit in a circle. Each wears a hat that is black or white. Each can see the other two hats but not their own, and they cannot communicate. They are told, truthfully, that at least one hat is white. A bell rings once a minute, and at each bell anyone who knows their own hat color must say it. In fact all three hats are white. What happens?

The answer. Nobody speaks at the first bell or the second bell. At the third bell all three say "white."

The general rule: if there are k white hats, the people wearing them announce at bell k. With 10 people and nine silent bells, all 10 hats are white and all 10 say so at the tenth bell.

In the interview. State the principle first: silence is information. Each bell that passes without an answer rules out one more case for everybody at once. Then build up from the smallest case. One white hat: its wearer sees only black and speaks at bell 1. Two white hats: each wearer sees one white, hears silence at bell 1, and speaks at bell 2. Three follows the same step. A different hat puzzle, where the group agrees a strategy in advance, is the Prisoners and Hats Problem further down.

The puzzle. A man looks at a photograph of a woman. A friend asks who she is. He says, "Brothers and sisters I have none, but this woman's father is my father's son." Who is the woman?

The answer. His daughter.

1. "My father's son" is either the speaker or one of his brothers.
2. He has no brothers, so "my father's son" is the speaker himself.
3. Replace the phrase: "this woman's father is me."
4. So the woman is his daughter.

In the interview. The usual wrong answer is "himself" or "his sister," which comes from solving the sentence in one pass. Say that you will start with the smallest phrase, resolve it, and substitute it back. The same riddle is often told with "that man's father is my father's son," and the answer is then his son. Listen for which version you were given.

These four use different skills from the ten above: backward reasoning, designing a group strategy, organizing an open question, and binary numbers.

The Five Pirates Problem (Backward Induction)

Five rational pirates, ranked by seniority, must split 100 gold coins. The most senior pirate proposes a split and everyone votes. If at least half of the pirates, including the proposer, vote yes, the split stands. If not, the proposer is thrown overboard and the next most senior pirate proposes. Each pirate wants to survive first, get as much gold as possible second, and would rather see a rival thrown overboard if everything else is equal. What should the most senior pirate propose?

Work backward from the end. Call the pirates A (most senior) to E (most junior). A pirate votes yes only if the offer beats what he would get in the next round.

The answer is 98 coins for the most senior pirate, 0 for the second, 1 for the third, 0 for the fourth and 1 for the fifth.

Say the two-pirate case out loud first, then add one pirate at a time. The mistake to avoid is starting with five pirates and guessing at a "fair" split.

The Prisoners and Hats Problem (Parity)

One hundred prisoners stand in a line, each wearing a red or blue hat. Each can see the hats of everyone in front of them but not their own. Starting from the back, each prisoner must call out a color, and everyone hears every answer. Those who name their own color are saved. The group may agree on a strategy beforehand. How many can be saved for certain?

The answer is 99. The prisoner at the back sees all 99 hats in front of him and says "red" if he sees an odd number of red hats and "blue" if he sees an even number. He has a 50% chance himself, but his answer tells everyone else whether the number of red hats among the other 99 is odd or even.

The second prisoner counts the red hats in front of him. If his count is odd or even in the same way as the announcement, his own hat is blue. If it differs, his hat is red. Each prisoner after him, moving toward the front of the line, does the same sum using the red hats he can see plus the "red" answers he has heard since the first one. Every one of those 99 is certain.

The idea to name is parity: one answer can carry one yes-or-no fact for the whole group. Test your strategy aloud on three prisoners before you claim it works for 100.

Why Are Manhole Covers Round?

There is no single correct answer. The question checks whether you can organize an open-ended answer instead of listing ideas as they occur to you.

Lead with the strongest reason: a round cover cannot fall through its own opening, because a circle has the same width in every direction. A square cover can be turned on edge and dropped through along the diagonal, which is longer than the side. Then add a second group of reasons under one label. Handling: a round cover can be rolled, and it fits the hole whichever way you put it down. Finish by saying which reason you think matters most.

The Poisoned Wine Problem (Binary Encoding)

You have 1,000 bottles of wine and exactly one is poisoned. Ten prisoners can taste any bottles they like, and the poison kills exactly one hour after drinking. You have one hour for the tasting, so you cannot test in rounds. How do you find the poisoned bottle?

Each prisoner is one binary digit. Number the bottles 0 to 999 and write each number in binary with 10 digits. Prisoner 1 sips from every bottle whose first digit is 1, prisoner 2 from every bottle whose second digit is 1, and so on. After the hour, write 1 for each prisoner who died and 0 for each who lived. That 10-digit number is the poisoned bottle. Ten prisoners give 2^10 = 1,024 different patterns, which is enough for 1,000 bottles.

