Logic Puzzles interview questions and answers

Question 1

You have two ropes. Each takes exactly 1 hour to burn, but they burn unevenly (half a rope might take 5 minutes or 55). Measure exactly 45 minutes.

Accepted Answer

Answer: Light rope A at both ends and rope B at one end, all at the same time. Rope A burns out after 30 minutes. At that moment, light the other end of rope B. Rope B burns out 15 minutes later. Total: 45 minutes. Why it works: Rope A is lit at both ends, so two flames travel toward each other and meet in the middle. Together they always finish one whole rope in half the time, which is 30 minutes. It does not matter that the rope burns unevenly, because you only care about the total amount burned, not where. When A finishes, rope B has been burning from one end for 30 minutes, so it has 30 minutes of rope left. Now you light B's other end too. Two flames again, so that leftover finishes in half the time: 15 minutes. 30 + 15 = 45. Analogy: the one trick to remember: lighting a rope at both ends makes it burn out in half its remaining time, no matter how uneven it is. In an interview: Say the trick first: burning from both ends halves the time. Then A both ends gives 30 minutes, and B both ends from that point gives another 15.

Question 2

You have a 3-litre jug and a 5-litre jug and a tap. Measure exactly 4 litres.

Accepted Answer

Answer: Fill the 5-litre jug. Pour it into the 3-litre jug until the 3 is full. That leaves 2 litres in the 5. Empty the 3-litre jug and pour those 2 litres into it. Fill the 5-litre jug again. Pour into the 3-litre jug until it is full, which needs only 1 litre. The 5-litre jug now holds exactly 4. Why it works: You are only ever allowed four kinds of move: fill a jug, empty a jug, or pour one jug into the other until the first is empty or the second is full. The plan uses the small jug to remove known amounts from the big jug. First you take 3 out of a full 5, leaving 2. You park that 2 in the small jug. Now the small jug has room for only 1 more. So when you fill the big jug again and pour into the small one, it takes just 1, and 4 stays behind. Analogy: track the pair of numbers (small jug, big jug) after every move. Each pour changes them by a known amount, so you can search step by step. In an interview: Walk through it as (small, big) states, naming each fill, empty, or pour. The idea is to use the small jug to draw off known amounts until 4 is left.

Question 3

With two jugs, which target amounts can you measure, and which are impossible?

Accepted Answer

Answer: You can measure any amount that is a multiple of the greatest common divisor (GCD) of the two jug sizes, up to the bigger jug. Nothing else is possible. Examples: Jugs of 3 and 5: their GCD is 1, so you can measure any whole number: 1, 2, 3, 4, 5. Jugs of 4 and 6: their GCD is 2, so you can only measure even amounts: 2, 4, 6. You can never get exactly 5 with these jugs. Why: every move adds or removes a whole jug of water, so any amount you can reach is (some 4s) plus or minus (some 6s). Maths says that set is exactly the multiples of the GCD. Analogy: each move changes your total by a whole jug, so you can only land on combinations of the two jug sizes, which means multiples of their GCD. In an interview: The general rule is the GCD of the two sizes. Give the 4-and-6 case to show why 5 is impossible there.

Question 4

Two doors: one leads to freedom, one to death. Two guards: one always lies, one always tells the truth. You do not know which is which. You may ask one guard one yes/no question. What do you ask?

Accepted Answer

Answer: Ask either guard, "If I asked the other guard which door leads to freedom, which one would he point to?" Then go through the other door, the one he did not name. Why it works: Your question passes through both guards, so it always contains exactly one lie. One lie flips the truth, so the answer you hear is always wrong, and you just do the opposite. If you asked the truth-teller: he honestly tells you what the liar would say, and the liar would point to the death door. You hear death. If you asked the liar: the truth-teller would point to freedom, but the liar lies about that and says death. You hear death. Either way you hear the death door, so pick the other one. Analogy: make the question go through one lie no matter who you ask. Then the answer is guaranteed wrong, so you flip it. In an interview: The core idea: route the question through one lie so the answer is always wrong, then take the opposite door.

