Once you’ve finished your UCAS Application, Personal Statement and admissions test, your interviews are the last major step of the Oxbridge Maths application process. Many will find that this is the hardest part of the process, whether it’s due to nerves, communication issues or not knowing what to expect.
The interviewers at Oxford and Cambridge interview applicants to assess a variety of things, including their mathematical ability, their motivation for studying at Oxbridge and their general skills as an academic. The pool of potential questions is near limitless, but that doesn’t mean you can’t be prepared.
In this guide, we’ll cover the essential information you need to know for your interviews, including the kinds of questions you’re likely to be asked and examples of what to expect. Let’s begin!
Oxbridge Interview Format
Before we start, here are answers to some of the most common questions relating to Oxbridge interviews. Our Oxford Interview Preparation Guide and Cambridge Interview Preparation Guide will offer more details about the interviews as a whole, so this will just be a brief overview.
When Are Oxbridge Maths Interviews?
All initial Oxbridge interviews are held in the first week of December. A timetable of interview dates sorted by subject is made available beforehand, but you will receive all of the times and dates you’ll need to be aware of in your invitation.
When Are Interview Invitations Sent Out?
Invitations from the colleges for interviews are sent out one to three weeks before the start of the interview period. This doesn’t give you much time to prepare, so it’s crucial that you don’t wait until you receive your invitation to begin preparing.
How Many Applicants Do Oxford and Cambridge Interview?
The two universities take different approaches to their shortlisting process. Oxford is far more selective, generally interviewing 20% – 30% of applicants depending on the course (30% of Maths applicants were interviewed from 2021 – 2023). Cambridge is much more generous with its interview slots, with roughly 70% of applicants being invited each year.
Where Are Oxbridge Interviews Held?
Since 2020, most interviews at Oxbridge have been conducted remotely, allowing you to attend from home or school via Microsoft Teams or Zoom (details on the required platform will be provided in your invitation).
Oxford has confirmed that all of its interviews will remain online for the foreseeable future. However, at Cambridge, a few colleges still offer in-person interviews, either as an option or a requirement. These take place on the college campus, with full instructions on where to go included in your invitation.
How Many Interviews Will I Attend?
At Oxbridge, most applicants can expect to attend at least two interviews in December. These interviews will often be split into different themes.
Who Will Be Interviewing Me?
Your interviewers are typically admissions tutors and lecturers from the college, usually from your subject department. Each interview will usually involve two interviewers, and it’s unlikely you’ll meet the same interviewer more than once.
What Format Are The Interviews?
Oxbridge interviews are always conducted in a traditional panel format, meaning you’ll be speaking with your interviewers in a back-and-forth conversation, responding to their questions as they come. At some point, you will be asked to work through one or more mathematical problems, which could involve analysing a reference.
If your interview is in person, you may need to write or draw on paper, and all necessary materials will be provided. For remote interviews, you’ll use a digital whiteboard, which will be available in the virtual meeting room during your interview.
How Long Are The Interviews?
Oxbridge interviews generally last 30 minutes. Interviewers don’t tend to give out extra time, so be sure to be concise when speaking.
What Happens After My Interviews?
After finishing your final interview, all that’s left is to wait for an offer or any further communication.
If your chosen college rejects your application, your application may be considered by another college, giving you a second chance of admission. This is more common at Cambridge, where the process has been named the Winter Pool. Some applicants placed into the pool will need to attend another interview in January, though some applicants will be admitted without additional interviews.
That’s all the basic information regarding Oxbridge interviews that you’ll need for now. Next, let’s explore the kinds of questions you can expect to find.
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Common Oxford and Cambridge Maths Interview Questions
In a broad sense, interview questions at Oxbridge can be broken down into six categories:
- Generic Questions
- Subject-Related Questions
- Academic Questions
- Reading-Related Questions
- Personal Statement Questions
- Thinking Questions
Each of these questions serves a distinct purpose that’s important to the admissions tutors, although some are more useful for maths interviews than others. Let’s go through each category to better understand them:
Generic Questions
These are the types of questions you’d expect in any interview. While they’re easy to predict, they’re not always easy to answer. Typically, these questions focus on your motivations, exploring why you’re attending the interview. Examples include:
- Why Maths?
- Why Oxford/Cambridge?
- Why this college?
