Equity Research puzzles, solved step by step
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087Monty Hall: there are three doors and a prize behind one. You pick a door. The host, who knows where the prize is, opens a different door with no prize and offers you a switch. Why is the intuitive answer of one half wrong, and what is the chance that switching wins?Squarepoint CapitalLondon · 2026
Try it first
Two doors are left. What is the chance that switching wins?
Show the worked solution
Switching wins two times in three. Your first pick is right with probability one third, and nothing the host does changes that. In the other two thirds of cases the prize is behind one of the two doors you did not pick, and the host, who must open an empty door, is forced to leave the prize door shut. One half is wrong because it treats the host's reveal as random when it is not.
Why does one half feel right?
Two closed doors, one prize: the picture in front of you looks like a coin toss, and intuition prices the picture rather than the process that produced it. One half would be right only if the two remaining doors were equally likely before the reveal and the reveal told you nothing about either of them. Neither is true. Your door was chosen blind; the other door survived a filter run by someone who knew the answer.
Starting from a pick of door 1, the prize is behind door 1, 2 or 3 with one third chance each, and in the two branches where the first pick is wrong the host is forced to open the only empty door, so switching wins in two of three equally likely branches. What exactly does the host's knowledge change?
Think of a shopkeeper who knows which of three mangoes is ripe. You pick one; she then removes an unripe one from the two you did not pick, always careful never to remove the ripe one. She has concentrated the two thirds chance that the ripe mango was among the other two onto the single one she left. Your own pick is untouched at one third, because she would have been able to remove an unripe mango whatever you picked. The reveal carries information about the other door, not about yours.
When would one half actually be the right answer?
This is the part the interviewer is asking for when they ask why intuition fails. If the host did not know and opened one of the other doors at random, and it just happened to be empty, then switching would win only one half of the time. A random reveal that could have shown the prize rules out the branches where it would have, and what is left is symmetric. So the answer depends on the host's rule, not on the doors. Scale it up to make the point land: with 100 doors, you pick one, and a knowing host opens 98 empty ones; nobody thinks the last closed door is a coin toss.
The analyst's version is reading a signal from someone who knows more than you do. A management team that chooses which numbers to disclose is running a filter, and what survives the filter is not a random sample.
Where candidates lose it
Saying one half is the obvious loss. The subtler one is saying two thirds and then being unable to explain why, which is exactly what this interviewer asked for: they want the reason intuition fails, not the memorised answer.
Have the three-branch tree ready and the ignorant-host variant as the contrast. Showing that the answer changes when the host's rule changes proves you understand the mechanism.
What the interviewer asks next
- The host opens a door at random and it happens to be empty. Now what is the chance switching wins?
- With four doors and a host who opens one empty door, what does switching win?
- Where in reading company disclosures do you meet the same kind of filtered evidence?
Asked at Squarepoint Capital, Quant Research Intern Interview, London, 2026 (Wall Street Oasis):
notably I was asked why the 'intuitive answer' was not true rather than just what the correct answer was, related to the Monty Hall problem
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.