Question 5

Same two doors, but now there is one guard and you do not know if he lies or tells the truth. One yes/no question?

Accepted Answer

Answer: Ask, "Would you say that the left door leads to freedom?" If he says yes, take the left door. If he says no, take the right door. Why it works: If he is a truth-teller, he answers honestly. "Yes" means left really is freedom. If he is a liar, he has to lie about his own answer. If left is freedom, he would normally want to say "no" (to mislead you), but the question asks what he WOULD say, so he lies about that and says "yes". Two lies cancel out. So for both guards, "yes" means left is the safe door. Asking "would you say X" instead of "is X" is the whole trick. Analogy: asking "would you say X" makes a liar lie twice, once about reality and once about his own answer, and two lies cancel back to the truth. In an interview: Use the "would you say X" form. A truthful guard answers straight, a lying guard lies twice and it cancels, so the answer is reliable either way.

Question 6

What is the three-gods version, and why is it so much harder?

Accepted Answer

Answer: It is still solvable in three questions using the same "would you say" trick, but it is famously called the hardest logic puzzle ever. For most interviews, knowing the two-guard versions above is enough. The setup: three gods, one always truthful, one always lying, one answers at random. They reply with the words "da" and "ja", and you do not know which word means yes. Two things make it brutal. First, the random god's answers are useless, so you must find a god who is definitely not random before you can trust anything. Second, you do not know their language, so a plain answer of "da" tells you nothing. The fix: the "would you say" trick cancels both the lying AND the unknown language, because the god's answer word ends up meaning the same thing no matter which word is yes. Your first question is designed to point you at a non-random god, and the next two pin down the rest. Analogy: the "would you say X" trick is powerful: it cancels lying and it cancels not knowing their language at the same time. In an interview: Name it as the hardest logic puzzle ever, explain that random plus unknown language is what makes it hard, and show the same "would you say" tool still works.

Question 7

You have 8 identical-looking balls. One is slightly heavier. Using a balance scale, find it in 2 weighings.

Accepted Answer

Answer: Split the balls into groups of 3, 3, and 2. Weigh the two groups of 3 against each other. If one side is heavier, the heavy ball is in that group of 3. If they balance, it is in the group of 2. Then one more weighing finds it. Step by step: Weighing 1: 3 balls vs 3 balls. - If a side drops, the heavy ball is among those 3. - If they balance, the heavy ball is one of the 2 you left off. Weighing 2: - From a group of 3: put 1 ball on each pan, keep 1 aside. If a side drops, that is the ball. If they balance, the one you kept aside is the ball. - From the group of 2: just weigh them against each other. Why groups of 3: a balance scale has three possible results (left heavy, right heavy, balanced), so two weighings can tell apart up to 9 cases. 8 balls fits. Analogy: each weighing gives three results, not two, so split into three groups, not two halves. In an interview: Lead with: a scale gives three outcomes, so divide into thirds. Groups of 3, 3, 2 solve it in two weighings.

Question 8

12 balls, one is the odd one out, but you do not know if it is heavier or lighter. Find it and say heavier or lighter, in 3 weighings.

Accepted Answer

Answer: Yes, it is possible, and 3 weighings is exactly the minimum. There are 12 balls times 2 (heavier or lighter) = 24 cases to tell apart, and 3 weighings give 3 x 3 x 3 = 27 results, so it just fits. The method in short: Weighing 1: 4 balls vs 4 balls. - If they balance, the odd ball is among the other 4. That is now an easy 2-weighing job against known-good balls. - If they do not balance, remember which side went down. The odd ball is among these 8. Weighing 2 and 3: move some of the suspect balls to the other pan and set some aside, following a fixed pattern. By watching whether the scale tips the same way, tips the other way, or balances, you work out both which ball is odd and whether it is heavy or light. The exact ball-swapping pattern is a known table. In an interview, the thing to show is the counting argument (27 results cover 24 cases) and the idea of swapping balls between pans to read the change. Analogy: do not just look at which side is heavier. Move suspects between pans and watch how the tilt changes. That change tells you the direction (heavy or light) too. In an interview: Start with the counting: 27 results cover 24 cases, so it is tight but possible. Then explain the swap-and-watch technique for getting the direction.

