Cambridge Maths Interview Questions 2026 with Model Answers
Six problems worked in full, one of them laid out the way the interview packs are.
Six problems worked in full, one of them laid out the way the interview packs are.
Reading a model answer is not the same as producing one under pressure, and this page can only give you the first. The Maths I pack is the second: the same class of problem, but set cold, with nothing to look at until you have made an attempt of your own. Maths is one of the nine subjects whose sample is free to read, so you can judge the standard before you decide.
Updated April 2026 for 2026/27 entry. Cambridge Maths interviews test how you think through problems, not just whether you know the answer. There are no formula sheets, and reasoning aloud is essential. Interviewers want to see your mathematical instincts, your willingness to explore, and how you respond when pushed beyond familiar territory. Six problems are worked below, and the sixth is set out in the three layers the paid interview packs use. The resources hub splits Maths into five packs at £180 each, and the table below says which one covers what.
The Maths I pack — £180
Ten questions across 26 pages. Each one is set out on its own to attempt first, with the hints held back until you need them, and the full worked answer last. One PDF, one payment, instant download.
Get the Maths I pack — £180Read the free sample firstCambridge Maths interviews are conducted by college Fellows, typically lasting 20 to 30 minutes per interview. Most candidates have two interviews at their college, and some receive a third at a different college if they are being considered as a pool candidate. The format is problem-based: you will be given questions at the board or on paper and expected to work through them in real time, talking as you go.
The core skill being assessed is mathematical thinking under guidance. Interviewers are not looking for instant correct answers. They are looking for candidates who can identify structure in a problem, make reasonable conjectures, test ideas, and adjust when something does not work. If you are guided toward a solution through hints, that is not a failure — it is part of the process. What matters is how you respond to those hints.
Cambridge draws on a wide spread of technique, and the hub carries five distinct £180 Maths sets to match it. Every row below is taken from that set's own card on the hub.
| Maths pack (£180 each) | What it covers | Free sample? |
|---|---|---|
| Maths I | The general set: algebra, calculus, proof, geometry, sequences, graph sketching | Yes |
| Maths II | The second general set, over the same areas as Maths I | Yes |
| Graph Sketching | Rational, modulus and parametric curves, plus potential-energy and Boltzmann sketches, each answered through six stages in order | No |
| Derivations | Series and algebraic derivations, kinematic and energy-method physics derivations, proofs by induction | No |
| Integration & Curve Sketching | By parts, substitution, partial fractions, areas and volumes of revolution, improper integrals, sketch-to-integral problems | No |
The free Maths sample is a single PDF, and it is the one both general sets offer. It runs to six pages and roughly 857 words: two questions on the first page, a page of hints on the second, then four pages of worked solutions. That is the same three-layer shape a paid set uses, at a fraction of the length and with two questions rather than ten.
Topics that appear regularly include: graph sketching, differentiation and integration, series and sequences, proof by induction, combinatorics, coordinate geometry, and introductory real analysis (such as limits and continuity). A-level Further Maths content is commonly assumed, particularly for questions involving matrices, complex numbers, and hyperbolic functions. Candidates who have worked through STEP I and STEP II problems will recognise the style immediately.
A printed pack cannot withhold, and that is its real limit
Every problem on this page can be solved on paper in a quiet room. A Cambridge interview cannot: you have to narrate the attempt, including the part where you try something and it fails, because a blank pause tells the interviewer nothing. That much a written resource can rehearse.
What it cannot rehearse is rationing. In the room the hint arrives only once you have stalled, and it is shaped by the thing you just said. In a PDF the questions, the prompts and the worked answers sit in one file, and nothing stops you scrolling to the last layer a minute in. So the discipline has to come from you, and it costs nothing: cover the lower layers, take the problem to a written answer of your own, and uncover the next layer only when you have genuinely stopped moving. Nobody can enforce that for you, and that is the honest limit of anything printed.
Step 1 — Domain and intercepts: The function is defined for all real x. At x = 0, y = 0. As x → −∞, x² grows but e^(−x) grows faster, so y → +∞. As x → +∞, e^(−x) decays faster than x² grows, so y → 0.
