Quant Puzzles: 12 Worked Interview Problems for 2026

A Diagnostic Set, Not a Question Leak

Jane Street's official Probability & Markets guide says its quantitative trading and research interviews can involve probability, expected value and making markets. SIG says it uses puzzles to let candidates experience decisions under uncertainty. Neither firm says every candidate gets the same 30 classics.

This collection selects 12 problems rather than repeating our dedicated quant brainteasers, probability interview questions and mental maths practice. It is a compact diagnostic set with checked arithmetic and a reusable idea in every solution.

Try each question before revealing the answer under realistic interview pressure. Ten quiet minutes with a wrong approach is more useful than recognising a solution in ten seconds.

Keep a pencil nearby. Several solutions are short only after the state space has been written correctly, and mental arithmetic cannot rescue a bad conditioning event. Say the assumption behind each answer aloud before calculating anything. All numerical answers below were independently recalculated and checked on 8 September 2026.

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Probability and Expectation

1. Monty Hall

There are three doors, with a prize behind one. You choose a door. A host who knows the locations must open an unchosen losing door and offer you the switch. Should you switch?

Solution: Yes. Your original choice wins with probability 1/3. The other two doors collectively hold probability 2/3; the host's constrained reveal moves that whole probability to the one eligible unopened door. Switching wins with probability 2/3.

The host's rule is essential. If the host opens a random door and happens to reveal a loss, the conditioning changes.

2. The birthday threshold

Ignoring leap days and assuming 365 equally likely birthdays, how many people are needed before a shared birthday is more likely than not?

Solution: 23. Compute the complement:

$P(\text{no match among } n) = \prod_{k=0}^{n-1} (365-k)/365$

For 22 people, the match probability is about 47.57%. For 23 it is about 50.73%. The threshold is therefore 23, not because 23 is near half of 365 but because there are 23 × 22 / 2 = 253 possible pairs.

3. Waiting for HT

Flip a fair coin until the consecutive pattern HT appears. What is the expected number of flips?

Solution: Four. Let E₀ be the expected remaining flips with no useful suffix and E₁ after the latest flip is H.

$E_0 = 1 + 0.5E_1 + 0.5E_0$

$E_1 = 1 + 0.5E_1$

The second equation gives E₁ = 2. Substituting into the first gives E₀ = 4. The useful move is to model states, including the overlap where another H leaves us in state E₁.

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Logic and Invariants

4. Ants on a triangle

Three ants begin at the vertices of a triangle. Each independently chooses clockwise or anticlockwise with equal probability and walks towards the next vertex. What is the probability of no collision?

Solution: 1/4. There are 2³ = 8 equally likely direction assignments. Only all-clockwise and all-anticlockwise avoid collision, so the answer is 2/8 = 1/4.

5. The bridge and torch

Four people take one, two, five and 10 minutes to cross a bridge. At most two cross together, they need the torch, and a pair moves at the slower person's pace. What is the minimum total time?

Solution: 17 minutes:

1. one and two cross: 2;
2. one returns: 1;
3. five and 10 cross: 10;
4. two returns: 2;
5. one and two cross: 2.

Total = 2 + 1 + 10 + 2 + 2 = 17. Sending the fastest person back twice is not enough; the trick is using the two fastest to shuttle the torch so the two slowest cross together.

6. Ants on a stick

Ants occupy arbitrary points on a one-metre stick, each walking left or right at one metre per second. On collision both reverse. What is the latest all ants can remain on the stick?

Solution: At most one second. Identical ants reversing on collision are observationally equivalent to ants passing through one another. Every path can therefore be treated as a straight walk to an end, no farther than one metre away.

One second is the upper bound, or supremum if starting points are strictly inside the stick. Saying exactly one second for every arrangement would be wrong.

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Market-Making Arithmetic

7. A conditional dice market

Two fair dice are rolled. You are told their sum exceeds eight. What is the conditional expected sum?

Solution: 10. Eligible sums and ordered outcomes are:

• nine: four outcomes;
• 10: three;
• 11: two;
• 12: one.