Question 9

You have 10 stacks of coins. Every coin weighs 10g, except one whole stack where each coin weighs 9g. You may use a digital scale once. Which stack is fake?

Accepted Answer

Answer: Take 1 coin from stack 1, 2 coins from stack 2, 3 from stack 3, and so on up to 10 coins from stack 10. Weigh all 55 coins together. If none were fake, they would weigh 550g. The reading will be 550 minus the fake stack's number. Example: if the scale shows 547g, you are missing 3 grams, so stack 3 is the fake one. If it shows 541g, stack 9 is fake. Why it works: each fake coin is 1 gram light. Because you took a different number of coins from each stack, the number of grams missing tells you exactly how many fake coins are in the pile, which is the stack number. Analogy: take a unique number of coins from each stack, so the missing weight directly spells out the stack number. In an interview: Sample a different count from each stack so the shortfall in grams is the stack number. One weighing, then simple subtraction.

Question 10

Four people must cross a bridge at night with one torch. The bridge holds at most two. They walk at 1, 2, 5, and 10 minutes. A pair walks at the slower person's speed. Cross everyone in 17 minutes.

Accepted Answer

Answer: 17 minutes. 1. Persons 1 and 2 cross together. 2 minutes. (total 2) 2. Person 1 walks back with the torch. 1 minute. (total 3) 3. Persons 5 and 10 cross together. 10 minutes. (total 13) 4. Person 2 walks back with the torch. 2 minutes. (total 15) 5. Persons 1 and 2 cross together. 2 minutes. (total 17) Why this beats the obvious plan: The obvious plan is to let the fastest person (1) walk everyone across and keep coming back. That costs 19 minutes, because the 10-minute person and the 5-minute person each cross alone. The better idea: send the two slowest people together, so you pay 10 once instead of 10 plus 5. The two fastest people act as torch-carriers who bring it back. Analogy: send the two slowest people at the same time so their walks overlap. Use the two fastest as the ones who carry the torch back. In an interview: Give the answer 17, then the key move: the two slowest cross together so you only pay 10 once, with the two fastest ferrying the torch.

Question 11

A farmer must cross a river with a wolf, a goat, and a cabbage. The boat holds the farmer plus one item. Alone, the wolf eats the goat, and the goat eats the cabbage. How do all four get across?

Accepted Answer

Answer: 1. Take the goat across. Come back alone. 2. Take the wolf across. Bring the goat back with you. 3. Take the cabbage across. Come back alone. 4. Take the goat across. Why it works: The goat is the problem. It cannot be left alone with the wolf (wolf eats goat) and cannot be left alone with the cabbage (goat eats cabbage). The wolf and cabbage are fine together. So the goat has to go first. Then, to bring the wolf over without leaving the wolf and goat alone, you carry the goat back with you. Now the goat is on the starting side again, safe on its own. You take the cabbage across (wolf and cabbage together, fine), then finally come back for the goat. Analogy: the goat is in both dangerous pairs, so it can never be left with the wolf or the cabbage. The move people miss is bringing the goat BACK on a trip. In an interview: The goat is in both bad pairs, so it goes first and last. The clever step is bringing the goat back on trip 3.

Question 12

You have 25 horses and a track with 5 lanes, so 5 race at a time. You have no stopwatch, only the finishing order. What is the fewest races to find the fastest 3 horses?