Walk through a small example to prove it. With eight bottles numbered 0 to 7 and three prisoners, bottle 5 is 101 in binary. Prisoners 1 and 3 sip from it and prisoner 2 does not. If exactly prisoners 1 and 3 die, the bottle is 5. If nobody dies, it is bottle 0.

Banking, trading and private equity interviewers like puzzles with a number at the end. They want to hear you set it up out loud before you calculate. Five that come up often:

1. A stock rises 50%, then falls 50%. Where does it end? Down 25%. Start at $100. Up 50% is $150. Down 50% of $150 is $75. The fall is taken from a bigger base, so equal percentage moves do not cancel.

2. How long does money take to double at 8% a year? About 9 years. The rule of 72 says divide 72 by the rate: 72 / 8 = 9. The exact answer is ln 2 / ln 1.08, about 9.01 years, so the rule is close at normal rates.

3. You roll a die and are paid its face value in dollars. You may roll once more and take the second roll instead. What is the game worth? $4.25. A fresh roll is worth 3.5 on average, so keep a 4, 5 or 6 and roll again on a 1, 2 or 3.

4. What is the chance of drawing two aces in a row from a full deck, without putting the first back? 1 in 221. The first card is an ace 4/52 of the time and the second 3/51, so 4/52 x 3/51 = 12/2,652 = 1/221.

5. How many times a day do a clock's hour and minute hands overlap? 22. They meet 11 times every 12 hours, not 12, because the minute hand gains one full lap on the hour hand every 12/11 hours. Twice a day gives 22.

For more spoken finance math, see the finance interview questions and answers.

Some employers set a timed online reasoning test before any interview. The skill is the same as in the puzzles above: use only the information you are given. Format and rules differ by test, so read the instructions you receive from the employer. Three methods carry over to most of them.

Verbal: True, False, Cannot Say

You read a passage and judge a statement against it.

• True: the passage proves the statement, and you can point to the words that do it.
• False: the passage contradicts the statement.
• Cannot say: the passage does not settle it either way.

Cannot say is where points are lost, because the statement often sounds reasonable. If you cannot underline the phrase that proves or disproves it, choose Cannot say. Try this passage:

"In the third quarter, the company launched new software, which contributed to a 15 percent increase in overall revenue. The marketing campaign was the most expensive in the company's history. Initial user feedback has been positive, but the long-term profitability of the software remains to be seen."

Numerical: Scan, Estimate, Calculate

Do not start with the arithmetic.

1. Scan. Read the question, then the table or chart. Which numbers do you need, and in what units (thousands or millions, revenue or profit)?
2. Estimate. Bracket the answer before you calculate.
3. Calculate. Do the exact sum. The estimate catches a wrong key or a misplaced decimal.

Example: a product sold $75,000 out of total sales of $250,000. What share is that?

Any answer option outside 25% to 50% can be crossed off before you calculate.

Deductive: Decide What Must Follow

A deductive test gives you a few statements and asks which conclusion must be true. Treat the statements as true even if they sound odd, and accept a conclusion only if no other reading is possible. Try these:

"All analysts at the firm passed the modeling test. Some people who passed the modeling test work in Chicago."

A second trap is reading "if" backwards. "If the deal closes, the team gets a bonus. The team got a bonus." You cannot say whether the deal closed, because the bonus could have another cause. You can say that if there was no bonus, the deal did not close.

Two habits help. Draw the groups as overlapping circles when the words say "all", "some" or "none". Rewrite each "if A then B" as "no B means no A" and check that before you answer.

Check for Negative Marking

Before you start, find out whether wrong answers cost points. If they do not, answer every question, and guess on any you cannot finish. If they do, answer only where you are fairly sure and leave the rest blank.

For practice in the format employers use, try the free SHL practice tests. Official LSAT prep (LSAC LawHub) and GMAT official prep are extra drilling for verbal and numerical reasoning, but those are admission tests in a different format from employer tests.

Reading an answer and producing it aloud under pressure are different skills. Cover the answers above, pick three puzzles, and explain each one to another person or to a recording in under two minutes. If you cannot, you know the answer but not yet the reasoning.

If your interviews include cases, estimation questions are the closest relative of the brain teaser, and they reward the same habit of stating assumptions before calculating. See our market sizing interview questions, try the free market sizing practice and case math drills, or browse the interview questions with example answers.

Soreno has 500+ practice questions across cases, technical and behavioral interviews, including market sizing and estimation questions. You can try it with a 7-day free trial.