Equity Research puzzles, solved step by step
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072A car leaves point A for point B, 100 miles away, at 50 miles an hour. At the same moment a bird leaves B flying towards A at 100 miles an hour. Each time the bird meets the car it turns back to B, and each time it reaches B it turns towards the car again, until the car reaches B. How far does the bird fly?BlackRockNew York · 2025
Try it first
How far does the bird fly?
Show the worked solution
200 miles. Ignore the zigzag and ask how long the bird is in the air. The car takes 100 / 50 = 2 hours to reach B, and the bird flies the whole time at 100 miles an hour, so it covers 2 x 100 = 200 miles. Summing the legs gives the same answer: 133.3 miles in the first round trip, then a third as much each round, and 133.3 / (1 - 1/3) = 200.
What is the shortcut, and why does it work?
Picture walking home while your dog runs back and forth between you and the front door. Ask how far the dog ran and the back and forth looks like the hard part, but the dog simply ran at its own speed for exactly as long as your walk took. When something moves at a constant speed for a known time, the distance is speed times time, however many times it turns around. The car's trip fixes the time at 2 hours, and the bird's speed does the rest.
The bird's zigzag shrinks by a third with each round trip, starting at 133.3 miles, but it flies for exactly the car's 2 hours at 100 miles an hour, so its total is 200 miles either way you count. How do you check it the long way?
The first meeting comes when the gap of 100 miles closes at 150 miles an hour, after 40 minutes, 33.3 miles from A. The bird has flown 66.7 miles and flies 66.7 back to B, a first round trip of 133.3 miles. By then the car is two thirds of the way, and every later round trip covers a third of the remaining gap, so each is a third of the last. A geometric series with first term 133.3 and ratio one third sums to 133.3 / (2/3) = 200, the same answer the shortcut gives in one line.
The relationshipv_bird the bird's speed, 100 miles an hour t_car the car's travel time, 2 hours 133.3 the bird's first round trip in miles 1/3 the ratio of each round trip to the one before What it says in wordsThe bird's distance is its speed times the car's travel time, and the geometric series of round trips agrees.Why would an investing firm ask a puzzle like this?
Because choosing the quantity that stays simple is most of the skill. Here time is simple and distance is messy, so solving in time turns an infinite sum into one multiplication. The same habit finds total cash generated over a period from an average rate instead of adding every quarter, or total interest paid on a loan from the average balance. Say the frame you chose before you give the number.
Where candidates lose it
Candidates start adding the legs, get the first round trip, and then run out of time or nerve at the second. The interviewer is watching whether you step back and find the frame where the problem is one line.
The other loss is saying the distance is infinite because the bird turns infinitely often. Infinitely many legs can have a finite total when each is a fixed fraction of the last; say that, then give 200.
What the interviewer asks next
- How many times does the bird turn round before the car arrives?
- Where is the car when the bird finishes its second round trip?
- If the bird flew at 150 miles an hour, how far would it go?
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
097Give a formula for the number of coloured stickers on an n by n by n Rubik's cube, and for the number of small cubes that show on the surface.T. Rowe PriceBaltimore · 2020
Try it first
For a standard 3 by 3 by 3 cube, how many small cubes show on the surface?
Show the worked solution
Stickers: 6n squared. Cubes on the surface: n cubed minus (n minus 2) cubed. Each of the six faces is an n by n grid of stickers, so there are 6n squared, which is 54 for n of 3. For the cubes, count the hidden core: peeling one layer off every side leaves (n minus 2) cubed, so 27 less 1 is 26 on show. Both formulas need n to be at least 2.
Why are stickers and surface cubes different counts?
A corner cube carries three stickers, an edge cube two and a face-centre cube one. The sticker count measures surface area, while the cube count measures how many pieces touch the surface, and the corners and edges make the first larger than the second. The surface area the question asks for is the sticker count: six faces, each n by n, so 6n squared unit squares.
What is the fast way to count the cubes on show?
Picture a box of laddoos packed three by three by three: to find how many touch the box, count the ones that do not. Only a core with one layer peeled from every side is hidden, so the hidden cubes form an (n minus 2) by (n minus 2) by (n minus 2) block, and everything else shows. For n of 3 that core is a single cube, so 26 show. The subtraction is quicker and harder to get wrong than adding corners, edges and faces.
A 3 by 3 by 3 cube has 27 small cubes and only the single centre cube is hidden, so 26 show on the surface, while six faces of nine stickers carry 54 stickers, and the table gives the same counts for larger cubes. The relationshipS(n) stickers, the surface area in unit squares C(n) small cubes with at least one face on the surface n small cubes along one edge, 2 or more What it says in wordsStickers are six square faces; surface cubes are the whole block less the hidden core, which expands to 6n squared less the overlap at edges and corners.How do you check the formula before you commit?
Test the smallest case you can picture. A 2 by 2 by 2 cube has no hidden centre, so all 8 pieces show, and the formula gives 8 minus 0, which is 8. Then look at the expanded form: 6n squared counts every face-cube once per face, minus 12n plus 8 removes the double and triple counts along the 12 edges and at the 8 corners. The formula breaks for n of 1, where it gives 2 for a single cube, so state that it holds from n of 2 upwards.
Where candidates lose it
Candidates confuse stickers with cubes and answer 54 cubes, or count six faces of n squared cubes and forget that edge and corner cubes sit on more than one face. Both come from starting at the surface.
The second loss is giving a formula without testing it. Plugging in n of 2 and n of 3 takes ten seconds and catches most slips, and noting that n of 1 fails shows care.
What the interviewer asks next
- How many small cubes show exactly two stickers on an n by n by n cube?
- For what n are more than half the small cubes hidden?
- How would the counts change if the cube were hollow?
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.