Accepted Answer

Answer: 7 races. Races 1 to 5: divide the 25 horses into 5 groups of 5 and race each group. Now you know the order within each group. Race 6: race the 5 group winners. Call them A1, B1, C1, D1, E1 in the order they finish. - A1 is the fastest horse overall. - Groups D and E are out completely. Each of their horses is behind at least 3 faster horses, so none can be in the top 3. Race 7: A1 is already known to be 1st. The only horses that could still be 2nd or 3rd are: - A2 and A3 (they were behind only A1) - B1 and B2 (group B lost only to A1) - C1 (group C lost only to A1 and B1) Race those 5. The top 2 finishers are the overall 2nd and 3rd. Analogy: after the race of winners, cut hard. Only horses that could still make the top 3, given what you already know, need one more race. That comes out to exactly 5. In an interview: Answer is 7. The part to explain is race 6 and then which horses are still able to be top 3, which is exactly 5.

Question 13

Monty Hall: 3 doors, a car behind one, goats behind the other two. You pick door 1. The host, who knows what is where, opens door 3 to reveal a goat. Should you switch to door 2?

Accepted Answer

Answer: Yes, switch. Switching wins the car 2 times out of 3. Staying wins only 1 time out of 3. Why: When you first picked, there was a 1 in 3 chance the car was behind your door, and a 2 in 3 chance it was behind one of the other two. The host then opens a goat door from those other two. He always shows a goat, he never opens the car, so he is giving you information. The 2 in 3 chance did not go away, it is now all sitting on the one other door he did not open. Easier to see with 100 doors: you pick 1, the host opens 98 goats, and offers you the single door he carefully avoided. You would switch instantly. Same logic with 3. Analogy: your first guess is probably wrong (2 in 3). The host's rule piles that whole 'probably wrong' chance onto the one door he leaves closed. In an interview: Switch, it wins 2 out of 3. The host's choice is informed, not random, so he moves the 2/3 onto the door he leaves shut. The 100-door version makes it obvious.

Question 14

A family has two children. You are told at least one is a boy. What is the chance both are boys?

Accepted Answer

Answer: 1/3, if all you were told is "at least one is a boy". (It changes to 1/2 if you were told about a specific child, like "the older one is a boy".) Why: For two children, the four equally likely combinations (by birth order) are: boy-boy, boy-girl, girl-boy, girl-girl. "At least one is a boy" only rules out girl-girl. That leaves boy-boy, boy-girl, girl-boy, all equally likely. Only one of those three is two boys, so the chance is 1/3. But "the older child is a boy" rules out both girl-girl AND girl-boy. That leaves boy-boy and boy-girl, so the chance is 1/2. The lesson: the exact wording of the clue matters, so always pin down what you were actually told. Analogy: write out the four equally likely combos, then cross out the ones the clue forbids, then count. In an interview: Answer 1/3 for the usual wording, then flag that 'a specific child is a boy' makes it 1/2. The puzzle is really about reading the condition carefully.

Question 15

Three ants sit on the three corners of a triangle. Each picks a direction along an edge at random and walks. What is the chance none of them collide?

Accepted Answer

Answer: 1/4. Why: The ants avoid each other only if they all walk the same way around the triangle: all clockwise, or all anticlockwise. Any other mix means two ants meet on an edge. Each ant has 2 choices, so there are 2 x 2 x 2 = 8 equally likely combinations. Exactly 2 of them are "all the same direction". So the chance of no collision is 2 out of 8, which is 1/4. The same idea works for n ants on an n-sided shape: the answer is 2 divided by 2 to the power n. Analogy: only two outcomes are safe: everyone goes the same way round. Count all outcomes, count the safe ones, divide. In an interview: 1/4. The only safe cases are all-clockwise or all-anticlockwise, 2 out of 8 equally likely direction choices.

Question 16

There are 1000 wine bottles, exactly one is poisoned. The poison kills in about 24 hours. You have 10 test strips and 24 hours. Which bottle is poisoned?

