Equity Research puzzles, solved step by step
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050Depreciation expense rises by Rs 10 crore and the tax rate is 25%. Walk the change through the income statement, the cash flow statement and the balance sheet.Millennium ManagementNew York · 2024
Try it first
What happens to cash?
Show the worked solution
Net income falls Rs 7.5 crore, cash rises Rs 2.5 crore, PP&E falls Rs 10 crore and equity falls Rs 7.5 crore. Pre-tax profit drops Rs 10 crore and tax drops Rs 2.5 crore, so net income is down Rs 7.5 crore. The cash flow statement adds back the Rs 10 crore of depreciation, so cash is up Rs 2.5 crore. Assets fall Rs 7.5 crore net, matching the fall in retained earnings.
Why does a non-cash expense raise cash?
A shopkeeper who can deduct the wear on her delivery van from her taxable income pays less tax, even though no money went out for the wear itself. Depreciation moves no cash, but it is deductible, so it cuts the tax bill, and the tax saved is real cash: 25% of Rs 10 crore, Rs 2.5 crore. That is the one cash effect in the whole question; everything else is accounting.
Rs 10 crore of extra depreciation cuts net income by Rs 7.5 crore, the add-back turns that into a Rs 2.5 crore rise in cash, and on the balance sheet cash up 2.5 and PP&E down 10 leave assets down 7.5, matching equity, because a non-cash expense raises cash by the tax it saves. How do you say it in the room in thirty seconds?
Go statement by statement, in the order the numbers flow. Income statement: pre-tax profit down 10, tax down 2.5, net income down 7.5. Cash flow: start at minus 7.5, add back 10, cash up 2.5. Balance sheet: cash up 2.5, PP&E down 10, assets down 7.5; equity down 7.5 through retained earnings; it balances. End on the balance, because that is the check the interviewer is waiting for.
Statement Line Change, Rs crore Income statement Pre-tax profit -10.0 Income statement Tax -2.5 Income statement Net income -7.5 Cash flow Operating cash flow +2.5 Balance sheet Cash / PP&E +2.5 / -10.0 Balance sheet Retained earnings -7.5 Every line follows from two facts: depreciation is deductible at 25%, and it moves no cash. Say the assumption: the tax return uses the same depreciation as the books. If the higher charge is only in the books, cash tax does not fall, the Rs 2.5 crore goes to a deferred tax asset instead, and cash does not move at all.
Where candidates lose it
The two classic slips are saying cash does not change because depreciation is non-cash, and letting net income fall the full Rs 10 crore by forgetting tax. Both come from rushing the first line.
The quieter loss is not closing the balance sheet. Candidates who say assets fall Rs 10 crore then cannot match equity. Cash up 2.5 is the piece that makes it balance.
What the interviewer asks next
- Now walk through a Rs 10 crore rise in inventory funded by cash.
- What if the extra depreciation is not deductible for tax?
- How does the change affect EBITDA and free cash flow?
Asked at Millennium Management, Investment Research, New York, 2024 (Wall Street Oasis):
Nothing as much, technical questions were super basic like $10 depreciation
071A timed aptitude item of the kind used on Wonderlic-style tests: if 3 analysts write 3 reports in 3 days, how many reports do 9 analysts write in 9 days?Point72New York · 2026
Try it first
Answer inside ten seconds.
Show the worked solution
27 reports. Find the unit rate first: 3 analysts writing 3 reports in 3 days means one analyst writes one report in 3 days, a third of a report per analyst-day. Nine analysts working nine days put in 81 analyst-days, and 81 x 1/3 = 27. Tripling the team triples the output and tripling the time triples it again: 3 x 3 x 3 = 27.
Why does the fast answer of 9 feel right?
The sentence has a rhythm, three, three, three, then nine and nine, and the ear wants to finish the pattern with another nine. It is the same pull as a kitchen puzzle where three cooks bake three cakes in three hours. Matching the pattern of the words is the trap; the fix is to reduce the problem to one worker and one day before scaling anything. Once you have the rate per analyst-day, the answer is multiplication.
The question's case, 3 analysts for 3 days, is 9 analyst-days and 3 reports, so one report takes 3 analyst-days; the full 9 by 9 grid holds 81 analyst-days and therefore 27 reports. How do you do it in ten seconds?
Scale each dimension on its own. Three times the analysts gives three times the reports, and three times the days gives three times again, so the answer is 3 reports x 3 x 3 = 27. The general rule is output equals the rate per worker per day, times the workers, times the days, and it handles any version of the item, including the ones where the numbers do not match so neatly.
The relationship3 / (3 x 3) reports per analyst-day, one third 9 x 9 analyst-days in the new case, 81 What it says in wordsReports equal the rate per analyst-day times the number of analyst-days.What is a timed test like this really checking?
Wonderlic-style tests put many short items against a tight clock, so the difficulty is the time, not any single item. The format, number of items and time allowed vary by version, so confirm what the firm uses. The habit that scores on the test is the same one that keeps a model right: write the unit rate, then scale it. On a timed test, skipping an item you cannot set up in a few seconds and coming back to it is usually worth more than grinding.
Where candidates lose it
Nine is the whole trap, and it comes from answering the rhythm of the sentence rather than the arithmetic. Under a clock, strong candidates say it because it sounds finished.
The second slip is 81, which is the number of analyst-days, not reports. Say the units out loud, reports per analyst-day, and both errors disappear.
What the interviewer asks next
- If 5 machines make 5 widgets in 5 minutes, how long do 100 machines take to make 100 widgets?
- Two analysts together write a report in 6 days; one alone takes 10. How long does the other take alone? (15 days)
- How would you pace a timed test where finishing every item is not expected?
Asked at Point72, Investment Research, New York, 2026 (Wall Street Oasis):
Final round was scheduled shortly after as well as a Wonderlic/Personality test.
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.