Step 2 — Turning points: Differentiate using the product rule: dy/dx = 2xe^(−x) − x²e^(−x) = xe^(−x)(2 − x). Setting this to zero gives x = 0 (minimum, since y = 0 here and the function is positive nearby for x > 0) and x = 2 (maximum, y = 4e^(−2) ≈ 0.54).
Step 3 — Concavity: The second derivative confirms x = 2 is a local maximum. There is an inflection point between x = 0 and x = 2, and another for x > 2.
Key features to describe aloud: "The function touches zero at the origin, rises to a maximum near x = 2, then decays toward zero as x increases. For negative x, it grows without bound because e^(−x) becomes large."
What the interviewer expects to hear: A candidate who identifies the asymptotic behaviour first, then finds turning points systematically, rather than jumping straight to differentiation without context.
Step 1: Use integration by parts with u = ln(x), dv = x dx. Then du = 1/x dx, v = x²/2.
Step 2: ∫ x ln(x) dx = (x²/2)ln(x) − ∫ (x²/2)(1/x) dx = (x²/2)ln(x) − x²/4 + C.
Step 3: Evaluate from 0 to 1. At x = 1: (1/2)(0) − 1/4 = −1/4. At x = 0: x²ln(x) → 0 (since x² → 0 faster than ln(x) → −∞). So the answer is −1/4.
What the interviewer expects to hear: An explicit comment on the limit at x = 0 — "I need to check this is not an improper integral that diverges" — before confidently stating the result.
Step 1: Recall that Σ xⁿ = 1/(1−x) for |x| < 1. Differentiate both sides with respect to x: Σ nxⁿ⁻¹ = 1/(1−x)². Multiply through by x: Σ nxⁿ = x/(1−x)².
Step 2: Substitute x = 1/2: Σ n(1/2)ⁿ = (1/2)/(1/2)² = (1/2)/(1/4) = 2.
What the interviewer expects to hear: "I recognise this as related to the geometric series — if I differentiate the standard result and multiply by x, I can generate the n·xⁿ form." Connecting the problem to a known technique is exactly the reasoning interviewers want to see verbalised.
Base case: n = 1: LHS = 1, RHS = 1(2)(3)/6 = 1. ✓
Inductive step: Assume true for n = k: Σ r² = k(k+1)(2k+1)/6. For n = k+1, add (k+1)² to both sides: k(k+1)(2k+1)/6 + (k+1)² = (k+1)[k(2k+1)/6 + (k+1)] = (k+1)[(2k²+k+6k+6)/6] = (k+1)(2k²+7k+6)/6 = (k+1)(k+2)(2k+3)/6. This matches the formula for n = k+1. ✓
What the interviewer expects to hear: A clear statement of the inductive hypothesis before the algebra begins, and a final sentence confirming the result follows by induction. Cambridge interviewers often push further — "Can you prove this without induction?" — so be ready to discuss a combinatorial or telescoping approach.
Assume √2 = p/q in lowest terms (p, q integers, no common factor). Then 2q² = p², so p² is even, meaning p is even. Write p = 2m. Then 2q² = 4m², so q² = 2m², meaning q is also even. This contradicts p/q being in lowest terms. Therefore √2 is irrational.
What the interviewer expects to hear: The candidate should name the method — "I'll use proof by contradiction" — before starting, and explicitly state where the contradiction arises rather than leaving it implicit.
Those five are finished solutions. A pack is assembled in the opposite order.
Above, the reasoning arrives already put together, which is the one arrangement an interview never gives you. A pack inverts it: a questions layer on its own, then a prompts layer holding the nudges an interviewer would ration out, and only then a suggested answer written first person as a walkthrough. Ten questions in that shape. Maths I is the set whose topics line up with the five problems above — algebra, calculus, proof, geometry, sequences and graph sketching. The next section is one question in that same order, for nothing.
Maths I on the resources hub — £180This one is set out the way the packs are, layers kept apart and named. Work it before you read on.
A fair coin is tossed repeatedly until two heads appear in a row. On average, how many tosses does that take?
Then, once you have an answer: the same coin, tossed until a head is immediately followed by a tail. Is the average wait longer, shorter, or the same?
The nudges, in the order an interviewer would release them. Take one, go back to the problem, and return for the next only once you have stopped moving.
The attempt that fails. The natural first move is to treat two heads in a row as an event of probability ½ × ½ = ¼, then quote the standard waiting time: if something happens with probability p, you wait 1/p attempts for it, so 1/(¼) = 4. It is quick, it is the obvious move, and it is wrong.
