Mutual Fund Mastery puzzles, solved step by step
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023A car leaves A for B, 100 miles away, at 50 miles an hour. At the same moment a bird leaves B toward the car at 100 miles an hour. Each time it meets the car it turns, flies back to B, then turns again toward the car, until the car reaches B. How far does the bird fly in total?BlackRockNew York · 2025
Try it first
How far does the bird fly?
Show the worked solution
200 miles. The car covers 100 miles at 50 miles an hour, so the journey lasts two hours. The bird flies the whole time at 100 miles an hour, so it covers 200 miles however many times it turns. The long way agrees: the bird's round trips are 133.3, then 44.4, then 14.8 miles, each a third of the last, summing to 133.3 times 1.5, 200.
What is the question really asking?
Think of a dog running back and forth between you and your front door while you walk home. You could trace every turn, or you could notice the dog runs the whole time you are walking. When something moves at a constant speed for a known length of time, distance is speed times time, and the path it takes in between does not matter. The bird's speed is given; the only thing you need is how long it flies, and that is the car's travel time.
The bird zigzags between the car and B in trips that shrink by two thirds each time, but it flies for exactly the two hours the car takes to cover 100 miles, so it covers 200 miles at 100 miles an hour. Can you prove it the long way, too?
Yes, and doing so briefly earns credit. The car and bird close a 100 mile gap at 150 miles an hour, meeting after two thirds of an hour, 66.7 miles from B. The bird flies 66.7 miles back to B, arriving at 1 hour 20 minutes, when the car is 66.7 miles from A. Each new chase starts with a gap a third of the last, so each round trip is a third as long: 133.3, 44.4, 14.8 and so on. A geometric series with ratio one third sums to the first term times 1.5, which is 200.
The relationshipv_{bird} the bird's speed, 100 mph v_{car} the car's speed, 50 mph d the distance from A to B, 100 miles 400/3 the first round trip, 133.3 miles What it says in wordsThe bird's distance is its speed times the car's travel time, and the infinite series of shrinking trips sums to the same number.Why would a fund interviewer ask this? It tests whether you look for the quantity that is fixed before diving into detail, the same habit that makes you ask what an investor actually earned before reconciling every trade. The limitation is that it is a pure puzzle; turning instantly at each meeting is an idealisation that makes the infinite number of turns harmless.
Where candidates lose it
The common failure is starting to sum the legs without noticing the shortcut, then getting lost in the second or third term under time pressure. The interviewer is watching whether you step back and ask what is constant.
The opposite failure is saying infinitely far because the bird turns infinitely many times. Infinitely many terms can have a finite sum when they shrink fast enough; here each is a third of the last.
What the interviewer asks next
- If the bird flew at 150 miles an hour instead, how far would it fly?
- How far from A is the car when the bird reaches B for the second time?
- How many round trips does the bird complete before the car is within one mile of B?
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
039Give an equation for the surface area of an n by n by n Rubik's cube, counted in small unit squares. Then give an equation for how many of the small cubes show at least one face.T. Rowe PriceBaltimore · 2020
Try it first
For a standard 3 by 3 by 3 cube, how many small cubes show at least one face?
Show the worked solution
The surface is 6n² unit squares, and n³ minus (n minus 2)³ cubes show at least one face. Six faces each carry n by n squares. For the cubes, count what is hidden: peeling one layer off every side leaves an (n minus 2) cube inside, so the visible ones are n³ minus (n minus 2)³, which expands to 6n² minus 12n plus 8. For a 3 by 3 by 3 cube that is 54 squares and 26 cubes.
Why are there two different counts here?
A house with a corner room has windows on two walls of that room, but it is still one room. Counting windows and counting rooms give different numbers. The surface counts squares, and a corner cube carries three squares and an edge cube two, so the square count is always larger than the number of cubes that show. Keeping the two counts apart is half the problem; candidates who blur them give 54 for both.
The surface is the easy part. Each of six faces is an n by n grid, so the area is 6n² unit squares: 54 for a standard cube. A quick check is to rebuild it from the cube types: 8 corners showing 3 squares, 12(n minus 2) edge cubes showing 2, and 6(n minus 2)² centre cubes showing 1. For n of 3 that is 24 plus 24 plus 6, which is 54 again.
On a 4 by 4 by 4 cube the skin carries 96 squares, but only 56 cubes show, because each corner cube shows three squares and each edge cube two; the dashed 2 by 2 by 2 core of 8 cubes is hidden, and 64 minus 8 is 56. Why count the hidden cubes instead of the visible ones?
The hidden cubes form one clean block, (n minus 2) on every side, so counting them and subtracting from n³ avoids every double count. Counting the visible cubes directly means adding corners, edges and faces while remembering that each edge already lost its corners. Both routes give the same answer, and the second makes a good check.
The relationshipn small cubes along one edge (n-2)^3 the hidden core after peeling one layer from every side V(n) cubes showing at least one face What it says in wordsVisible cubes are all the cubes minus the core you cannot see; expanded, it is the surface area less a correction for corners and edges.The expanded form is worth a sentence because it links the two answers. 6n² is the square count; the minus 12n plus 8 removes the extra squares that edge and corner cubes contribute. Say the edge case too: the formula needs n of at least 2. For n of 1 it gives 2, but a single cube is one cube.
Where candidates lose it
The trap is giving 6n² as the answer to both questions. The interviewer is checking whether you notice that one cube can show up to three squares. Name the two counts before you write anything.
The second loss is trying to count visible cubes from the surface and getting tangled in double counts at the edges. Go to the hidden core first, then offer the corner, edge and face breakdown as the check.
What the interviewer asks next
- How many small cubes show exactly two faces, as a formula in n?
- For which n is the hidden core larger than the visible shell?
- How would the counts change for a 4 by 5 by 6 box?
Asked at T. Rowe Price, Equities, Baltimore, 2020 (Wall Street Oasis):
Give an equation that yields the surface area of an n by n by n Rubic's cube based on number of blocks per side
