Cambridge Computer Science Interview Questions 2026 with Model Answers

Cambridge Computer Science interviews are among the most mathematically rigorous at any UK university, combining algorithm design, discrete mathematics, and formal logic under genuine time pressure. Interviewers expect structured reasoning, not just correct answers. Updated April 2026 for 2026/27 entry. The same questions, answered in full, are in the Computer Science pack at £180.

What Cambridge Computer Science Interviews Test in 2026

Cambridge CS interviews are conducted by college Fellows, typically in two separate sessions on the same day or across two days. Each interview lasts around 25–40 minutes. The panel is not looking for a polished performance — they are assessing how you think when you encounter a problem you have never seen before.

The core areas tested in 2026 include:

• Algorithm design and complexity — constructing efficient solutions and reasoning about time and space complexity using Big O notation
• Discrete mathematics — combinatorics, graph theory, modular arithmetic, and number theory
• Formal logic and proof — including proof by induction, proof by contradiction, and logical equivalence
• Probability and counting — often embedded within algorithmic or combinatorial problems
• Recursion and recurrence relations — defining and solving problems recursively

Candidates are expected to have strong A-level Mathematics. Further Mathematics is not formally required, but the depth of mathematical reasoning expected — particularly around sequences, series, and proof — means that students without it will need to do significant independent preparation. The Cambridge CS admissions test, the TMUA (Test of Mathematics for University Admission), is sat in October and tests similar mathematical territory. Strong TMUA performance signals readiness for the interview's mathematical demands.

Algorithm Design and Complexity: 3 Worked Problems with Full Solutions

Problem 1: Finding duplicates in an array

Question: You are given an array of n integers where each value lies between 1 and n. Describe an algorithm to find whether any value appears more than once. What is the time and space complexity of your approach?

Model answer: A naïve approach compares every pair of elements — O(n²) time, O(1) space. A better approach sorts the array first (O(n log n) time) and then scans for adjacent duplicates. The optimal approach uses a hash set: iterate through the array, and for each element check whether it already exists in the set. If it does, a duplicate has been found. If not, add it. This runs in O(n) time and O(n) space.

Reasoning note: Interviewers want to see you generate multiple solutions and compare them explicitly. Jumping straight to the optimal answer without acknowledging trade-offs is a missed opportunity. Always state your assumptions — for example, whether the array is sorted or whether memory is constrained.

Problem 2: Counting paths in a grid

Question: How many distinct paths are there from the top-left corner to the bottom-right corner of a 4×4 grid, moving only right or down?

Model answer: To reach the bottom-right corner of a 4×4 grid (i.e., moving from position (1,1) to (4,4)), you must make exactly 3 moves right and 3 moves down — 6 moves in total. The number of distinct orderings of these moves is the number of ways to choose which 3 of the 6 moves are rightward: C(6,3) = 20.

Reasoning note: This is a combinatorics problem dressed as a grid problem. Recognising the underlying structure — that the problem reduces to choosing positions in a sequence — is exactly what Cambridge interviewers reward. A dynamic programming solution (building up a table of path counts) is also valid and demonstrates algorithmic thinking.

Problem 3: Recursive definition of a function

Question: Define a recursive function that computes the sum of digits of a positive integer. What happens if the integer is 0?

Model answer: Define f(n) as follows: if n = 0, return 0 (base case). Otherwise, return (n mod 10) + f(n div 10). For n = 253: f(253) = 3 + f(25) = 3 + 5 + f(2) = 3 + 5 + 2 + f(0) = 10. The base case f(0) = 0 is essential to terminate the recursion. For n = 0 as an input, the function correctly returns 0, since 0 has a digit sum of 0.

Reasoning note: Always define your base case explicitly and justify it. Interviewers will probe edge cases — what about negative integers? What about single-digit numbers? Anticipating these shows mathematical maturity.

