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  1. 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?Logic and brainteasersWarm upProp 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.

    Two flames burn a rope's 60 minutes twice as fast, wherever they meetRope 1Rope 2both ends: 60 min of burn in 30goneone end: 30 min of burn usedboth ends: 150 min15 min30 min45 min60 minLight rope 1 at both endsand rope 2 at one endRope 1 out: lightrope 2's other endRope 2 out:45 minutes
    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 relationship
    t=602+60−302=30+15=45 minutest = \frac{60}{2} + \frac{60 - 30}{2} = 30 + 15 = 45 \text{ minutes}
    60/2rope one, burned from both ends
    (60 - 30)/2rope 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?