## Oxbridge Mathematics Interview Questions

**Imagine a ladder leaning against a vertical wall with its feet on the ground. The middle rung of the ladder has been painted a different colour on the side, so that we can see it when we look at the ladder from the side on. What shape does that middle rung trace out as the ladder falls to the floor?**

** Show that if n is an integer, $\text{n}^3 – \text{n}$ is divisible by** **6**.

**Is mathematics a language?**

**Differentiate $x^x$, then sketch it.**

**What is the volume of the largest cube that fits entirely within a sphere of unity volume?**

**I have a chess board, but two of the corner squares have been removed. Is it possible to cover the board with dominoes if the two squares removed are adjacent corners? What if they were diagonally opposite corners?**

**Integrate $\frac{1}{(x^2)}$ between -1 and 1. Describe any difficulties in doing this?**

**Describe a complex number to a non-mathematician.**

**If a cannon is pointed straight at a monkey in a tree, and the monkey lets go and falls towards the ground at the same instant the cannon is fired, will the monkey be hit? Describe any assumptions you make.**

**Integrate $x\text{log}(x)$.**

**If you could have half an hour with any mathematician past or present, who would it be?**

**Do you know where the multiplication sign came from?**

**If $(\cos(x))^2 = 2\sin(a)$, what are the intervals of values of a in the interval $0 ≤ \text{a} ≤ \pi$ so that this equation has a solution?**

**If a round table has n people sitting around it, what is the probability of person A sitting exactly k seats away from person B?**

**You are given that $y = t^t$ and $x = \cos t$. What is the value of $\frac{\text{d}y}{\text{d}x}$?**

**Differentiate $y = x$ with respect to $x^2$?**

**Prove by contradiction that $2a^2 – b^2$ is true only if a and b are both odd?**

**What is the significance of prime numbers?**

**Tell us about some maths books that you have been reading.**

**What areas of maths do you like?**

**How many 0s does the number 30! have?**

**What was the most beautiful proof in A-Level Mathematics?**

**What do you think is beautiful in maths?**

**How many solutions to $kx=e^x$ for different values of k?**

**Prove by contradiction that when $z^2 = x^2 + y^2$ has whole number solutions that x and y cannot both be odd.**

**Derive the formula for the volume of a sphere.**

**How would you prove that any integer can be expressed factors or is itself a prime number?**

**Prove pythagoras theorem.**

**Sketch $\sin \text{h}(x)$ or $\cos \text{h}(x$).**

**There are 30 people in one room. What is the probability that of them have the same birthday?**

**What are modular functions?**

**What do know about Fermat’s last theorem?**

**What is your favourite number?**

**How many ways can I cover a $2 \times \text{n}$ grid with $2 \times 1$ tiles?**

## Oxbridge Computer Science Interview Questions

**Tell me about binary searches. What about their efficiency?**

**Algebraic references with respect to summation formulae and proofs by induction.**

**It is a fact that, apart from the peripherals, the whole of a computer can be made from NAND gates. The Egyptians created NAND gates using marbles rolling down shutes and used the them for booby trapping pyramids. Did, then, the Egyptians invent the computer? If not, explain fundamentally why not.**

**What is the one fundamental difference between a spreadsheet and a database?**

**Why is the number 2.7182818… used in mathematics?**

**Explain the principle of the global positioning system (GPS). What factors contribute to its accuracy?**

**What was it about computer science that sparked your interest?**

**10 boxes large enough to contain exactly 10 building blocks, and 100 blocks in 10 different colours. So, how can you pack the blocks into the boxes so that each box contains a maximum of two different colours. (There may not be equal numbers of each colour).**

**There are 11 lily pads, numbered 0 to 11. A frog is sitting on pad 0, and wants to end up on pad 10. Each jump the frog takes moves him forward by one or two places. There are many ways and movement orders to get to pad 10. How many different ways can the frog take?**

**You and your nemesis are locked in a room. There is a table with a bar of chocolate on it. Green paint marks one square of chocolate as a square containing poison. You must take turns breaking off at least one square of chocolate (breaking it off in a line) and eat whatever you break off. If the green square is left, you must eat it and die. You can decide who goes first/second. What’s your strategy?**

**You have a list of just fewer than 1,000,000 numbers. These are different and are between 0 and 999,999. What is your strategy to find a number between 0 and 999,999 that isn’t in the initial list?**

**An urn contains 23 white beans and 34 black beans. A monkey takes out two beans; if they are the same, he puts a black bean into the urn, and if they are different, he puts in a white bean from a large heap he has next to him. The monkey repeats this procedure until there is only one bean left. What colour is it?**