Accepted Answer

Answer: Number the bottles 0 to 999 and write each number in binary (10 digits is enough, since 2 to the power 10 is 1024). Give each strip one digit position. Strip k gets a drop from every bottle that has a 1 in digit k. After 24 hours, the strips that turned positive are the 1s of the poisoned bottle's number. Read that binary number to get the bottle. Why it works: Each strip answers one yes/no question: "is digit k of the poisoned bottle's number a 1?". Ten yes/no answers are enough to pick one number out of 1024. Example: if strips 0, 1, 3, 4, 6, 7, 9 come back positive, the binary number with 1s in exactly those places is 1011011011, which is 731. Bottle 731 is poisoned. Analogy: each strip is one bit of the final answer. Put a drop on every bottle whose number has a 1 in that bit. In an interview: Binary encoding: each strip reads one bit of the bottle number, and the pattern of positive strips is the answer in binary.

Question 17

The birthday problem: how many people do you need in a room before there is a better-than-even chance two of them share a birthday?

Accepted Answer

Answer: 23 people. Why it feels wrong: People imagine "someone shares MY birthday", which really does need about 253 people for even odds. But the question is about ANY two people matching. With 23 people there are 23 x 22 / 2 = 253 different pairs, and each pair is a chance to match. The chances pile up fast. You calculate it backwards: the chance everyone has a different birthday is 365/365 x 364/365 x 363/365 and so on. For 23 people that drops just below 50%, so the chance of at least one match is just above 50%. Analogy: count pairs, not people. 23 people make 253 pairs, and every pair is a fresh chance to match. In an interview: 23. The trick people miss: it asks about any pair, and 23 people already form 253 pairs, so the odds build much faster than the 'shares my birthday' feeling.

Question 18

You call three friends in another city to ask if it is raining. Each friend tells the truth 2/3 of the time and lies 1/3 of the time. All three say it is raining. What is the chance it is actually raining?

Accepted Answer

Answer: About 96% (26/27), under the usual simplifying assumptions. The trick is to work with the chance that it is NOT raining, which means all three friends lied. Each friend lies with probability 1/3, and (assuming they lie independently) all three lying is (1/3) x (1/3) x (1/3) = 1/27. So the chance they did not all lie, meaning it really is raining, is 1 - 1/27 = 26/27, which is about 0.963, or 96%. Note: this version ignores the prior chance of rain and assumes a friend only lies to flip the true answer. A fully rigorous version would use Bayes' theorem with a base rate. In an interview, state the assumption and give 26/27. Analogy: for 'at least one' questions, flip it: the only way it is NOT raining is if every single friend lied. Multiply those small probabilities and subtract from 1. In an interview: About 96%. I compute the complement: all three lying is one third cubed, which is 1/27, so the chance it is raining is 26/27. I would flag that this ignores the prior probability of rain, which a full Bayesian answer would include.

Question 19

There are 3 light switches outside a room and 3 bulbs inside. The door is shut and you cannot see in. You can flip switches as much as you like, then open the door once. Match each switch to its bulb.

Accepted Answer

Answer: Turn switch 1 on and wait about 10 minutes. Then turn switch 1 off, turn switch 2 on, and open the door right away. - The bulb that is lit is switch 2. - The bulb that is off but warm to the touch is switch 1. - The bulb that is off and cold is switch 3. Why it works: You only get one look, so that look has to tell you more than just on or off. A bulb also gets hot when it has been on. Leaving switch 1 on for a while heats its bulb. By the time you open the door, that bulb is off but still warm, which tells switch 1 and switch 3 apart. Analogy: one look must carry more than one clue. A bulb has two things you can check: is it lit, and is it warm. In an interview: The unlock is that a bulb gives off heat as well as light. Warm one up, swap, and one glance sorts all three.

