Free Interview Resources
Free practice questions, preparation advice, and expert insights for Mathematics and Computer Science interviews at Imperial College London.
Real Mathematics and Computer Science interview questions in the style Imperial College London asks. Try answering each one aloud before you reveal the hint.
How many binary strings of length n contain no two consecutive 1s? Derive a formula.
Problem-Solving
Hint
Condition on the last bit: strings ending in 0 versus ending in 1. This yields a recurrence you should recognise as the Fibonacci numbers, so aim to set up and justify the base cases cleanly.
You are given a 2D array of size n by n where every row is sorted left-to-right and every column is sorted top-to-bottom. Describe an algorithm to search for a target value in better than O(n squared) time.
Problem-Solving
Hint
Start from a corner where one direction increases and the other decreases (top-right or bottom-left). Each comparison should let you discard a whole row or column, giving O(n) steps.
Prove that the square root of 2 is irrational, then explain where your argument would break if you tried to reuse it for the square root of 4.
Problem-Solving
Hint
Use proof by contradiction with a fraction in lowest terms and a parity argument. The follow-up tests whether you understand which step relies on 2 being prime, since the parity step fails for a perfect square.
Given n coins, one of which is counterfeit and lighter, and a two-pan balance, what is the minimum number of weighings that guarantees you can find it? Justify the bound.
Problem-Solving
Hint
Each weighing has three outcomes, so k weighings distinguish at most 3^k cases. Argue both that this is a lower bound and that a ternary-splitting strategy achieves it.
What does it mean for an algorithm to run in O(n log n) time, and why is that meaningfully better than O(n squared) as n grows large? Can an O(n squared) algorithm ever beat an O(n log n) one in practice?
Conceptual Reasoning
Hint
Define big-O as an asymptotic upper bound ignoring constants. A strong answer notes that constant factors and small n can make the asymptotically worse algorithm faster in practice.
Structured interviews that combine technical problem-solving with motivation and personal statement discussion.
Imperial interviews vary by department. Engineering and Computing tend to be technical with problem-solving elements. Medicine uses a Multiple Mini Interview (MMI) format with several short stations. Most interviews last 15-30 minutes and may include a presentation or group exercise.
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
Explain the difference between a function being continuous and being differentiable. Can you sketch a function that is continuous everywhere but differentiable nowhere?
midHint
Anchor continuity and differentiability in limits, and note differentiability is the stronger condition. The classic pathological example is a Weierstrass-type function; even describing 'infinitely wrinkled' behaviour shows understanding.
In propositional logic, what is the relationship between a statement, its converse, its inverse, and its contrapositive? Which are logically equivalent, and why does that matter for writing proofs?
midHint
A statement is equivalent to its contrapositive, and the converse to the inverse, but not across those pairs. Tie this to why proof by contrapositive is a legitimate strategy.
What is a recursive function, and how does induction justify that a recursive definition actually computes what we intend? Give an example where naive recursion is far slower than an iterative approach.
midHint
Link the recursion's base and recursive cases to the base case and inductive step of a proof. Naive recursive Fibonacci, with its exponential repeated subcalls, is the canonical example.
You mention an interest in both mathematics and computer science. Give an example of a problem that genuinely sits in the overlap, and explain what each discipline contributes to it.
entryHint
Pick something concrete such as cryptography, algorithm correctness, or graph theory, and separate the mathematical structure from the computational concern. Vague enthusiasm without a real example is the trap to avoid.
You listed a programming project on your application. Talk me through a design decision you got wrong the first time, and what changed your mind.
midHint
Choose a real, specific decision and show reflective reasoning rather than a polished success story. Interviewers value honesty about the debugging or redesign process.
You cited a book from your reading. Pick one idea from it that you did not fully understand at first, and explain how your understanding developed.
midHint
Show intellectual honesty by naming a genuine difficulty and how you worked through it. Being able to explain a hard idea plainly matters more than claiming to have found it easy.
Why the joint Mathematics and Computer Science degree specifically, rather than pure Mathematics or pure Computing? What would you lose by taking either single-subject route?
hardHint
Articulate what the intersection offers you that neither alone does, and show awareness of the trade-off in depth. A strong answer reflects genuine course knowledge, not a generic 'I like both'.
Roughly how many distinct games of noughts and crosses are possible? Estimate it and explain your reasoning about what to count.
midHint
Start with a crude upper bound like 9 factorial for move orderings, then discuss why symmetry and early-terminating games reduce it. The interviewer wants to see you reason about overcounting, not recall a figure.
Imagine a version of chess played on an infinite board. Which pieces become more powerful, which become weaker, and can you even define checkmate? Reason it through.
hardHint
There is no single right answer; the point is to reason about mobility, the loss of edge constraints, and whether cornering a king remains possible. Comfort with an unfamiliar, ill-defined problem is what is being assessed.
If you could ask a computer a single yes-or-no question and get a truthful answer, but you had to phrase it entirely in mathematics, what would you ask and why?
hardHint
This probes what you find genuinely interesting; a thoughtful reference to an open problem or an undecidable question shows depth. The reasoning behind the choice matters far more than the choice itself.
12-16 weeks
8-12 weeks
4-8 weeks
2-4 weeks
1-2 weeks
the week of
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Gives applicants a department-level feel for undergraduate mathematics at Imperial.
Explains Imperial undergraduate admissions themes, including applications, admissions tests, and personal statements.
A student-life perspective on the joint course; use as informal context rather than official admissions evidence.
A concise primer on the test format; compare tactical advice with UAT-UK guidance.
Useful for study-technique ideas, but official UAT-UK materials should remain the primary source.
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Further Reading
by Kevin Houston
Useful for bridging from school mathematics to proof, definitions, and rigorous reasoning.
by T. W. Körner
A strong enrichment text for mathematical modelling, estimation, and problem-solving culture.
by Anany Levitin and Maria Levitin
Builds algorithmic reasoning in a puzzle format suitable for mathematics-computing applicants.
by A. K. Dewdney
Introduces a broad range of computer science ideas without requiring a full undergraduate background.
by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik
A demanding but relevant bridge text for applicants interested in the mathematical foundations of computing.
by UAT-UK
Primary source for test format, sitting dates, scoring notes, fees, and candidate support.
by UAT-UK
Links to the specification, specimen and practice tests, notes on mathematics, notes on logic and proof, and past-paper archive.
by UCAS
Backup source for course code, qualification, duration, start date, deadline, and published entry requirements.
by Imperial College London
Confirms which Imperial departments and courses require TMUA.