**What is the relevance of philosophy in science as opposed to everyday life?**

**Write an algorithm to find all factors of a number.**

**How would you define infinity?**

**Discuss the time complexity of sorting algorithms and why this is important in practical applications.**

## Oxbridge Physics Interview Questions

**A ball bearing is flying through space (vacuum and no overall gravitational field). It heads towards a doughnut, through it’s centre and out the other side. Draw graphs of 1) speed versus time and 2) acceleration versus time.**

**Draw the graphs of $y=\frac{1}{x} +x$ and $y=7+3\cos(2x+\frac{\pi}{2})$.**

**Describe a heat engine.**

**Draw graph of weight versus time for the following: 1) man stands on scale very gently and then gets off again very gently; 2) man jumps onto scales and then jumps off again; 3) man stands on scale and lets his knees unlock so that he drops, then stops. For each of these describe the reasoning behind the graph.**

**The wall of death fairground ride: it’s spinning in a horizontal circle. Then the floor that people are standing on falls away. Calculate how fast it has to spin before the floor can fall away without the people dropping out given that : coefficient of friction=m and radius of the thing=r.**

**Derive an expression for the separation of fringes caused by Young’s Double Slit apparatus.**

**Why do particle accelerators have to be so large. How do they work?**

**Why can’t you light a candle in a spaceship?**

**Why does fire go upwards?**

**Does a cup of hot coffee cool quicker if milk is added before or after stirring?**

**What angle should you aim when trying to shoot a monkey that will fall out the tree the moment you shoot.**

**What forces act on you were you at the centre of the earth?**

**When an ice cube melts in a glass of water, does the water level increase, decrease or stay the same?**

**If you dig a hole right through the Earth and jump into it, what is your motion?**

**If you leave a fridge turned on in a thermally isolated room, what happens to the room?**

**Two identical beakers with the same volume of water are placed on each pan of a double-pan balance. A steel ball is suspended from a string and submerged in the water of one of the containers. A hollow plastic ball of the same volume is submerged in the water of the other container and fastened to the bottom of the beaker by a string. Will the balance move, and if so in which direction?**

**This cricket jumps off this plank. Can you calculate the trajectory of the cricket as a function of the initial speed and angle? Can you differentiate this to find the maximum range of the cricket?**

**What can you tell me about the tides and how they vary with the position of the moon?**

**A ball is initially at rest. It is then pushed upwards by a constant force for a given amount of time. Can you sketch the velocity of the ball as a function of time (from the start to when it hits the ground)?**

**Describe what would happen if the atmosphere was cooled to 20 Kelvin?**

**Can you explain how glow sticks work?**

**Explain how a syringe is able to take water out of a cup.**

**A magician performs a trick in which he starts off with a table full of cutlery with a cloth underneath the table. He then rapidly pulls out the cloth and all of the cutlery appears to stay in its original position. Discuss the principles of physics that enable the magician to perform this trick.**

## Oxbridge Engineering Interview Questions

**What are the main differences between the engines in jet fighters and the engines in jet airliners; which type of engine is the more efficient, and (qualitatively) why?**

**How would you design a gravity dam for holding back water?**

**Why did they used to make the mill chimneys so tall?**

**What are the fundamental differences between Engineering and Physics?**

**If you had a cylinder, sealed at both ends, with the pressure rising inside, would it blow at the end or split along the side first?**

**A rectangular sheet dimensions $\text{a} \times \text{b}$ is to be made into an open-topped box by cutting a square of side h from each corner and folding the 4 sides up. Find the value of h which allows the maximum volume of the box?**

**Sketch a velocity time graph for a skydiver jumping out of a plane.**

**Why does a bullet spiral?**

**Why do sausages split lengthways, rather than around the circumference?**

**How does a fridge work?**

**What challenges do you think you would be facing as a formula 1 engineer in 10 years time?**

**How does a helicopter fly?**

**Someone places a 30cm ruler on top of one finger from each of your hands, so that you have one finger at each end of the ruler, and the ruler is resting on your fingertips. What happens if you bring these fingers together?**

**Here is a drawing of a rollercoaster. What are the forces experienced by the passengers at the bottom and the tops of the loops? Can you estimate the number of G’s of force that they would experience?**

**A box is sliding down a hill at a given angle. A projectile is shot from the box at a given time. It then perfectly lands in the box some time later. What is the distance that the box slid between the projectile being shot and landing?**

**Why do large ships not sink despite weighing thousands of tonnes?**

**How do aeroplanes stay in the air?**

## Oxbridge Economics Interview Questions

**Consider a production line. What could be done to help the worker to get away from the routine?**

**Are large or small companies more successful?**

**Why do Rolls-Royce build cars by hand, and Toyota by machine?**

**How would you market a rock band?**

**Why do governments tax their citizens? Given this equation, can you draw a graph, and tell me what the optimal rate of tax would be to maximise revenue? How could we work this out?**

