Portfolio Management puzzles, solved step by step

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  1. 003A car leaves town A for town B, 100 miles away, at 50 miles an hour. At the same moment a bird leaves B at 100 miles an hour, flies to meet the car, turns back to B, then turns again toward the car, and keeps shuttling until the car reaches B. How far does the bird fly?Logic brainteasersWarm upBlackRockNew York · 2025

    Try it first

    Answer inside ten seconds.

    Show the worked solution

    200 miles. The car needs 100 miles at 50 miles an hour, which is two hours. The bird flies without stopping for those two hours at 100 miles an hour, so it covers 200 miles, however many times it turns. Summing the legs gives the same answer: 66.7 there and back, then 22.2 there and back, each pair a third of the one before, which adds to 200.

    Why is summing the legs the slow way?

    Picture a dog on a walk that runs ahead to the gate and back to you, over and over, until you reach the gate. Nobody counts the dog's sprints; you just ask how long the walk took and how fast the dog runs. When something moves at a constant speed for a known time, distance is speed times time, whatever path it traces. The bird's zig-zag looks like the hard part of the question. It is a distraction. The only thing that matters is when the flying stops, and that is when the car arrives.

    Plot position against time: the bird simply flies for as long as the car drivesAB5000.511.52 hoursTimefirst meeting: 40 min, 33.3 miles from Acar, 50 mphbird, 100 mphAsk how long, not how farCar: 100 miles / 50 mph = 2 hBird: 2 h x 100 mph= 200 milesCheck: sum the legs66.7 + 66.7 = 133.3then 22.2 + 22.2, 7.4 + 7.4 ...each pair a third of the last: 200
    The car takes two hours to cover 100 miles at 50 miles an hour, and the bird zig-zags between B and the car for exactly those two hours, so at 100 miles an hour it flies 200 miles; the shrinking legs of 66.7, 66.7, 22.2, 22.2 and so on add to the same total.

    How do you check 200 the long way, in case the interviewer asks?

    The bird and the car close at 150 miles an hour, so they first meet after 100 / 150 of an hour, 40 minutes, when the car is 33.3 miles from A and the bird has flown 66.7 miles. The bird flies 66.7 miles back to B. By then the car is 66.7 miles along, 33.3 miles from B, and the same geometry repeats on a gap a third as large. Each round trip is a third of the one before, so the legs form a geometric series: 133.3 times one over one minus a third, which is 200.

    The relationship
    d=vbird×Dvcar=100×10050=200check: 133.3×11−13=200d=v_{bird}\times\frac{D}{v_{car}}=100\times\frac{100}{50}=200 \qquad \text{check: } 133.3\times\frac{1}{1-\tfrac{1}{3}}=200
    Dthe distance from A to B, 100 miles
    v_{car}, v_{bird}the speeds, 50 and 100 miles an hour
    What it says in wordsThe bird's distance is its speed times the car's travel time; the geometric sum of its legs confirms it.

    Why ask this on a quantitative research desk? Because the same move, stepping back from the path to the total, is how you price anything path-dependent in your head: ask what is conserved or fixed before tracing every step.

    Where candidates lose it

    Candidates start computing the first meeting point, then the second, and lose the room in arithmetic. The interviewer is watching for the moment you ask how long the bird flies; some interviewers stop you once you begin summing legs.

    The second trap is saying infinite because the bird turns infinitely often. A sum of infinitely many shrinking terms can be finite, and here it is.

    What the interviewer asks next

    • What if the bird flew at 150 miles an hour?
    • If both towns sent a car toward each other at 50 miles an hour, how far does the bird fly?
    • How many times does the bird touch the car?

    Asked at BlackRock, Quantitative Research, New York, 2025 (Wall Street Oasis): A car starts at point A going 50 miles an hour towards point B, and a bird starts at point B going towards point A at 100 miles per hour

  2. 030You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn from one end to the other, but it burns unevenly, so half the length does not mean half the time. How do you measure exactly 45 minutes?Logic brainteasersWarm upAsset managementReal assets

    Try it first

    What does lighting a rope at both ends give you, if it burns unevenly?

    Show the worked solution

    Light rope 1 at both ends and rope 2 at one end, together. When rope 1 burns out, light the other end of rope 2; when rope 2 burns out, 45 minutes have passed. Rope 1 lasts 30 minutes because two flames share its 60 minutes of burning. At that moment rope 2 has 30 minutes left, and lighting its other end halves that to 15. 30 plus 15 is 45.

    Why does uneven burning not spoil the halving?

    Two people eating one plate of food from opposite sides finish it in half the time one person would take, whether the food is piled high on one side or spread evenly. They simply meet off-centre. A rope holds 60 minutes of burning in total, and two flames consume it twice as fast, so it is gone in 30 minutes wherever the flames happen to meet. The unevenness decides the place, never the time. That is the only fact the puzzle needs.