So $E[S \mid S > 8] = (9\times4 + 10\times3 + 11\times2 + 12\times1) / 10 = 100/10 = 10$.

A quoted market needs a spread, but no unique bid and offer follows from fair value alone. Risk limits, opponent information and inventory matter.

8. Red cards after new information

Make a market on the number of red cards among the first 26 cards of a shuffled deck. You then learn that eight of the first 10 were red. What is the new conditional expectation for the first 26?

Solution: 104/7, or about 14.86. There are 18 red cards left among 42 unknown cards, and 16 unknown positions remain in the first half:

$8 + 16 \times (18/42) = 8 + 48/7 = 104/7 \approx 14.86$

A common mistake is to use 16 remaining red cards and obtain about 14.1. Observing eight reds means 26 - 8 = 18 remain.

9. Updating after a trade

You quote 50 bid, 52 offered. Another informed participant buys at 52. What should happen to your next quote?

Solution: There is no numeric answer without a model. The trade is evidence that your valuation may be low, while your new short inventory increases the cost of further selling. Both can move the quote, and uncertainty may justify a wider spread.

The test is whether you distinguish information from inventory, not whether you recite "51 at 53". Ask what the counterparty knows and state how your risk limit affects size.

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Mental Maths and Options

10. Fast multiplication

Calculate 47 × 63 mentally.

Solution: Use a nearby round pair:

$(50 - 3)(60 + 3) = 3000 + 150 - 180 - 9 = 2961$

Cross-check with 47 × 60 + 47 × 3 = 2820 + 141 = 2961. A fast answer with a fast independent check is safer than one clever expansion.

11. Doubling at 7%

How many annual periods does £1 take to exceed £2 at 7% compound interest?

Solution: Solve $1.07^n = 2$:

$n = \ln 2 / \ln 1.07 \approx 10.245$

After 10 periods the value is about £1.967; after 11 it is about £2.105. The Rule of 72 gives 72/7 ≈ 10.29 as a useful estimate, but 11 whole annual compounding periods are needed to exceed £2.

12. At-the-money call delta

Under Black-Scholes, assume spot equals strike, zero rates and dividends, 30% annual volatility and one month to expiry. What is the call delta?

Solution: About 0.517. For a spot-ATM call,

$d_1 = 0.5\sigma\sqrt{T} = 0.5 \times 0.30 \times \sqrt{1/12} \approx 0.0433$

and delta is $N(d_1) \approx N(0.0433) \approx 0.5173$.

Rounding this to 0.53 or 0.54 is too coarse for the stated inputs. Delta would also change if "ATM" meant forward-ATM or if rates and dividends were non-zero.

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A Useful Second Pass

Label each miss with the move that would have helped:

• conditional sample space;
• complement probability;
• state recursion;
• symmetry;
• invariant;
• expectation after partial observation;
• information versus inventory;
• approximation plus cross-check.

Then change one premise. Make the coin biased. Change the dice threshold. Add a dividend to the option. A memorised answer breaks; a method survives.

For a book-length workbook see our Green Book guide. For implementation rather than puzzles, move to quant coding interview questions.

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Frequently Asked Questions

Are these real leaked interview questions?

No. They are representative practice problems. Firms change questions and processes, and no named employer has confirmed this exact set.

Why are there only 12 puzzles?

Each problem tests a distinct move, while the linked probability, mental-maths and brainteaser pages provide more volume. Twelve fully worked problems are more useful here than a repetitive catalogue.

Should I memorise the solutions?

No. Re-solve each problem after changing a condition. The transferable method matters more than the original number.

Do all quant interviews use puzzles?

No. Role and firm determine the format. Jane Street publicly discusses probability and markets, while research and engineering interviews may focus more on statistics, coding or system design.

How should I speak during a puzzle?

State assumptions, define variables and explain why you chose the method. If the problem is underspecified, say what extra information would determine the answer.

What should I practise next?

Use a role-specific set: probability and market-making for trading, statistical research design for researcher roles and timed coding for developer roles.