These are easy enough to prepare for, although be sure not to fully script your answers as you may come off as passionless or robotic. Memorise the key points of motivation for each of these questions and build an answer around them – this will come off as more authentic, even if your delivery isn’t perfect.
If you need to discuss your personal motivations, be sure to keep your answer consistent with what you wrote in your Personal Statement.
Subject-related Questions
For STEM subjects like mathematics, questions relating to the subject will be the most common. This first variety is the more general type of question when the interviewer may ask about your understanding of a topic or your thoughts on something specific.
With questions like these, they don’t expect you to have a complete grasp of every concept in maths. Instead, they want you to utilise your existing knowledge to discuss a concept or creatively tackle a problem.
The key to such questions is to think out loud while processing your thoughts. Since you won’t have a fully formed answer in advance, it’s important to take time to carefully consider the question. A brief pause is acceptable, but the primary goal of these questions is to understand how you approach problem-solving. Articulating your thought process in real time is the best way to showcase this.
In the next section, we’ll examine some examples of these types of questions, as each one is unique and requires various skills for effective preparation.
Academic Questions
This is the type of question that you will also commonly see in your interviews. Essentially, this is when the interviewer will give you a problem to solve, similar (though not exact) to what you would find in an exam. Since mathematics is such a binary subject, this kind of question is much easier to ask than a question about your opinions on a topic.
Depending on how skilled you are in the topic being tested, these questions can also be a lot easier to answer too, although it’s important to remember that you’re still in an interview. Remember to explain your process while working through it, as this is much more valuable to the interviewers than just getting the question right. You should be given paper or a digital whiteboard to work on (depending on the type of interview you’re attending), so be sure to visualise everything you’re doing.
Reading-Related Questions
These questions pertain to any wider reading you’ve done or mentioned in your Personal Statement. They are relatively straightforward to answer; you’ll simply need to discuss what you’ve been reading, your thoughts on it, and any interesting concepts or facts you’ve discovered. These questions are important because of how much Oxbridge values independence in its students, including their ability to find and read relevant texts on their own time.
- Tell me about something you’ve read recently.
- What did you think about X?
These questions aren’t restricted to books, so feel free to discuss articles, papers, news stories, or documentaries that you’ve recently seen if they’re relevant.
Personal Statement Questions
The idea behind these questions is straightforward: the admissions tutor wants to discuss something in your Personal Statement. However, they are not as common because Oxford and Cambridge admissions tutors typically focus less on your Personal Statement and are more interested in testing your mathematical ability.
If an interviewer does reference your statement, they will likely be seeking more information about an experience you’ve had or a super-curricular activity you’ve participated in. These questions are easy to address; just offer additional context about what you wrote and be honest in your response.
In some cases, they may use an excerpt from your statement to launch into a Subject-related Question, such as if you mention a specific topic you’re interested in.
Thinking Questions
These are sometimes called the “weird” questions because they may appear quite abstract and unrelated to the course. However, they are intended to assess your general thinking skills, such as critical thinking and problem-solving, in unconventional contexts. They aren’t meant to trip you up, but they will likely make you pause and think for a moment (just remember not to remain silent for too long).
You may have heard some horror stories involving some very obscure general questions or scenarios given to some applicants, but thankfully, these are not given to Mathematicians. The only questions that will be asked of you in your interview that have nothing to do with solving some hard mathematical problem will likely be the generic questions mentioned before.
If you do get a question that seems odd at first, it’s more than likely just a mathematical problem in disguise, so just be prepared to work in more detailed scenarios and contexts than just a standard equation.
Now that we understand the types of questions they like to ask at Oxbridge interviews, it’s time to explore some worked examples of math interview questions.
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Example Cambridge And Oxford Maths Interview Questions
Below are six questions that represent common question types in an Oxbridge maths interview. These likely aren’t going to be asked word-for-word in your interviews, but you can use them to understand the processes and strategies you can implement on the questions you’ll be asked. These questions are all focused on Mathematics, but understanding the way they’re structured will be the most challenging aspect.
Oxbridge Maths Interview Question 1
a) How big is the number (6 − √35)^{20}?
b) Show that (6 + √35)^{20} is very close to an integer.
How to Answer
Let’s consider the part a) first. Mathematicians are often quite blaisé in their usage of words like ‘big’ and ‘small’. These terms are comparatives and, by themselves, are pretty much meaningless: for example, 2 is big in comparison to 0.02, but most people wouldn’t consider 2 a ‘big’ number. Nonetheless, it is generally understood that a ‘small’ number is very close to 0, and a ‘big’ number is considerably larger than 100.
