Oxford University college skyline at dusk

What interviewers are checking

How you think out loud: They want your reasoning pathway, not just the final answer.
Problem-solving under constraint: Can you simplify, approximate, or model quickly?
Fluency with first principles: Algebra, calculus, mechanics, electricity, vectors, probability.
Resilience: Handling curveballs, regrouping after a dead end, and using hints effectively.

Core resources (official + trusted)

Oxford PAT past papers (official): PAT archive
Cambridge ESAT info: ESAT overview
STEP past papers + solutions: STEP archive
MAT past papers (Oxford Maths/CS): MAT archive
NSAA past papers (Cambridge NatSci): NSAA archive
NRICH maths problem sets: NRICH (focus on “Warm-up” and “Olympiad” levels for structure/rigour)
Isaac Physics: Isaac Physics graded problems and “problem solving in physics” boards.
I Want to Study Engineering: IWSE short, interview-style problems with hints/solutions.

Pick 3–5 at random, speak your reasoning aloud, time yourself.

🔬 Physics Interview Problems (20)

Why does the boiling point of water increase when salt is dissolved in it?
Estimate how many air molecules are in this room.
What makes some chemicals explosive while others burn slowly?
How would the period of a pendulum change if the amplitude is large rather than small?
Why does a ball bouncing on the ground never return to its original height?
Explain how an airplane wing generates lift.
Why does charge accumulate on sharp points in conductors?
How much energy is required to boil a kettle of water?
If the gravitational constant doubled overnight, what observable effects would we notice?
Explain what happens at a physiological level during a hot flush in menopause.
Why is the sky blue, and why are sunsets red?
If the Earth stopped spinning suddenly, what would happen to objects on the surface?
How would you design an experiment to measure the speed of sound using minimal equipment?
Why do heavier objects not fall faster than lighter ones in a vacuum?
How does a fiber-optic cable transmit light over long distances without significant loss?
Why do satellites stay in orbit rather than fall back to Earth?
How can you estimate the height of a building using a barometer?
Explain how a refrigerator cools items inside it, and what happens to the energy removed.
Why does adding more resistors in parallel reduce the total resistance?
Describe the energy transfers that occur when a car brakes to a stop.

📐 Mathematics Interview Problems (20)

Prove there are infinitely many prime numbers.
Show that 0.999... = 1.
How many different routes are there from (0,0) to (5,5) on a grid if you can only move right or up?
Find the maximum value of f(x) = x e^{-x}.
Prove whether √2 is rational or irrational.
Sketch y = 1/(1 + x²) and describe its key features.
Find the area under y = x² between 0 and 1 without using the fundamental theorem of calculus.
A curve has derivative f'(x) = 3x² - 4x + 1. What can you deduce about the curve's shape?
How many trailing zeros are there in 100!?
Is there a largest real number?
Solve the differential equation dy/dx = y.
Prove that the sum of the first n odd numbers equals n².
How many handshakes occur if 30 people all shake hands once with each other?
Does there exist a function that is continuous everywhere but differentiable nowhere?
Estimate the value of π using a geometric or probabilistic method.
How many ways can you arrange the letters in the word "OXFORD"?
Find the limit of sin x / x as x → 0 without using L'Hôpital's rule.
Can the sum of the reciprocals of all prime numbers converge?
If a square and a circle have the same perimeter, which encloses more area?
Is 1 a prime number? Justify your answer.

Mechanics: projectile motion with/without air resistance, circular motion, energy vs. force approaches.
Electricity: series/parallel reasoning, potential dividers, RC time constants.
Waves: interference conditions, Doppler shifts, boundary reflections.
Thermo & gases: ideal gas approximations, energy distributions, simple cycles.
Estimation: Fermi problems (e.g., “How many photons hit a solar panel per second?”).

Functions & graphs: sketching with asymptotes/turning points; odd/even; periodicity.
Calculus: optimisation, related rates, series approximations, integration tricks.
Algebra: inequalities, sequences/series, binomial expansions, functional equations.
Vectors/geometry: projections, shortest distance, locus questions.
Probability/combinatorics: conditional probability, expected value with constraints.

PAT/ESAT/NSAA: practise mixed sections; focus on rapid diagramming and units.
MAT: practise full solutions with clear exposition—show elegance, not just correctness.
STEP: timebox by question; learn when to pivot if algebra explodes.
Debrief: After each paper, write three bullets: (a) fastest win, (b) biggest trap, (c) a cleaner alternative method.

Light review of your own solutions, not new papers.
Write a one-page cheatsheet of heuristics and common pitfalls (units, sign errors, forgetting constraints).
Sleep, hydration, and a test of your mic/camera if online.

Run structured mock interviews with timed prompts, guided hints, and post-review of your reasoning.
Build a personalised problem pack targeting your weak themes (mechanics/graphs/probability).
Calibrate your “thinking aloud” style so it’s clear, concise, and confident under pressure.

Ready for a mock? Share your target college/course and the test you’re sitting via the contact page, and I’ll set up a focused interview rehearsal.