Cambridge Mathematics interview

The Cambridge Mathematics interview is, in the Faculty's words, "maths, followed by some more maths". You work through unfamiliar problems out loud with College Fellows, who deliberately set problems you'll get stuck on, then give hints to see how you use them. Most applicants have two interviews of about 25 to 30 minutes, often one pure and one applied, in December 2026 (main period 7 to 18 December). Whether they're online or in person depends on the College.

Key facts

What happens in the interview

Almost all the time goes on unfamiliar problems, mostly numbers, sequences, integrals and graph sketching, and sometimes probability and basic mechanics. Mathematics with Physics applicants always get at least one mechanics question. There are no trick questions. The University says interviewers may also ask about your personal statement, but the Faculty describes Maths interviews as all maths.

Interviewers run it as a conversation, in an informal atmosphere:

• They deliberately set problems where you'll get stuck, then give hints and watch whether you can use them. They don't leave you stuck.
• They push with "why?" and ask for a sketch.
• They test a special case, change a condition or ask "can this be extended?"
• If you solve something straight away, they move on briskly.

Not finishing a problem, making a slip or saying "I don't know yet" is normal. You can ask the interviewers to explain anything you don't understand.

You'll work on paper, or on a tablet or shared whiteboard online, and may be asked to share your working. Have paper and a pen beside you, or a stylus and tablet if your College asks for one.

Colleges run the interviews, so practice varies. Trinity gives one interview of 40 to 45 minutes straight after a short written test, Churchill usually one of 30 to 40 minutes, and Christ's sets a problem sheet before one of its two interviews.

What interviewers look for

The Faculty of Mathematics looks for three qualities:

• Mathematical knowledge and skills: school maths used accurately and fluently to solve a problem.
• Mathematical ability: thinking logically, making connections, and picking up a new idea quickly and using it on challenging problems.
• Motivation: real enthusiasm for maths, and the perseverance to keep going when you're stuck.

Toby's practice interview scores these three, plus mathematical insight (seeing structure and connections, sketching and predicting behaviour, and asking "what if?" to generalise) and explaining your thinking (talking through what you're doing and why, precisely, including what you're unsure of).

How the interview is marked

Cambridge doesn't publish a marking scale for Maths interviews. The Faculty says it doesn't simply grade and add up the parts of an application. Colleges weigh the interview together with your grades, reference, personal statement and TMUA, and offers usually depend on STEP grades in the summer. About a third of places each year go to people who missed their STEP grades, after Colleges look at their whole application and their STEP scripts.

Anyone can have an interview that doesn't go to plan, and a bad interview isn't the end of an application.

Example questions

These are problems published by Cambridge and Oxford sources, in the form Toby asks them. Interviewers set their own questions, so yours will differ.

Root 101 by hand

Published on a Cambridge Physics (Cavendish Laboratory) page: Physics and Maths for Natural Sciences sample paper

Find √101 to 4 significant figures, without a calculator. Talk through how you get there.

• Why interviewers ask it: it shows whether you can make a quick, controlled approximation, and say which way the error goes and roughly how big it is.
• A hint an interviewer might give: "Which perfect square is close to 101? So how big is the little bit you need to add to 10?"
• A strong answer makes a quick first approximation, then checks the fourth figure by squaring nearby values instead of trusting the approximation, and says straight away whether the estimate is slightly too big or too small.
• A common slip is thinking a 1% increase in a number gives a 1% increase in its square root.

Tiling a 2 by n grid

Published by Oxford: sample interview questions

How many ways are there to cover a 2 × n rectangular grid with 2 × 1 tiles? Each tile covers two neighbouring squares, with no gaps or overlaps. Don't worry about the general answer yet: sketch the cases n = 1 to 4 on paper and count them.

• Why interviewers ask it: it shows whether you can organise small cases systematically, spot a pattern and then prove it with a clean case split, rather than trusting the first few terms.
• A hint an interviewer might give: "Look at the left-most column, the two squares at the left-hand end. In how many different ways can a tile cover the top one?"
• A strong answer lists the small cases in an organised way, for example by what happens at the left-hand end, then proves the pattern with cases that don't overlap and checks the starting values.
• A common slip is guessing the pattern from four terms and never justifying it.

Three points chasing each other

Published by Downing College, Cambridge: sample maths test

Suppose that P, Q, and R are three points in a plane, such that PQ = PR = QR = 1. At each point in time, P is moving toward Q, Q is moving toward R, and R is moving towards P, all with speed v = 50. At what time T will all the points reach the centre of the triangle?

• Why interviewers ask it: it shows whether you use symmetry to tame a hard-looking motion, and resolve velocities sensibly.
• A hint an interviewer might give: "What shape do the three points make a moment later? And a moment after that?"
• A strong answer uses the symmetry before doing any algebra, and resolves the velocities carefully instead of guessing what the paths look like.
• A common slip is assuming each point travels one side length, or heads straight for the centre.

Practise with Toby

Toby is an AI interviewer that runs practice interviews by voice. The Cambridge Mathematics interview is built from Cambridge's admissions pages, the Faculty of Mathematics' guidance and College admissions pages, and follows the common pattern of two interviews:

• Pure maths interview. Unfamiliar pure problems talked through aloud: numbers, sequences and series, inequalities, graph sketching, proof and counting.
• Applied maths interview. Calculus, approximation, probability and basic mechanics, often with a sketch. Mathematics with Physics applicants always get a mechanics question.

Not every College splits pure and applied this way.

It draws on 38 problems, each with hints and follow-ups. Fifteen are published problems from Cambridge and Oxford sources, and the rest were written to the same style and level, with every result independently re-derived.

Toby puts the problem on screen, gives you time to think and asks you to talk it through. He gives a hint only when you're stuck, asks why, and changes a condition to see if your reasoning holds. Like a real interviewer, he doesn't praise every answer.

You can add your personal statement so Toby can ask about it briefly, and choose a 10, 20 or 30-minute interview. Your prep page has the format, what interviewers look for and practice problems with hints and worked solutions. Afterwards, your debrief gives you a practice mark out of 10 (6 is acceptable, 7 good, 8 strong), a score against each criterion, notes on every question and the full transcript. A practice mark is a guide, not a prediction.

Frequently asked questions

Sources

Last checked October 2026.

• University of Cambridge: Mathematics
• University of Cambridge: Interviews
• University of Cambridge: Preparing for interview
• Cambridge Faculty of Mathematics: How to apply
• Cambridge Faculty of Mathematics: Admissions FAQ
• Cambridge Faculty of Mathematics: Guide to Admissions
• Cambridge Physics (Cavendish Laboratory): Physics and Maths for Natural Sciences sample paper
• University of Oxford: Sample interview questions
• Downing College, Cambridge: Sample maths test

Related interviews

• Oxford Mathematics interview
• Cambridge Computer Science interview
• Cambridge Natural Sciences interview

All Oxford and Cambridge interviews·University admissions

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