Maths interview problems
Proofs, curve sketching and counting, worked out loud. Type your thinking, take a hint when you are stuck, then answer the follow-ups.
The problems
Zeros at the end of 50!
Free50! means 50 × 49 × 48 × … × 2 × 1. How many zeros are at the end of 50!? Talk me through how you would find out without multiplying it out.
Try it freeSketch y = x / (1 + x²)
In the rehearsalSketch the graph of y = x / (1 + x²). Describe it to me as you go: where it crosses the axes, what happens for large x, and where it turns.
n³ − n and 6
In the rehearsalShow that n³ − n is divisible by 6 for every whole number n.
Breaking a stick
In the rehearsalBreak a stick of length 1 at two points chosen at random along it. What is the probability that the three pieces can form a triangle?
Is √2 + √3 rational?
In the rehearsalIs √2 + √3 rational? Convince me.
Which is bigger, e^π or π^e?
In the rehearsalWithout a calculator, which is bigger: e^π or π^e?
A sequence that grows slowly
In the rehearsalLet a₁ = 1 and aₙ₊₁ = aₙ + 1/aₙ. Is a₁₀₀ bigger than 14?
Points on a circle
In the rehearsalPut n points on a circle and join every pair with a straight chord, with no three chords meeting at one point inside. Into how many regions is the disc cut? Try n = 1 to 5 first.
Rehearse all 8
Hints, follow-ups and Mia's marks on every attempt, until 31 December.
Oxbridge Interview Rehearsal: two subjects
£199 once. No VAT is charged.
Rehearse every problem in two subjects you choose, with hints, follow-ups and Mia's marks, until your interviews.
- All 8 problems in each of your two subjects, chosen after you pay
- Hints when you are stuck, then the tutor's follow-up questions
- Mia's marks on every attempt: thinking aloud, using hints, checking, reasoning
- Your next problem picked from your weakest skill
- Every attempt saved, with re-attempts side by side
Access until 31 December 2026. Choose your subjects after paying. Full refund within 14 days, for any reason.
By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.Oxbridge Interview Rehearsal: all subjects
£299 once. No VAT is charged.
Every problem in all seven subjects, for joint courses and for practising beyond your own subject.
- All 56 problems: Maths, Computer Science, Engineering, Physics, Economics, Law and PPE
- Hints when you are stuck, then the tutor's follow-up questions
- Mia's marks on every attempt: thinking aloud, using hints, checking, reasoning
- Your next problem picked from your weakest skill
- Every attempt saved, with re-attempts side by side
Access until 31 December 2026. Full refund within 14 days, for any reason.
By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.What the universities say
“You may be asked to explore unfamiliar ideas, think through problems aloud or consider new perspectives.”
University of Oxford: Interviews“Students do make mistakes, and that's fine”
“It's not assumed that a less-talented student will need more help on any given problem”
University of Oxford: Sample interview questions“we may ask you to solve a problem and talk us through your thought processes”
Oxford Mathematical Institute: Interviews“Our tutors are not mind readers, so if you don't say anything our tutors can't tell what you are thinking!”
Oxford Computer Science: How to apply“the people interviewing you are interested in how you work out your answers and how you think”
University of Cambridge: Preparing for your interview