Why it is wrong matters more than the number, because that is what gets asked next. The 1/p result counts independent attempts, and these pairs are not independent: tosses 1 and 2, 2 and 3, 3 and 4 overlap, so one toss finishes a pair and starts the next at the same time. The units warn too. Four pairs would be eight tosses; four tosses would be four. If you cannot say which your 4 counts, the model is not doing what you think.
The attempt that works. Carry only the part of the history that changes what happens next. At any moment you are in one of three situations: no progress, because you have just started or just thrown a tail; one head in hand; or finished. Write E₀ for the expected number of further tosses from no progress and E₁ for the expected number from one head.
From no progress, throw once. Half the time a head takes you to the one-head situation; half the time a tail leaves you where you began:
E₀ = 1 + ½E₁ + ½E₀, so ½E₀ = 1 + ½E₁, so E₀ = 2 + E₁.
From one head, throw once. Half the time the second head arrives and you stop; half the time a tail does not merely fail to help — it destroys the head you were holding and returns you to no progress:
E₁ = 1 + ½(0) + ½E₀.
Substitute the first line into the second: E₀ = 2 + 1 + ½E₀, so ½E₀ = 3, so E₀ = 6. The average wait is six tosses, not four, and E₁ = 4 — once a head is in hand, four more tosses on average.
The second question, which is what the first one was for. Run the same two lines for head-then-tail. The no-progress line is unchanged: E₀ = 2 + E₁. The one-head line is not: a tail now finishes the job, and a head costs nothing at all, since it leaves you holding a head, which is where you already were:
E₁ = 1 + ½(0) + ½E₁, so E₁ = 2, and E₀ = 4.
Four tosses for head-then-tail against six for head-then-head, from the same coin, both patterns having probability ¼. The difference is entirely what a failure costs: chasing two heads, a tail destroys your progress; chasing head-then-tail, a head costs nothing. Which is why the discarded 4 is not simply wrong — it is right for one pattern and wrong for the other, and a candidate who gave 4 to both cannot tell.
If the interviewer wants a check rather than a repeat: for a fair coin the expected wait for a pattern is the sum of 2k over every k where the pattern's first k letters match its last k. HH scores on both k = 1 and k = 2, giving 2 + 4 = 6. HT scores only on k = 2, giving 4. Both answers again, without a single state.
What is being tested. Not probability. The question is whether you will abandon a remembered formula for a model you build on the spot once the formula stops fitting, and whether you can name the quantity your own equation is about. The state argument is four lines long and uses nothing past GCSE algebra; what it needs is the decision to stop reciting and start defining. That decision, and how long you take to make it after the first answer is questioned, is precisely the response-to-a-hint this page describes at the top as the thing being marked.
The hint is the part of the interview no worked solution contains
A Cambridge interviewer pushes until you are stuck, then offers a hint, and what is learned is how fast the hint is taken up. A finished solution cannot put you through that, because by the time you are reading one the stall is already over. The prompts layer is shaped for exactly that gap: nudges rather than answers, so a stuck candidate can take one and go back to the problem instead of to the ending.
Of the five Maths sets, Maths I and Maths II are the two with a free sample, so the arrangement can be judged before anything is paid for. Graph Sketching, Derivations, and Integration and Curve Sketching have none, and each says so on its own card.
See Maths I on the hub — £180Getting stuck is not unusual — it is built into the interview format. Interviewers will often present problems that are genuinely difficult, then offer hints to see how you respond. The worst thing you can do is go silent. Here is a practical script for when you do not know where to start:
"Let me think about what kind of problem this is. I can see it involves [topic]. My first instinct is to try [approach], because [reason]. I'm not sure that will work immediately, but let me see what happens if I..."
If you are completely stuck after a minute: "I'm not immediately seeing a route in. Could I ask — is this related to [technique]? I want to make sure I'm not missing something obvious before I try something more complicated."
This approach demonstrates intellectual honesty, mathematical awareness, and the ability to ask productive questions — all qualities Cambridge tutors value highly.