    Light both ends and time halves, however unevenly the rope burns0 min15 min30 min45 minRope 1both endsflames meet here, not in the middlegone at 30Rope 2one end,then bothone end: 30 min of burning usedboth ends30 left, halvedlight rope 2's other endRope 2 gone at45 minutesStopwatch
    Rope 1, lit at both ends, is gone after 30 minutes even though its flames meet away from the middle. Rope 2, lit at one end, has 30 minutes of burning left at that moment, and lighting its other end halves that to 15 minutes, ending at 45.

    How do you know rope 2 has exactly 30 minutes left at the half-hour?

    Rope 2 has been burning from one end for 30 minutes, so it has used 30 of its 60 minutes of material, whatever length that turned out to be. You never measure length; you only ever track time used and time left. Lighting the far end of what remains halves the 30 minutes left, and the rope finishes 15 minutes later. Say the timeline in that order and the answer is audible.

    Interviewers often extend it: with the same two ropes you can also time 15, 30, 60 and 90 minutes, each by deciding which ends are burning at which moment. Answering one extension shows you own the principle rather than a memorised trick.

    Why would a real asset desk ask this? It is a clean test of whether you separate the thing you can measure from the thing that is noisy. A rent roll is lumpy month to month; the annual total is what the valuation rests on. Saying that link in one sentence costs nothing.

    Where candidates lose it

    Candidates try to cut or fold the ropes, or to reason about lengths, which the uneven burning makes useless. The whole question is set up to see whether you let go of length and think only in minutes of burning.

    The second slip is lighting rope 2's second end at the start. Rope 2 must be lit at one end at minute 0 so that exactly 30 minutes of it are used up when rope 1 finishes.

    What the interviewer asks next

    • Using the same two ropes, how do you measure 15 minutes?
    • With three ropes, what is the longest time you can measure beyond 60 minutes?
    • Can you measure 20 minutes with two ropes? Why or why not?
  3. 053You have nine bags of coins that look identical. Eight weigh the same and one is lighter. Using a balance scale with no weights, what is the fewest number of weighings that always finds the light bag?Logic brainteasersWarm upAsset management

    Try it first

    What is the fewest weighings that is guaranteed to work?

    Show the worked solution

    Two weighings. Put three bags on each side and leave three aside. Whichever side rises holds the light bag; if the pans balance, it is among the three on the table. Take that group of three and weigh one bag against another: the one that rises is light, and if they balance, it is the third. A balance gives three outcomes per weighing, and three times three covers nine bags.

    Why is halving the pile the wrong instinct?

    Halving is what you would do with a question that answers yes or no, like guessing a number between 1 and 100. A balance tells you more than that. A balance scale has three outcomes, left pan rises, right pan rises, or level, so each weighing should split the suspects into three equal groups, not two. The group left on the table is not wasted: a level balance is information too. With four and four plus one aside you would learn little from a level result except that the odd bag is the spare one, and an unlucky run needs three weighings.

    A balance has three outcomes, so split the bags into three every timeWeighing 1: bags 1-3 vs 4-6bags 7-9 stay on the tableLeft side riseslight bag in 1, 2 or 3Balancedlight bag in 7, 8 or 9Right side riseslight bag in 4, 5 or 6Weighing 2: bag 7 vs bag 87 risesbag 7balancedbag 98 risesbag 8Same second stepunder the othertwo branches1 weighing:3 outcomes2 weighings:3 x 3 = 9
    Weighing three bags against three splits nine suspects into three groups of three, whichever way the pans move, and weighing one bag against one then names the light bag, so two weighings give nine end points, one for each bag.

    How do you prove two is the minimum?

    Count outcomes. One weighing produces at most three different results, and three results cannot point to nine different bags. With k weighings you can separate at most 3 to the power k bags, so nine bags need at least two weighings, and the method above shows two is enough. The same count answers the natural follow-up: three weighings handle up to 27 bags, four handle 81.

    The relationship
    3k≥n⇒32=9≥93^k \ge n \quad\Rightarrow\quad 3^2 = 9 \ge 9
    kthe number of weighings
    nthe number of bags, one of which is light
    3outcomes of one weighing: left light, right light, balanced
    What it says in wordsEach weighing multiplies the number of distinguishable outcomes by three, so you need enough weighings for the outcomes to cover every bag.

    Why would an asset manager ask this? It is a test of whether you use all the information a measurement gives you. A risk report, a performance attribution or a data screen can be read in more ways than pass and fail, and the candidate who spots the third outcome here is the one who reads the level result as a finding.

    Where candidates lose it

    The trap is answering three or four because you split into halves. It is the natural instinct, and the interviewer is watching whether you notice that a balance can come out level, which is a result in its own right.

    The second loss is giving two without the counting proof. Say the 3 to the power k rule in one sentence: it shows you know the answer is a minimum, not just a method that happened to work.

    What the interviewer asks next

    • How many weighings do you need for 27 bags? For 100?
    • Now you do not know whether the odd bag is heavier or lighter, and there are twelve. How many weighings?
    • You may take any number of coins from each bag and weigh once on a scale that shows grams. How do you find the light bag?