Question 20

You are on a narrow island covered in dry grass. A fire starts at the west end. The wind blows east, so the fire will sweep across and reach you at the east end in a few hours. You cannot swim and have no tools except the fire. How do you survive?

Accepted Answer

Answer: Take some burning grass and start your own small fire a short way to the east of where you stand. The wind pushes it east, and it burns a strip of the island down to bare ground. Once that strip has burned out, walk onto it. When the big fire arrives from the west, it reaches your burned strip, finds no grass to burn, and stops. Why it works: You cannot fight the fire and you cannot leave the island. But a fire needs fuel. If you burn away the grass in front of you first, in a controlled way, you create a safe patch of bare earth. The main fire dies when it reaches it. Analogy: you cannot beat the fire or escape, but you can remove its fuel ahead of time. Burn your own safe zone before the big fire gets there. In an interview: This is the 'backfire' answer: use a small controlled fire to burn away your own fuel, then stand on the cleared ground.

Question 21

A man lives on the 10th floor. Every morning he takes the lift down. Coming home he rides to the 7th floor and walks the stairs the rest of the way, except on rainy days, when he rides all the way to 10. Why?

Accepted Answer

Answer: He is very short. He can reach the ground-floor button and the 7th-floor button, but not the button for the 10th floor. On rainy days he carries an umbrella and uses it to poke the 10 button. How to get there: Ask what is special about floor 7 (it is as high as he can reach) and what changes when it rains (he has an umbrella, or in some versions the lift is more crowded and someone presses it for him). Analogy: look for a hidden fact about the person. 'Why only floor 7' points at height. 'What changes when it rains' points at the umbrella. In an interview: He is too short to reach the 10 button, and the umbrella lets him on rainy days. I would get there by asking what is special about floor 7.

Question 22

Why are manhole covers round?

Accepted Answer

Answer: Because a round cover is the only common shape that cannot fall through its own hole. Why: A square cover can be turned on its edge and dropped through the hole diagonally, because the diagonal of a square is longer than its side. A circle is the same width in every direction, so no matter how you turn it, it cannot slip through the slightly smaller circular opening. Bonus reasons often mentioned: a round cover can be rolled instead of carried, it does not need to be lined up a certain way when you put it back, and a circle is a strong shape for resisting pressure from traffic and soil. Analogy: the real question is 'what does a circle have that a square does not'. The answer is the same width in every direction. In an interview: A circle cannot fall through its own hole, because it has the same width in every direction. Then add the practical extras like rolling.

Question 23

100 prisoners stand in a line, each wearing a red or blue hat. Each sees every hat in front but not their own or the ones behind. Starting from the back, each says one word, 'red' or 'blue', trying to guess their own hat. They can agree a plan beforehand. How many can be guaranteed to survive?

Accepted Answer

Answer: 99 are guaranteed to survive. The person at the very back has a 50/50 chance. The plan: The back person counts the red hats they can see. If that count is even, they say "red". If it is odd, they say "blue". This is a coin flip for them, but it tells everyone else one useful fact: whether the number of red hats ahead is even or odd. Now every other prisoner, on their turn, knows that starting fact, can see all the hats in front of them, and has heard every guess behind them. From that, they can work out their own hat colour exactly. Each new guess keeps the running count updated for the people still to go. Analogy: one person gives up their own certainty to announce a single fact, whether the red hats add up to even or odd. That one fact lets everyone else be sure. In an interview: 99 for sure. The back prisoner spends their guess to announce the even/odd count of red hats, and everyone else uses that plus what they see and hear.

Question 24

What is the angle between the hour and minute hands of a clock at 3:15?

