Skip to main content
Cambridge Mathematics interview preparation

Free Interview Resources

Cambridge Mathematics Interview Questions

Free practice questions, preparation advice, and expert insights for Mathematics interviews at Cambridge.

1-2 interviews · supervision-styleFormat

Sample Cambridge Mathematics Interview Questions

Real Mathematics interview questions in the style Cambridge asks. Try answering each one aloud before you reveal the hint.

01

Four towns lie on a straight line, and you must make separate return journeys from one depot to each town. Where should the depot be placed to minimise the total distance driven?

Problem-Solving

mid

Hint

Model the town positions as four numbers on a line and write the total distance as a sum of absolute values.

02

One coin in a pile of 129 coins has heads on both sides; all the others are fair. A randomly chosen coin is tossed eight times and lands heads every time. What is the probability of heads on the next toss?

Problem-Solving

hard

Hint

Use Bayes' theorem to update the probability that the chosen coin is double-headed after seeing eight heads.

03

An urn contains five balls of each of four colours. Three balls are drawn without replacement. Write expressions for the probabilities of drawing exactly one red ball, of drawing three different colours, and of drawing more blue balls than yellow balls.

Problem-Solving

mid

Hint

Count favourable selections using combinations before dividing by the total number of three-ball selections.

04

A dodecahedron is shown to you. Without counting every edge one by one, work out the number of edges and vertices, then reason about how many embedded cubes could be formed using its vertices.

Problem-Solving

hard

Hint

Use faces, incidences and symmetry; Euler's formula may help with the first part.

05

How many axis-aligned squares can be formed using points from a 10 by 10 grid of dots?

Problem-Solving

entry

Hint

Count squares by side length: a square of side k has fewer possible positions as k increases.

Supervision-style interviews with problem-solving and academic discussion, often with two interviewers.

Cambridge interviews usually happen at your first-choice college. Most applicants have two interviews, with some subjects requiring a third at the pooled college. Cambridge interviews tend to involve two interviewers and may include a written assessment or pre-interview task sent on the day.

20-45 minutes per interview2 interviews at first-choice college, possibly 1 more if pooled
  • -Cambridge often sends a pre-reading or stimulus material 20-30 minutes before the interview. Use that time wisely.
  • -At Cambridge, you may be given a piece of paper and asked to work through a problem. Write clearly and explain as you go.
  • -The supervision system at Cambridge is about collaborative learning, so interviewers want to see if you can be "taught" during the session.

Invitation → Decision: the interview timeline

Interview Invitation

Late Nov

Arrival to Interview

Early Dec

Technical Question

Mid Dec

Decision

Early Jan

Problem-Solving

3 questions
01

Evaluate the integral below.

entry

Hint

Look for a substitution where the derivative of the inner function is already present.

02

Find a general expression for the nth derivative of the function below.

mid

Hint

Compute the first few derivatives and look for a factorial and sign pattern.

03

How many ways can you tile a 2 by n board with 1 by 2 dominoes?

entry

Hint

Think about what can happen at one end of the board, then relate the answer for length n to the answer for shorter lengths.

Conceptual & Discussion

5 questions
01

Sketch the graph below, identifying intercepts, asymptotes and any removable discontinuities.

entry

Hint

Factor the numerator and denominator first, then check where the denominator is zero.

02

Compare the behaviour near zero and far from zero for the three functions below.

mid

Hint

Think separately about the bounded oscillation of sine and the multiplying powers of x.

03

Sketch the curve below and explain how you decide its symmetries and key points.

hard

Hint

Look first for symmetry in x and y, then examine what happens when x² or y² equals 0, 1 or 2.

04

Sketch the graph below, describing its range, turning points and behaviour as x becomes large.

mid

Hint

Differentiate after checking the denominator is always positive.

05

Sketch the two cosine curves below and interpret the shaded signed area represented by the integral.

entry

Hint

Be careful about the order of the limits and about where cosine is positive or negative.

Personal Statement

4 questions
01

Which proof from A-level Mathematics do you find most elegant, and what makes it mathematically satisfying?

entry

Hint

Choose a proof you can reconstruct, not just name; focus on the key idea rather than memorised steps.

02

If you could spend thirty minutes with one mathematician, past or present, who would you choose and what would you ask them?

entry

Hint

Pick someone connected to a mathematical idea you can explain, not only a famous name.

03

Tell us about a mathematical idea from your personal statement that changed how you think about proof or problem solving.

entry

Hint

Use one specific example and be ready to explain the mathematics, not only why you enjoyed it.

04

Which area of mathematics have you explored beyond school, and what question did it leave you wanting to answer?

mid

Hint

Connect your reading or independent work to a precise mathematical question.

Curveball & Modelling

3 questions
01

Estimate how far a golfer could hit a ball if the club head completes roughly one full circle in half a second.

mid

Hint

Start with an order-of-magnitude estimate for club-head speed, then decide what idealised projectile assumptions are reasonable.

02

As a ladder slides down a vertical wall, what path is traced by the midpoint of the ladder?

mid

Hint

Represent the ladder endpoints as (a,0) and (0,b), then use the fixed ladder length.

03

Colour the faces of a cube so that no two adjacent faces have the same colour. How many colours are needed, and how would you justify your answer?

mid

Hint

Think about opposite faces and adjacency before trying to list every colouring.

