Hedge Funds puzzles, solved step by step
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007You have two ropes. Each burns completely in exactly 60 minutes, but unevenly, so half a rope need not take 30 minutes. With a lighter and nothing else, how do you measure exactly 45 minutes?Prop and quant trading firmsLong-short equity funds
Try it first
What is the first move?
Show the worked solution
Light rope one at both ends and rope two at one end at the same moment; when rope one burns out, light rope two's other end, and it burns out at 45 minutes. Two flames always meet after burning 60 minutes of rope between them, so rope one takes 30 minutes however uneven it is. Rope two then has 30 minutes left, which two flames finish in 15.
Why does lighting both ends halve the time on an uneven rope?
Picture two people eating a long, uneven sandwich from opposite ends, each chewing through whatever is at their end. However the filling is spread, they meet once the whole sandwich has been eaten between them, and together they finish in half the time one would take. Two flames consume the rope's total burn time twice as fast, so a 60 minute rope lit at both ends is gone in 30 minutes, wherever the flames happen to meet. Length tells you nothing here; burn time is the only quantity you can trust.
Rope one, lit at both ends, is gone at 30 minutes; rope two, lit at one end at the start, has 30 minutes of burn left at that moment, and lighting its other end finishes it 15 minutes later, at 45 minutes. The relationship60/2 rope one, burned from both ends (60 - 30)/2 rope two's remaining burn time, burned from both ends What it says in wordsEvery step halves a known amount of burn time; nothing depends on where along the rope the time is stored.Why is this really a question about information?
The rope hides where its time is stored, much as an order book hides how much size is waiting behind a price. The solution uses only what is known, the total burn time, and never what is not, how it is spread along the rope. Anyone who cuts a rope in half is assuming evenness that the first sentence ruled out. Say that out loud before you give the method: naming what you may not assume is half of a good answer.
Expect the follow-up. The same trick measures 15 minutes as an interval, the gap between rope one going out and rope two going out. Each rope lit from its second end at a known moment halves whatever burn time it has left, and chaining those halvings is how you reach times such as 52.5 minutes with a third rope. Walk through the chain in order, one lighting at a time.
Where candidates lose it
The instinctive answer cuts or folds a rope, which quietly assumes it burns evenly. The question rules that out in its first sentence, and an interviewer will stop you there.
The subtler slip is lighting rope two late. It has to be lit at the very start, alongside rope one, so that exactly 30 minutes of its burn time are gone when rope one finishes. Say that both lightings happen together.
What the interviewer asks next
- How would you measure 15 minutes?
- With one rope, which times can you measure?
- With three such ropes, how do you measure 52.5 minutes?
057One glass holds 200 ml of wine and another holds 200 ml of water. You pour 50 ml of wine into the water glass and stir, then pour 50 ml of the mixture back into the wine glass. Is there more wine in the water glass or more water in the wine glass?Wolverine TradingChicago · 2025
Try it first
Which is larger at the end?
Show the worked solution
They are exactly equal: 40 ml of wine sits in the water glass and 40 ml of water sits in the wine glass. After the first pour the water glass holds 200 ml of water and 50 ml of wine. A 50 ml pour of that mix is one fifth wine, so 10 ml of wine and 40 ml of water go back. Each glass ends at 200 ml, which forces the two amounts to match.
What is the argument that needs no arithmetic?
Two classrooms hold 30 students each. Five walk from room A to room B, and then any five people at all walk back. Both rooms hold 30 again. Every seat in room A left empty by a room A student who stayed away must now be filled by a room B student, so the room B students in room A always equal the room A students in room B. The glasses work the same way, because both finish at 200 ml.
The wine glass goes from 200 ml of wine to 150 ml, then gets back 10 ml of wine and 40 ml of water; the water glass ends with 160 ml of water and 40 ml of wine, so each glass holds exactly 40 ml of the other liquid. How do the numbers confirm it?
After the first pour the water glass holds 250 ml, of which 50 ml, one fifth, is wine. Stirred evenly, a 50 ml pour back carries 10 ml of wine and 40 ml of water. The wine glass ends with 150 plus 10, 160 ml of wine, and 40 ml of water. The water glass keeps 200 minus 40, 160 ml of water, and 40 ml of wine. Forty and forty.