Model Answer
We know that exponentiation to large powers (like 20) will make numbers x such that |x| < 1 very small and numbers x such that |x| > 1 very large. Therefore, to answer this question, we merely need to ascertain the size of the number inside the brackets. This is easily done, since we have that 5 = √25 < √35 < √36 = 6, and therefore that 0 < 6 − √35 < 1. We conclude that (6 − √35)^{20} will be a very small number.
For part b), the first question to ask yourself is: what does it mean for a number to be close to an integer? Clearly, 1.1 is as close to an integer as 1.5 is (because 1.5 is 0.5 away from one, and 1.1 is 0.1 away from one), with 1.0001 being closer to one still. We conclude that a number x is close to an integer if there exists an integer y such that |x − y| is a very small number.
But weren’t we just dealing with a very small number in part a)? And, indeed, the number we’re dealing with here looks suspiciously similar to the number from that part. When answering this, write out the two numbers from both parts of the question like so. We’ll call the number from part a) just a, and the number from part b) just b:
Adding these two numbers together, we see that all terms involving √35 cancel out, and we’ll be left with some integer! Let’s call this integer n. We have that b = n − a, i.e. that b is equal to an integer minus a very small number and thus is close to an integer itself.
Oxbridge Maths Interview Question 2
You are a knight in the castle of Camelot, seated at the round table with 9 other knights (making there 10 people at the table in total). You are all about to shake hands with exactly 1 other person. However, it is an understood custom in Camelot that no arms should cross when you do this. How many ways can you shake hands?
How to Answer
This is a question that seems pretty odd at first, but it has a clear mathematical background once you think about it. The solution for this is quite a long one, though, so let’s go through it in full before considering how we could condense it down.
Let’s focus on you to start with. When you reach across the table and shake the hand of someone else, you will cut the table into two halves. The remaining knights can only shake hands within the halves you have created, as otherwise, they will cross arms with you.
How many ways different ways can you cut the table in half? Let’s work it out:
- If you shake hands with the person one to your left (the person adjacent to you), the table will be split into a half of 0 people and a half of 8 people. The same thing happens if you shake hands with the person one to your right.
- If you shake hands with the person two to your left or right, the table will be split into a half of 1 person and 7 people. But then the one person you have isolated will have to cross your arms to shake hands with someone, so you cannot shake hands like this.
- If you shake hands with the person three to your left or right, the table will split into a half of 2 people and 6 people.
- You cannot shake hands with the person four to your left or right, as you would create a half of 3 people and run into the same problem as before.
- If you shake hands with the person five to your left (who is the same person as the person five to your right!), you will cut the table into two halves of 4.
Once you have cut the table in half, the remaining people must all shake hands. Therefore, to answer the question, we need to work out how many ways 2, 4, 6 and 8 people can shake hands at a round table so that no arms cross.
Obviously, if there are only 2 people at a round table, they must shake hands with each other. Similarly obvious is that there are 2 ways that 4 people at a round table can shake hands with each other.
To work out how many ways 6 people at a round table can shake hands, we once again focus on just one person and how many ways they can cut the table in half. They may shake hands with the person directly to their left, leaving a half of 4 people who, as we just worked out, can shake hands in 2 different ways. This is the same if they shake hands with the person directly to their right.
Alternatively, you can shake hands with the person 3 to your left (who is the same person 3 to your right!) and split the table into two halves of 2 people, who must each shake hands with each other. Therefore, there are 2 · 2 + 1 = 5 ways of 6 people shaking hands so that no cross at a table of 6 people. You work out the number of ways 8 people can shake hands in an identical way; you conclude there are 14.
To complete the question then, we just have to do some calculations. There are two ways to cut the table into halves of 0 and 8, and there are 14 ways the remaining 8 people can shake hands. There are two ways to cut the table into halves of 2 and 6, and there are 5 ways the 2 and 6 people remaining can go about shaking hands. Finally, there is one way you can split the table into two halves of 4, and there are 2 · 2 = 4 ways these halves can shake hands. Therefore, in total, there are 2 · 14 + 2 · 5 + 4 = 42 ways you can all shake hands.
Model Answer
This was a long process to get to, and it likely wouldn’t be viable to explain all of this in your interview. However, this explanation was extremely comprehensive and planned out in advance, while you will be answering the question on the fly and likely won’t consider every aspect of the question.
