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Interviews
Interviews are not always required as part of the admissions process to a maths degree. However, the following general links provide useful information when preparing for a Mathematics degree course interview.
General guidance on the application process for The University of Cambridge
An overview of mathematics at the University of Cambridge
FAQs about Interviews and admissions at Cambridge
Includes the following about potential.
There are many qualities that are indicative of mathematical potential, for example: the ability to make connections between different mathematical ideas, the ability to think logically, the flexibility to understand new concepts quickly and use them to solve challenging problems, the mathematical curiosity to see standard problems from different angles and to explore possible generalisations, or different applications. In an interview, if you are able to solve a problem straight away, this tells us that you're good at solving that particular problem (and perhaps that you have seen it before). On the other hand, if you get stuck, then we will give you hints and we'll have the opportunity to see if you can use them, think logically, apply concepts that you haven't seen before. That's why we will always ask some questions where you will get stuck somewhere, which really helps us judge your mathematical potential (and don't worry: we won't let you be stuck and waste your time, but we will guide you and give you hints!).Maths puzzle site mentioned by Cambridge 'Cut-the-knot' (aka Interactive mathematics Miscellany and Puzzles)
Also for Cambridge, look at page 13 of their admissions guide.
The advice includes the following.
The best ways of preparing for interview are:
Practise lots of maths problems, including material from the STEP Support Programme foundation modules at https://maths.org/step/assignments, but also maths quizzes and fun problems from websites such as www.cut-the-knot.org/.
Practise sketching functions.
Practise solving problems saying aloud to a friend or parent what you’re doing (so you’ll be used to saying aloud what you’re thinking during the interview).
When looking at mathematical statements and problems, practise asking yourself questions such as: “What if ...?" (for example what if, instead of all natural numbers in this problem we look at only even numbers?), or “Can this be extended ...?" (for example, something valid for a particular function, which happens to be an even function, can it be extended to all even functions? Yes/no - why?).
Tom Rocks Maths - a video using Cambridge admissions questions to interview Dr Tom Crawford
A tutor’s eye view of the Admissions Process at Oxford University
An overview of mathematics at Oxford University
Preparing for interviews from NRICH
Colmanweb interview questions
2019 Oxford interview question
Videos
This website has some examples (scripts) of interviews to consider
Examples of Maths Practice Questions from Cambridge interview questions
How many 0s does the number 30! have?
How long does a mirror have to be for you to see your whole body?
A body with mass m is falling towards earth with speed v. It has a drag force equal to kv. Set up a differential equation and solve it for v.
Show that if n is an integer, n^3 – n is divisible by 6.
A body with mass ‘m’ is falling towards earth with speed v. It has a drag force equal to kv. Set up a differential equation and solve it for v.
You have a 3 litre jug and a 5 litre jug. Make 4 litres
What was the most beautiful proof in A-Level Mathematics?
If x is odd prove x^2 - 1 is always a multiple of 8
What do you think is beautiful in maths?
Prove that any number consists of prime factors or is a prime number.
Consider two identical, frictionless slopes, down which we send two identical particles. If each particle starts the same height up each slope, but one rolls whereas the other simply translates down the slope, which particle will reach the bottom first?
Differentiate x^x, then sketch it.
Is it possible to cover a chess-board with dominoes, when two corner squares have been removed from the chessboard and they are (a) adjacent corners, or conversely, (b) diagonally opposite.
Integrate 1/(x^2) between -1 and 1. Describe any difficulties in doing this?
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 xlog(x).
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.
Sketch y=ln(x) explaining its shape.
Compare the integrals between the values e and 1: a) int[ln(x^2)]dx; b) int[(lnx)^2]dx and c) int[lnx]dx. Which is largest?
Sketch y=(lnx)/x.
Differentiate x^x and (x^0.5)^(x^0.5).
Sketch y=cos(1/x).
What is the square root of i?
If each face of a cube is coloured with one of 6 different colours, how many ways can it be done?
If you have n non-parallel lines in a plane, how many points of intersection are there?
Sketch y = (x^4 – 7x^2 + 12)/(x^4 – 4x^2 +4).
Sketch y^2 = x^3 – x.
Integrate from 0 to infinity the following: Int[xe^(-x^2)]dx and Int[(x^3)e^(-x^2)]dx.
If you could have half an hour with any mathematician past or present, who would it be?
What is the most pieces of pizza I can get from ‘n’ cuts?
I am an oil baron in the desert and I need to deliver oil to four different towns which happen to lie on a straight line. In order to deliver the correct amounts to each town, I must visit each town in turn, returning to my warehouse in between each visit. Where should I position my warehouse in order to drive the shortest distance possible? Roads are no problem since I have a friend who is a sheikh and will build me as many roads as I like for free.
Sketch y = sinx and y = (sinx) - 1
I drove to this interview at 50 kmph and will drive back at 30kmph because of the traffic. What is my average speed?
A body with mass ‘m’ is falling towards earth with speed v. It has a drag force equal to kv. Set up a differential equation and solve it for v.
Derive the formula for the volume of a sphere.
Do you know what a hyperbolic function is?
Draw the graph of y= (x-3)(x-2) / (x 2)(x-1)
Estimate the fifth root of 1.2
e^x = y^x. For what value of x is there only one solution?
How would you prove that any integer can be expressed factors or is itself a prime number?
How many 0s has 30 factorial?
How would you derive pi?
How would you prove that e is irrational?
How would you prove that the square root of 3 is irrational?
How would you write down 0.1 recurring as a fraction?
If X is odd prove X squared – 1 is always a multiple of 8
Integrate ysiny with respect to y.
Plot 1/x
Plot e x
Plot In x
Plot x^2 + x
Prove Pythagoras' theorem.
Sketch sinh(x) or cosh(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 do you think is beautiful in maths?
What is the significance of Euler’s equations?
What is your favourite number?
Why do we approximate many functions in maths as cosine?
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