The relationship50/250 the share of wine in the water glass after the first pour 200/250 the share of water in it What it says in wordsThe wine left behind in the water glass equals the water carried back, because each glass ends at its starting volume.What does the interviewer learn from how you answer?
Whether you look for something that stays fixed before you reach for arithmetic. The conservation argument survives every variation: uneven stirring, several pours back and forth, any spoon size, as long as both glasses finish at their starting volumes. Desks use the same move on inventory: if two accounts hold the same totals after a string of transfers, whatever left one must have been replaced from the other, whatever the route. Give the one-line argument first, then the 40 and 40 as the check.
Where candidates lose it
Most people say there is more wine in the water, because the first pour was pure wine and the return pour was diluted. That instinct tracks the pours instead of the end state, and only the end state matters.
The other way to lose it is to announce that the answer depends on stirring. It does not: the equality holds for any mix, because both glasses finish at 200 ml. Unequal pours are what break it, which is the usual follow-up.
What the interviewer asks next
- Repeat both pours a second time. How much of each liquid is in each glass now?
- If you do not stir at all before pouring back, what changes?
- The pour back is only 25 ml. Are the two amounts still equal?
Asked at Wolverine Trading, Equity Hedge, Chicago, 2025 (Wall Street Oasis):
Variation on the wine glass question
082A car drives 60 miles at an average speed of 30 mph. How fast must it drive the 60 miles back to average 60 mph over the whole round trip?Man GroupLondon · 2016
Try it first
Answer inside ten seconds.
Show the worked solution
It cannot be done at any finite speed. Averaging 60 mph over the 120-mile round trip means finishing in 2 hours. The outward 60 miles at 30 mph already took 2 hours, so the return leg would have to take no time at all. Driving back at 90 mph, the tempting answer, gives an average of only 45 mph.
Why is 90 mph the wrong instinct?
Averaging 30 and 90 to get 60 treats the two speeds as if they counted equally. They do not, because the car spends far longer at the slow speed. Think of a student who scores 30% on a three-hour paper and 90% on a ten-minute quiz: nobody would call that a 60% performance. Average speed is total distance over total time, so the slow leg carries more weight because it takes up more of the clock.
A 60 mph average over 120 miles allows 2 hours, and the outward leg at 30 mph uses all of them, so a return at 90 mph ends 40 minutes late for a 45 mph average and even 300 mph ends 12 minutes late for 54.5 mph. How do you prove it cannot be done?
Work in time, not speed. At 60 mph, 120 miles takes exactly 2 hours, and the first leg has already spent those 2 hours. Any return speed, however fast, adds some time, which pushes the average below 60 mph. At 90 mph the return takes 40 minutes and the average is 45 mph; at 300 mph it takes 12 minutes and the average is 54.5 mph. The average creeps towards 60 but never reaches it.
The relationshipv the speed on the return leg, in mph 2 hours already spent on the outward leg 60/v hours the return leg takes What it says in wordsThe average is the whole distance over the whole time, and the whole time is always more than the 2 hours a 60 mph average allows.Where does the same mistake show up on a desk?
It appears whenever numbers are averaged without the right weights. A position that falls 50% and then rises 50% does not break even, because the second move works on a smaller base; the average that matters is the one weighted the way the thing actually compounds. Speeds over equal distances call for the harmonic mean, which sits below the simple average whenever the numbers differ. It is the same reason that putting a fixed rupee amount into a fund each month buys units at an average cost below the average price over those months.
Where candidates lose it
90 mph is the whole trap, and it comes from averaging the speeds instead of dividing distance by time. The interviewer asks it quickly precisely so that the symmetric answer comes out first.
The second loss is saying impossible without the reason. Give the time budget in one line: 2 hours allowed, 2 hours already used.
What the interviewer asks next
- What return speed gives a round-trip average of 45 mph?
- The car drives the first 60 miles at 40 mph instead. What speed back gives 60 mph overall?
- Why is the average cost of buying a fixed rupee amount each month below the average price?
Asked at Man Group, Equity Hedge, London, 2016 (Wall Street Oasis):
A car travels a distance of 60 miles at an average speed of 30 mph.