The key to an effective answer is to just verbally explain your thought process as you go, not worrying about getting straight to the solution or covering every possible outcome. Just say what’s on your mind without dwelling too long, and the answer should actually come.
Oxbridge Maths Interview Question 3
a) Differentiate x^{x} and b) determine the behaviour of this function as x → 0
How to Answer
This is an all-time classic interview question, with variations of it asked in more recent years (i.e. differentiate xf(x) for some other f(x)). It requires a trick: we need to write x as eln x. This may seem like a silly thing to do, but we then have:
x^{x} = (e^{ln x})^{x} = e^{x ln x}
This gives us a much easier function to go about differentiating. We simply have to use the chain rule:
For this part b) of the question, we need to be aware of two technical results. These are:
The second of these results is known as L’Hˆopital’s Rule, and it often comes in useful in interviews. Let’s now use these results and the trick from part a) to answer the question. Using the first result, we have:
Using the second result:
Putting these together, we have lim_{x}→0 x^{x} = e^{0} = 1.
Oxbridge Maths Interview Question 4
Differentiate tan x. Differentiate sec x and b) Integrate sec x.
How to Answer
The first part of this question is completely standard; the answers should be second nature to you:
(d/dx)/tan x = sec^{2} x
(d/dx)/sec x = sec x tan x
The second part of this question is harder. There are a variety of ways we could go about doing it – including substitution techniques that you may have seen in school. But these are long, and the interviewer wants us to be cleverer than this… we should be thinking about why the interviewer has just got us to do something very easy. Perhaps we can use those results to solve this integral?
If we add the two results together, we notice we can take a factor of sec x out on the right-hand side:
d/dx (sec x + tan x) = sec x(sec x + tan x)
This seems promising. We are integrating sec x, and we have now found a rather interesting way to express this function:
The right-hand side looks promising. Remember, the differential of ln(f(x)) is f′(x)/f(x). It should be clear then how to perform the integral are asked to:
(Having seen this, it might be good to think about how you could integrate csc x).
Oxbridge Maths Interview Question 5
At a strange rally, protesters who believe bananas are an evil fruit are chanting ‘Ban, ban bananas’! How many ways are there of arranging the letters in the phrase BANBANBANANAS?
How to Answer
Here’s another scenario-based question, but the relevance of this one is easy to understand. We are being asked how many distinct ways there are of arranging the letters B-A-N-B-A-N-B-A-N-A-N-A-S (note that there is no way of distinguishing between the same letter). The first thing we want to do is count the number of letters: there are 13 in total, with 5 As, 4 Ns, 3 Bs and 1 S. Then we will imagine that we make phrases by putting letters into 13 empty slots, in a row next to each other (a bit like the slots letters are put into in the game show Countdown).
How many ways are there of putting the As into the slots? Well, there are 13 slots in total and 5 As to put into them, so the answer is (^{13} _{5}) = 1287. We will then put the Ns into the remaining 8 slots: there are 4 Ns, so there are (^{8} _{4}) = 70 ways of doing this. We then put the Bs into the remaining 4 slots: there are 3 Bs, so there are (^{4} _{3}) = 4 ways of doing this. Finally, there is one slot remaining, so the S has to go there.
We conclude that there are 1287 · 70 · 4 = 360360 ways of organising this letter (the interviewers probably wouldn’t get you to calculate the explicit form of this answer. Just telling them the answer is 287 · 70 · 4 would almost certainly suffice).
Oxbridge Maths Interview Question 6
Prove that all the numbers between 0 and 1 are not countable (we say that a set is countable if it can be listed. For example, the set {a, b, c} is countable as it can be listed like 1) a, 2) b, 3) c. Similarly, the integers are countable as they can be listed like 1) 0, 2) 1, 3) -1, 4) 2, 5) -2, …)
How to Answer
This is a hard interview question designed for candidates whose interviews are going very well. Since we want to prove the opposite of something, we should immediately think of using proof by contradiction (recall the proof that there are infinitely many prime numbers or that √2 is irrational).
So, let’s assume that all the numbers between 0 and 1 are countable. Then, we can list them. When we do this, we will write out their decimal expansions so that the digit in the mth decimal place of the nth number on our list is a_{n,m}, i.e.
- 0.a_{1},_{1}a_{1},_{2}a_{1},_{3}a_{1},_{4}...