Accepted Answer

Answer: 7.5 degrees. The minute hand at 15 minutes points exactly at the 3, which is 90 degrees from the top. The hour hand is not on the 3 any more. It moves 360 degrees in 12 hours, so 30 degrees per hour, so 0.5 degrees per minute. At 3:15 it has moved 15 * 0.5 = 7.5 degrees past the 3, so it is at 97.5 degrees. The gap is 97.5 minus 90 = 7.5 degrees. General formula: angle = |30 * H - 5.5 * M|, where H is the hour and M the minutes. For 3:15: |90 - 82.5| = 7.5. Analogy: the trap is assuming both hands are on exact numbers. The hour hand creeps forward every minute, and that creep is the whole answer here. In an interview: 7.5 degrees. The minute hand is on the 3 (90 degrees) but the hour hand has crept 15 x 0.5 = 7.5 degrees past the 3. The formula is |30H - 5.5M|.

Question 25

A lily pad in a pond doubles in size every day. After 30 days it covers the whole pond. On which day did it cover half the pond?

Accepted Answer

Answer: Day 29. If it doubles every day, then the day before it was full it must have been half full. One more doubling from half gives full. The puzzle plays on the feeling that half the pond should take about half the time. With doubling (exponential growth) almost all the change happens right at the end: on day 25 the pond is only about 3 percent covered. This is the lesson people miss about anything that compounds: it looks slow for a long time, then finishes almost all at once. Analogy: exponential growth is quiet until it is sudden. Half the pond on day 29 feels wrong only because our intuition expects growth to be steady. In an interview: Day 29. One doubling before full is half full. The point is that exponential growth stays tiny for most of the time and then completes almost entirely in the final steps.

Question 26

How do you cut a round cake into 8 equal pieces with only 3 straight cuts?

Accepted Answer

Answer: Make two cuts straight down from the top, crossing at the centre, to get 4 quarters. Then make the third cut horizontally through the middle of the cake's height, slicing it into a top layer and a bottom layer. 4 pieces times 2 layers = 8 equal pieces. The trick is that the cuts do not all have to be vertical. Using the third dimension (the height of the cake) is what makes 8 possible with 3 cuts. If the puzzle insists all cuts are vertical from above, the best you can do is 6 or 7 pieces, not 8 equal ones. Analogy: everyone pictures cutting a flat circle. The cake is a cylinder, and the horizontal cut through its side is the move people do not think of. In an interview: Two vertical cuts through the centre for 4 quarters, then one horizontal cut through the middle of the cake's height to split every piece into two. The insight is to cut through the third dimension.

Question 27

Jane's mother has four children. The first is named Spring, the second Summer, the third Autumn. What is the fourth child's name?

Accepted Answer

Answer: Jane. The question already told you: 'Jane's mother has four children'. Jane is one of them. The Spring, Summer, Autumn pattern is a distraction designed to make you say 'Winter'. These riddles test whether you listen to the exact wording instead of pattern-matching. The answer is in the first sentence. Analogy: the puzzle hands you the answer at the start, then spends three sentences trying to make you forget it. Re-read the first line. In an interview: Jane. The first sentence names her as one of the four children. The seasons are bait to make you answer Winter. The lesson is to listen to the literal wording.

Question 28

You are at the bottom of a 30-foot wall. Each hour you climb 3 feet, then slide back 2 feet. How long to reach the top?

Accepted Answer

Answer: 28 hours. Your net progress is 1 foot per hour (up 3, back 2). It is tempting to say 30 hours, but that ignores the last climb. After 27 hours you are at 27 feet. On hour 28 you climb 3 feet and reach 30, the top. You are OUT of the well before the slide happens, so you never slide back that time. General rule for these: climb until (height minus one climb) is reached at net speed, then one more move finishes it. Here: reach 27 in 27 hours, then hour 28 climbs the last 3. Analogy: the trap is applying the 'net 1 foot per hour' rule all the way to the top. The slide only happens if you are still in the well, and on the winning hour you have already escaped. In an interview: 28 hours. Net progress is 1 foot per hour, so after 27 hours you are at 27 feet, and on hour 28 the 3-foot climb reaches the top before any slide. The mistake is applying the net rate to the final move.