**What is the basis of performance related pay?**

**Why is income per head in the United States between 50 and 100 times larger than in countries such as Burundi and Malawi? (Essentially: why are some countries rich and some countries poor?)**

**Would it be feasible to have an economy entirely based on the service sector?**

**If a Martian drops money across the country, does inflation go up?**

**Are there too many people in the world?**

**What is the point of using NHS money to keep old people alive?**

**Explain why teachers might be changing jobs to become plumbers.**

**What is the difference between buying and selling of slaves and the buying and selling of football players?**

**Are bankers paid too much for what they do? Should the government be making efforts to limit how much money they get?**

**What would you say if Gordon Brown were to take a report which shows that people who go to university earn more than those who do not, and then proclaim that going to university causes you to earn more?**

**A new country forms in Europe, and along with it, they introduce a new currency. How would the international market value the worth of this currency?**

**Why is a film actor wealthier than a theatre actor?**

**I give you £100. You have to offer part of this £100 to someone. Unless they accept the offer, you get nothing. How much will you offer?**

**What is the optimal way to win at the board game Monopoly? Could you do something similar in real life?**

**Why is deflation a scare to the UK?**

**Would you support the privatisation of the NHS?**

**What is a public good?**

**Is it really possible to measure GDP?**

**Would it be right for a government to intervene in the market?**

**Should governments subsidise agriculture?**

**What are the consequences of changing interest rates?**

**What is the point of privatisation?**

**How would you make a hypothetical sandwich stall more profitable?**

**What is a monopoly? What are the advantages and disadvantages?**

**Summarise Economics into one word.**

**Why do firms exist?**

## FAQs

You will be interviewed by academic tutors, usually from a college. They teach and research at the University and decide who studies here. Normally you will be interviewed by two tutors, occasionally more.

Tutors will understand that you may be nervous and will try to put you at your ease. They want you to feel able to be yourself in the interview, and to allow you to demonstrate your skills and abilities. They will probably ask you a few simple questions to begin with: perhaps about something in your personal statement or why you have applied for a particular course. They will then move on to questions about your subject.

Depending on what is relevant for the course you are applying for, you may be given a text, a poem, a graph, or an object, and then asked to answer questions and comment on it. You may be given these before the interview, and will be advised if there is anything in particular on which you need to focus. Tutors may also refer to any written work that you were asked to submit.

Oxbridge interview questions may be about the subjects that you are currently studying at school or college. However, you will also be offered opportunities to show whether you have read around the subject and to demonstrate your knowledge and interest beyond your school or college syllabus.

There may well be more than one right way to answer an Oxbridge interview question in which case tutors will be more interested in exploring your thought process. Remember they are trying to find out how you think, so anything you say will interest them.

Many Oxbridge interview questions are designed to test your ability to apply logic and reason to an idea you may never have encountered before. If you think you don’t know the answer to a question, don’t panic, but try and apply your mind to it – you may surprise yourself! You may also wish to explain that you haven’t covered that topic yet, but the key thing is to appear interested. Don’t play it cool because you’re nervous. Tutors love their subject and they want to teach people who feel likewise.

Sometimes tutors may suggest an alternative way of looking at a problem. They are looking for evidence that you are willing to engage with new ideas, and that you can be flexible in your thinking. Often your answers will lead to a discussion and students sometimes feel they learn a lot in interviews – despite their nerves. Interviewers are not trying to make you feel ignorant or catch you out, but to stretch you in order to assess your potential.

Top tip: don’t try and second guess what tutors are looking for – just be yourself.

The Oxbridge interview questions may seem difficult, but many of the topics you will cover do not have simple ‘right’ or ‘wrong’ answers. Interviewers are not trying to make you feel ignorant or catch you out, but to stretch you in order to assess your potential.

Sometimes tutors may suggest an alternative way of looking at a problem. They are looking for evidence that you are willing to engage with new ideas, and that you can be flexible in your thinking.

Tutors may ask you about extra-curricular activities which you have mentioned in your personal statement, particularly to help you settle into the interview. However extra-curricular activities will not be assessed unless they help to demonstrate how you meet the selection criteria for your course.

Please don’t feel as though you must ask a question. If you are given the chance to ask your own questions at the end of the interview, this does not form any part of your assessment.

- You should practise speaking your thoughts out loud when solving the Oxbridge interview questions.
- If you get stuck during the interview, don’t just sit silent and think to yourself. Keep communicating with the tutors – tell them what you’re thinking and why those ideas aren’t working.
- Remember that it’s okay to ask for help if you get stuck.
- Make sure you understand the question properly before you start working on them.