Mathematical Logic and Discrete Mathematics Questions

Worked Example: Proof by Induction

Question: Prove that the sum of the first n positive integers equals n(n+1)/2.

Model answer:

Base case: For n = 1, the sum is 1. And 1(1+1)/2 = 1. ✓

Inductive hypothesis: Assume the statement holds for some k ≥ 1, i.e., 1 + 2 + … + k = k(k+1)/2.

Inductive step: We must show it holds for k+1. The sum 1 + 2 + … + k + (k+1) equals k(k+1)/2 + (k+1) by the inductive hypothesis. Factoring: (k+1)[k/2 + 1] = (k+1)(k+2)/2. This is exactly the formula for n = k+1. ✓

Conclusion: By the principle of mathematical induction, the formula holds for all positive integers n.

Reasoning note: Cambridge interviewers expect you to state all three components — base case, hypothesis, and step — clearly and separately. Skipping the base case or being vague about what you are assuming are common errors that cost marks.

Logic Question

Question: Is the following statement true or false? "If a number is divisible by 4, then it is divisible by 2." Can you write the contrapositive?

Model answer: The statement is true — any multiple of 4 is also a multiple of 2. The contrapositive is: "If a number is not divisible by 2, then it is not divisible by 4." This is logically equivalent to the original and is also true. The converse — "if divisible by 2, then divisible by 4" — is false (e.g., 6 is divisible by 2 but not 4).

Reasoning note: Interviewers are checking whether you distinguish the contrapositive — which is always logically equivalent to the original statement — from the converse, which is not, rather than treating any rearrangement of a conditional as interchangeable. Producing a concrete counterexample to disprove the converse, rather than just asserting it is false, is what separates a rigorous answer from a lucky guess.

Cambridge CS vs Oxford CS: Interview Style Differences

Both universities conduct rigorous CS interviews, but the emphasis differs in important ways. The table below summarises the key distinctions for 2026 entry.

Oxford tends to build problems incrementally — you might start with a simple case and be guided toward a generalisation. Cambridge is more likely to present a formal mathematical statement and ask you to prove or disprove it from the outset. Neither style is harder in absolute terms, but they reward different preparation.

How to Prepare for the Most Mathematical CS Interview in the UK

Preparation for Cambridge CS interviews should begin well before the TMUA in October. The following approach is structured around the areas most likely to appear.

1. Master proof techniques — induction, contradiction, and direct proof. Work through examples from A-level Further Mathematics or introductory discrete mathematics texts.
2. Study combinatorics and graph theory — permutations, combinations, the pigeonhole principle, trees, and basic graph traversal (BFS, DFS).
3. Practise algorithm design out loud — the ability to articulate your reasoning as you work is as important as reaching the correct answer. Record yourself or work with a partner.
4. Review Big O notation — understand how to derive and compare time complexities, not just quote them.
5. Work through past TMUA papers — the mathematical content overlaps significantly with interview questions.

For structured practice with realistic problems, Cambridge Computer Science interview questions with worked solutions cover the full range of topics — from formal proof to algorithm design — with annotated model answers that show the reasoning process, not just the final result.

One of the most effective preparation habits is to practise explaining your thinking to someone who will push back. Cambridge interviewers are not passive — they will redirect you, introduce constraints, and ask you to generalise. Practising under that kind of pressure is essential.

Cambridge's own Computer Science department points applicants at Oxford's sample problems rather than publishing its own. We traced what Cambridge actually publishes.

Frequently Asked Questions

Cambridge Computer Science interviews are genuinely demanding, but they are also fair — every candidate is given problems they are unlikely to have seen before, and the process rewards clear thinking over memorised answers. With thorough preparation in discrete mathematics, proof technique, and algorithm design, candidates can approach the interview with real confidence. None of it depends on having met the specific problem before; it depends on treating an unfamiliar proof the way the interview does — built in front of another person, one step and one check at a time, rather than recalled.

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