Question 29

How would you estimate how many ping pong balls fit inside a Boeing 747 (a Fermi estimation question)?

Accepted Answer

Answer: You are not expected to know the number. The interviewer wants to see you break an impossible question into a chain of rough estimates you CAN reason about, stating each assumption. The method: 1. Estimate the usable volume of the plane. Treat the cabin as a cylinder: roughly 60 m long, 6 m across, so radius 3 m. Volume is about pi times r squared times length, roughly 3 times 9 times 60, about 1,600 cubic metres. Round to 1,000 to 2,000. 2. Estimate the volume of one ping pong ball. Diameter about 4 cm, so radius 2 cm. A ball's bounding cube is 4 cm each side, about 64 cubic cm, or 6.4 times 10 to the minus 5 cubic metres. 3. Account for packing: spheres in a box waste about 25 to 35 percent of space, so multiply the count by about 0.7. 4. Divide: 1,500 divided by 0.000064, times 0.7, gives roughly 16 million. Same approach for 'pennies to the top of the Empire State Building' (building height divided by penny thickness) or 'piano tuners in a city'. Analogy: you cannot weigh the elephant, but you can estimate its parts. Break the question into a few numbers you can each guess within a factor of 2, and combine them. In an interview: I would not guess a number, I would build a chain: estimate the plane's usable volume as a cylinder, estimate one ball's volume, apply a packing factor of about 0.7, and divide. State every assumption. The answer they want is the method and an order of magnitude, around 10 million.

Question 30

How should you approach a logic puzzle in an interview?

Accepted Answer

Short version: restate the problem, think out loud, try a smaller version, work out a lower bound, then commit to an answer and check it. 1. Restate the puzzle and confirm the rules. Ask about anything unclear. Interviewers often leave a gap on purpose. 2. Think out loud. They are watching how you reason, not just whether you get the answer. Silence tells them nothing. 3. Try a smaller version. Two ants instead of three, one weighing instead of three, 4 bottles instead of 1000. The pattern usually shows up in the small case. 4. Count the information. How much does each action tell you? A balance scale gives 3 results, a yes/no question gives 2. That gives you the minimum number of steps and often hints at the method. 5. Give an answer, then check it against every rule in the puzzle. 6. If you are stuck, say what you have ruled out and why. That is still progress, and interviewers will often nudge you. Analogy: the interviewer is grading your thinking process. A clear, spoken attempt that does not quite finish beats a silent correct answer. In an interview: Restate the rules, think out loud, try a tiny version, use a counting argument for the minimum, then commit and verify. Show the process.

Question 31

What tricks show up again and again in these puzzles?

Accepted Answer

Most interview puzzles are one of about seven ideas in disguise: 1. Information per action. Each weighing gives 3 results, each yes/no question gives 2. Count the cases you must separate to find the minimum number of steps. 2. Binary encoding. To find one item among many with a few tests, number the items in binary and let each test read one digit (1000 bottles, the fake coin stack). 3. Parity. Announcing one even/odd fact lets a whole group coordinate (the 100 hats). 4. Overlap, do not add. Do slow things at the same time (both rope ends, the two slowest on the bridge together). 5. Embedded questions. "Would you say X" cancels lying and unknown languages. 6. Work backwards from the goal. 7. Use a second signal. A bulb gives heat as well as light. Missing weight can spell out a number. Analogy: if you know the handful of standard tricks, you spend your time matching the puzzle to a trick instead of inventing a method from scratch. In an interview: List the recurring tools: counting information, binary encoding, parity, overlapping slow tasks, embedded questions, working backwards, shrinking the problem. Naming the tool usually cracks the puzzle.

Logic puzzles and brain teasers

Measuring and pouring

Liars, truth-tellers and doors

Weighing puzzles

Crossing and scheduling

Probability puzzles

Lateral thinking classics

How to approach puzzles