12+ weeks

foundational fluency and problem habits

  • Audit A-level and Further Mathematics topics that commonly appear in problem-solving interviews: algebra, functions, calculus, sequences, probability and graph sketching.
  • Start a weekly problem notebook using STEP Support Programme, NRICH and selected harder school problems.
  • For each problem, write one line on the key idea rather than only recording the final answer.
  • Choose two personal-statement mathematical topics that you can explain without notes.

8-12 weeks

TMUA timing and reasoning accuracy

  • Practise TMUA-style multiple-choice questions under timed conditions and without a calculator.
  • Review mistakes by type: algebra slip, missed condition, poor diagram, weak probability model or slow method.
  • Alternate timed practice with slower exploratory problems so that speed does not replace depth.
  • Begin explaining one solved problem aloud each week to a teacher, mentor or peer.

4-6 weeks

supervision-style interview practice

  • Practise graph-sketching questions where the first step is to identify domain, symmetry, intercepts, asymptotes and limiting behaviour.
  • Work through combinatorics and probability prompts using clear case separation before calculation.
  • Do mock interviews where the interviewer is allowed to interrupt, redirect or offer hints.
  • Prepare short explanations of two proofs or mathematical ideas mentioned in your personal statement.

1-2 weeks

flexibility and communication

  • Redo selected problems from your notebook, this time focusing on spoken structure rather than novelty.
  • Practise saying when you are unsure and proposing a smaller case or simpler model.
  • Check the College interview invitation for timing, platform, equipment and any written-task instructions.
  • Prepare paper, pen, calculator-free working habits and a quiet setup if the interview is online.

the week of

consolidation and calm recall

  • Review key mistakes from your problem notebook rather than starting large new topics.
  • Do one light graph sketch, one probability problem and one personal-statement explanation.
  • Sleep properly and avoid replacing clear thought with last-minute memorisation.
  • On the day, focus on defining the problem, explaining your first step and responding carefully to hints.

Unlock the full guide

  • The full Mathematics question bank, by category, with hints
  • A week-by-week preparation roadmap
  • The common mistakes that cost offers — and how to avoid them

Free Resource

The Complete Cambridge Mathematics Interview Guide

Enter your email to unlock the full question bank, worked approaches, a week-by-week prep roadmap, and the mistakes that cost offers.

No spam. Unsubscribe anytime.

Want a Real Mock Interview?

Practice with a Cambridge Mathematics graduate who's been through it.

Book Mock Interview →

Watch & Learn

Cambridge Mathematics Interview Videos

Simon Singh on Fermat’s Last Theorem

A Numberphile video linked from the Cambridge reading-list ecosystem and useful for mathematical storytelling.

Simon Singh on mathematics in The Simpsons

A Numberphile video useful for accessible mathematical curiosity and discussion beyond the syllabus.

All videos are the property of their respective creators.

Further Reading

Recommended Resources

Book

How to Solve It

by George Pólya

A classic problem-solving text; use it to practise planning, trying special cases and checking solutions.

Book

How to Think Like a Mathematician

by Kevin Houston

Useful for moving from school-style methods toward proof, definitions and university-style mathematical thinking.

Book

Mathematics: A Very Short Introduction

by Timothy Gowers

A concise route into how mathematicians think about abstraction, proof and structure.

Book

Closing the Gap

by Vicky Neale

Good for applicants interested in prime numbers and in how modern mathematical arguments develop.

Book

Advanced Problems in Mathematics

by Stephen Siklos

A strong bridge into STEP-style extended problem solving and careful written solutions.

Tool

STEP Support Programme

by University of Cambridge / MEI

Structured problem-solving material that also helps with interview-style reasoning and post-offer preparation.

Tool

NRICH Mathematics

by University of Cambridge

A Cambridge-based source of exploratory problems, useful for practising multiple solution routes.

Website

Plus Magazine

by University of Cambridge

Accessible mathematical articles that can support wider reading and personal-statement discussion.

Website

Cut-the-knot

by Alexander Bogomolny

A large collection of mathematical puzzles and proofs for applicants who want regular short problem practice.

Tool

Lucy Cavendish College Interview Question Pack (Sciences & Maths)

by Lucy Cavendish College, University of Cambridge

A published source of Oxbridge-style mathematical prompts for oral problem-solving practice.

Frequently Asked Questions

Yes. Cambridge Mathematics applicants are required to take the TMUA, with the standard applicant test window recorded as 12 to 16 October 2026.
Yes, normally. Cambridge Mathematics offers usually include STEP II and STEP III, commonly with grade 1 requirements, as a post-offer condition.
Most applicants should plan for one or two interviews of around 20-30 minutes each. The College invitation is the definitive source for the exact number and timing.
They are academic conversations centred on unfamiliar mathematical problems. Typical areas include numbers, sequences, integrals, graph sketching, probability and sometimes mechanics.
No, not usually. Mathematics applicants are not normally asked to submit written work, although College-specific interview tasks may vary.
Say what you know, define the obstacle, and try a smaller case or a simpler model. Interviewers often use hints to see how you learn and adapt, not simply whether you already know the answer.
No. Sample questions are best used to practise reasoning aloud, checking assumptions and responding to follow-up prompts. Memorised answers are fragile when the interviewer changes a condition.
Choose the mathematical items you mention most prominently and make sure you can explain them precisely. Be ready to reconstruct an example, define key terms and answer a follow-up question.

Related Guides

Get Expert Cambridge Mathematics Interview Coaching

1-to-1 mock interviews with Cambridge graduates who know exactly what Mathematics interviewers look for.

Book a Free Session →