- 0.a_{2},_{1}a_{2},_{2}a_{2},_{3}a_{2},_{4}
- 0.a_{3},_{1}a_{3},_{2}a_{3},_{3}a_{3},_{4}
- 0.a_{4},_{1}a_{4},_{2}a_{4},_{3}a_{4},_{4}
And so on. Now, we want to find a contradiction to the statement that every single number between 0 and 1 is on our list. The natural thing to do, therefore, is to find a number not on our list! We can construct this in the following way. Consider the number 0.b1b2b3b4… such that:
- if a_{m, m} = 5 then b_{m} = 4
- otherwise b_{m} = 5
Notice that we have forced the mth digit of our number 0.b_{1}b_{2}b_{3}b_{4}… to be different from the digit am,m. Therefore, 0.b_{1}b_{2}b_{3}b_{4} cannot equal any of the numbers on our list. But our list should contain all the numbers between 0 and 1, and therefore should contain 0.b_{1}b_{2}b_{3}b_{4}… This is a contradiction, and we’re done.
These are just six examples of questions that could arise in a maths interview at Oxbridge. Since mathematics is such a vast subject, the likelihood of encountering these specific questions is low, but the purpose of these examples is to illustrate how you can effectively tackle any maths-based interview question to provide a thorough and insightful response.
At this stage, it will be head to be fully skilled in the concepts behind every potential question, so don’t be discouraged if you don’t grasp everything discussed in these examples. All of these questions will be based upon concepts that you should have learnt during your A-Levels, so everything you’re asked will be achievable at your level of knowledge.
Oxbridge Maths Interview Tips
Lastly, here are some general tips to help you make the most of your Oxbridge interview and ensure you perform well.
Be Early
As with any significant interview or appointment you may have, you should strive to arrive 10 to 20 minutes early, depending on the type of interview. For remote interviews, make sure your computer is set up, your camera and microphone are tested and that you won’t be interrupted during the interview. If your interview is in person, allow yourself ample time to reach the college, as it can sometimes be challenging to locate the room where you need to be.
Remain Calm
You will often hear this in relation to interviews, and it’s true, although feeling nervous is normal and hard to completely cure. However, it’s essential to stay calm and composed when entering the interview, as this will aid in maintaining clear communication and thoughtful responses. Remember, it’s not just about answering questions correctly; a significant part of what they’re assessing is your personality and motivation for studying there. These aspects are the most crucial to convey.
Think Out Loud
We’ve already touched on this, but it’s crucial to emphasise the importance of avoiding long pauses while thinking. This may be difficult to do as some of these questions require a lot of thought, but you should try to make a habit of just describing what you’re thinking (although you shouldn’t just say a bunch of filler that doesn’t actually add to anything).
Even if you’re unsure how to answer a question, it’s essential to share your thought process and the knowledge you’re drawing from. If you can show that you’re considering the right elements when responding, it will matter less if you make an error during your explanation.
Opinions Don’t Matter
This tip likely isn’t going to factor into a Maths interview due to how objective the subject is. However, when answering questions about your thoughts on a topic, your feelings are not as important as your ability to support your reasoning with logic and facts, along with your personal interpretations of more subjective issues. Be honest in your responses, even if you believe your viewpoint is less conventional.
Use Visuals
This is essential for a maths interview. Mathematics is such a visual subject, with many elements in equations and diagrams that can’t be verbally explained. When combined with the need to provide comprehensive explanations of what you’re doing, writing and drawing your process is the only way to effectively demonstrate this to the admissions tutors.
When practising for your interviews, work on increasing the speed of your work while ensuring everything can still be read. It doesn’t need to be perfect as you can still verbally explain things as well, but it’s best practice to remain neat when drawing and writing your method.
That wraps up our guide to the types of questions you may encounter in maths interviews at Oxford and Cambridge. This is only a starting point, as this guide provides a solid foundation of what to expect, but your next step should be to continue revising your subject while also starting to practice for your interview through mock interviews. Participating in several mock interviews, preferably with someone you’re not close with, will help you refine your technique and alleviate your nerves in preparation for the real thing.
Remember: don’t wait until you’ve gotten your invitation to start preparing; it takes more than a couple of weeks to get yourself to a comfortable position for your interviews! While there are a lot of other things to do during the months leading up to December, it’s important to balance everything effectively to ensure you don’t leave any weak points that could cost you your place.
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