Free Interview Resources
Free practice questions, preparation advice, and expert insights for Mathematics and Philosophy interviews at Oxford.
Real Mathematics and Philosophy interview questions in the style Oxford asks. Try answering each one aloud before you reveal the hint.
A ladder slides down a vertical wall; track the midpoint of a marked rung as it falls. What curve does the point trace?
Problem-Solving
Hint
Model the ladder, wall and floor as a right-angled triangle and write the midpoint coordinates in terms of the intercepts.
How many ways can a 2 by n rectangle be tiled using 2 by 1 dominoes?
Problem-Solving
Hint
Start with n = 1, 2, 3 and 4, then look for a recurrence from the first tile placement.
If a marked point is one-third of the way along a falling ladder rather than at the midpoint, how does the traced curve change?
Problem-Solving
Hint
Reuse the midpoint method but replace the midpoint coordinates by a weighted average of the endpoints.
After solving the 2 by n domino problem, what changes if the grid is 3 by n and the tiles are 3 by 1?
Problem-Solving
Hint
Separate cases by the first column and ask what remains after each legal first placement.
Find the normal to the curve below at x = 1, then use its intercepts to determine the length between the axes.
Problem-Solving
Hint
Differentiate to find the tangent gradient, take the negative reciprocal for the normal, then find the two intercepts.
Tutorial-style interviews with subject-specific problems, often involving unfamiliar material.
Oxford interviews typically take place at the college you applied to. You will usually have two or three interviews of around 20-30 minutes each, sometimes at different colleges if you are pooled. The atmosphere is meant to resemble a tutorial: the interviewer gives you a problem and watches how you reason through it.
Interview Invitation
Late Nov
Arrival to Interview
Early Dec
Technical Question
Mid Dec
Decision
Early Jan
Interview Invitation
Late Nov
Arrival to Interview
Early Dec
Technical Question
Mid Dec
Decision
Early Jan
Find the total area enclosed between the curve below, the x-axis and the vertical lines x = -2 and x = 2.
midHint
Locate where the curve crosses the axis before integrating, because signed area is not the same as total area.
Evaluate the finite sum below.
entryHint
Compute the first few terms and look for periodic behaviour rather than expanding all 39 terms.
Find the largest angle in the stated interval that satisfies the trigonometric equation below.
hardHint
Try rewriting the equation in terms of sin(2x) or use identities to reduce the expression to a standard form.
Give an example of an argument with false premises but a true conclusion.
entryHint
Separate validity from truth: the conclusion can be true even if the support offered for it is bad.
What exactly is involved when one person blames another?
midHint
Start with a simple proposal, then test it against cases where someone judges wrongdoing but does not blame.
If a machine could give you every enjoyable experience while hiding that none of it was real, would you choose to enter it?
midHint
Distinguish pleasure, value, reality, agency and whether a good life is subjective or objective.
Are our deaths bad for us?
hardHint
Separate the process of dying from the state of being dead, then ask who or what is harmed.
How would you distinguish lying, deceiving and misleading?
midHint
Try definitions, then test them with cases where someone tells a literal truth but creates a false impression.
What mathematical idea from your current studies would you choose to explain, and why is it deeper than it first looks?
entryHint
Choose a topic you genuinely know well, then move from technique to the underlying idea or proof.
Which idea in logic, set theory or philosophy of mathematics first made the connection between mathematics and philosophy feel natural to you?
midHint
Link a specific book, problem or argument to the bridge papers on logic, set theory or philosophy of mathematics.
Take a definition you have recently learned in mathematics and work out one consequence that is not immediately obvious.
hardHint
Pick a definition with examples and non-examples, then test edge cases before trying to prove a general claim.
Can a perfectly valid mathematical proof be unexplanatory?
hardHint
Separate proof as verification from proof as explanation, then look for examples such as brute-force cases or non-constructive arguments.
If two different axiomatic systems prove different things, what should a mathematician or philosopher conclude?
hardHint
Avoid treating axioms as ordinary empirical claims; ask what follows within a system and what motivates choosing a system.
Is an individual's plane journey morally harmless if the flight would depart anyway?
midHint
Ask what counts as a moral reason and compare the case with voting or participation in collective actions.
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Book Mock Interview →Watch & Learn
Shows the style of an Oxford mathematics interview and tutor-candidate interaction.
Useful for seeing a single problem developed through discussion.
Provides course-specific insight into the joint degree.
Useful legacy problem-solving practice because Oxford notes continuity between MAT-style problem solving and TMUA preparation.
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Further Reading
by Kevin Houston
Good for shifting from school-style calculation to proof, definitions and mathematical explanation.
by T. W. Körner
Offers problem-led mathematical stories that encourage flexible reasoning.
by Ian Stewart
Useful for reflecting on what mathematical study at university feels like.
by Simon Blackburn
A broad introduction to philosophical questions suitable for developing argumentative confidence.
by Stewart Shapiro
A direct bridge into the philosophical side of mathematics and foundations.
by University of Oxford Mathematical Institute
Course-family-specific explanation of TMUA use, dates and preparation.
by UAT-UK
Free specifications, specimen/practice tests and past paper archive.
by UAT-UK
Especially relevant to TMUA Paper 2 and the course's logic/foundations overlap.
by University of Oxford Mathematical Institute
Free recorded support for TMUA-style problem solving and interviews.
by University of Oxford
Helps applicants build genuine subject exploration for personal statement and interview discussion.