STEP preparation and interview readiness: Cambridge explicitly expects applicants to engage with STEP (Sixth Term Examination Paper) problems as part of their preparation. STEP I and STEP II questions are excellent interview practice because they require extended reasoning, creative approaches, and the ability to work without a template. If you find full STEP questions overwhelming at first, work through the individual parts rather than the whole question, and focus on understanding why each step works rather than memorising solutions. The STEP Support Programme, run freely by Cambridge's Faculty of Mathematics, provides structured problem sets that build exactly the kind of thinking interviewers are looking for.
Both universities use problem-based interviews, but the emphasis differs in important ways. Oxford Maths interviews tend to lean toward pure mathematics — proof, abstraction, and formal reasoning. Candidates may be asked to work with definitions they have not seen before and construct arguments from first principles. The style rewards candidates who are comfortable with rigour and precision in mathematical language.
Cambridge Maths interviews place greater emphasis on applied problem-solving and the ability to work through unfamiliar problems with guidance. Questions are more likely to involve concrete calculations — integration, series, graph sketching — alongside proof. The interviewer's role is more explicitly collaborative: they will nudge, hint, and redirect. Cambridge also places significant weight on STEP performance at offer stage (typically STEP II and STEP III grades are required), which means the interview and the STEP paper are part of a coherent preparation strategy.
In short: Oxford rewards abstract rigour; Cambridge rewards adaptive, guided problem-solving with strong technical foundations. Strong candidates prepare for both, but should calibrate their practice accordingly.
No. STEP is sat in the summer after offers are made, and for Cambridge Maths it normally appears as a condition of the offer rather than as a shortlisting test before interview. What carries over is the style of thinking, not the paper: STEP questions run long, start from a setup you have not met, and reward a candidate who builds a method instead of recalling one, which is what an interview problem asks for too. The STEP Support Programme, run free by Cambridge's Faculty of Mathematics, is the most direct place to practise that without spending anything.
Most Cambridge Maths applicants have two interviews, both usually at their applied college. If a college does not make an offer but considers the candidate strong, they may be placed in the winter pool and interviewed by a second college. A small number of candidates have three interviews in total. Each interview typically lasts 20 to 30 minutes.
It is not a formal requirement, but in practice the vast majority of successful applicants have studied Further Maths at A-level. Interview questions regularly draw on Further Maths content — complex numbers, matrices, hyperbolic functions, and proof by induction. Candidates without Further Maths will need to cover this material independently and should be upfront with their school about doing so as early as possible.
Start with individual parts of STEP I questions rather than full problems. The STEP Support Programme (free, run by Cambridge) provides scaffolded problem sets designed to build up gradually. You can also use TMUA (Test of Mathematics for University Admission) questions as a stepping stone — they are harder than A-level but more accessible than STEP, and they develop the same style of thinking. Work on understanding the reasoning behind each step, not just reaching the answer.
Cambridge Maths interviews are genuinely challenging, but they are designed to be a conversation, not an interrogation. The candidates who perform best are those who have practised thinking aloud, who treat hints as useful information rather than signs of failure, and who have spent time working through problems that do not yield to standard methods. That kind of preparation takes time, but it is entirely achievable before the 2026/27 application cycle.
Maths I. It is the general set, and its topics — algebra, calculus, proof, geometry, sequences and graph sketching — cover the range a Cambridge Maths interview draws on. Maths II runs a second general set over the same areas. The other three are narrow by design: Graph Sketching, Derivations, and Integration and Curve Sketching each take one technique a long way, which makes them a follow-up rather than a starting point. Maths I and Maths II are also the two that carry the free sample; the other three state on their own cards that they have none.
It may not, and that is close to why it was chosen. A Cambridge Maths interview is not built to walk a syllabus, so the topic a problem wears is nearly incidental; what recurs is the demand underneath it. The coin question needs no result past GCSE algebra, and candidates still lose it by reaching for a formula they already knew instead of defining what they were counting. That same failure would look identical dressed as a geometry problem, or a series, or a sketch. Prepare for the demand rather than the topic list.
Official Resources
The free Maths sample is two questions, a page of hints and four pages of worked solutions. If that shape is not how you want to work, no paid set will change your mind. If it is, Maths I and Maths II run ten questions each in that same three-layer shape at £180 each, and the hub takes 10% off two packs or 20% off three or more automatically at checkout.
Download the free Maths sampleThe interview packs are written by specialist tutors and are sold on the resources hub, not here.