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This section will go through three different maths interview questions, with a scripted model interaction between the interviewer and the student working through the solution. Each question will touch on a different topic, giving you a feel for the broad variety of questions you may get asked in an interview.

### Question 1 [Geometry/Spatial Reasoning]

Question: Consider the unit cube. Find the shortest distance to go between the points (0.2, 0.3, 0) and (0.5, 0.5, 1), travelling along the surface of the cube. (If you aren’t familiar with the term ‘unit cube’, the interviewer may clarify what they mean by this, e.g., “the cube with the vertices (0,0,0), (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,1,0), (1,0,1), (1,1,1).” Alternatively, the interviewer may have drawn up a unit cube for you on a piece of paper or a whiteboard before you enter the room).

Top tip: the interviewers aren’t trying to trip you up with terminology. They are primarily looking to see how you think in response to these questions

Student: I see that the path must go along the face of one of the other sides of the cube, and the two faces which the points are on. I’ll try drawing the cube first to visualise it.

Interviewer: Ok. What should the path look like on each of the faces it crosses?

Student: The path must be a straight line of each of the faces.

Interviewer: Good. [pause] How will you go about calculating the distance?

Student: I guess that I should use Pythagoras’ equation to calculate the distance between two points across a face but I’m not sure which face it should go across.

[silence]

Interviewer: Can you rule out any possibilities?

Student: If we go along the ‘bottom’ face, the distance will be

whereas if we go along the ‘left’ face, the distance will be

so the answer is

, which is….

Interviewer: That is a good enough answer, you don’t need to simplify it any more than that. Why have you excluded going along the ‘right’ or ‘top’ faces?

Top tip: Unlike in A-level the interviewers are more interested in the process not the exact numerical answer; an answer in the form above should suffice.

Student: Clearly the point (0.2, 0.3) is in the bottom left hand corner of the square, so for the shortest distance we should be travelling along the ‘bottom’ or ‘left’ faces.

### Further hints:

• “Would a diagram be useful?”

You should not be afraid to start drawing on the paper in front of you while you think about the question. From experience, I think that it is better to think out-loud when one has an instinct about the geometry of the problem. In this situation it would be helpful to draw a net of the cube and the possibilities for where the point (0.2, 0.3) is, or alternatively, to draw four separate rough sketches. A helpful diagram might look something like:

• “What is special about the point (0.5, 0.5, 1)?”

You might notice that the second listed point is in the centre of the opposite face, which should make the question a little easier.

• If you are struggling to get started with this, the interviewer may offer a simpler version of the problem For example: “What is the distance between the points (0.5,0.5,1), (0.5,0.5,0), travelling along the surface of the cube?”

### Extending the question:

Depending on how quickly you arrive at the answer and how much prompting the interviewer needs to give you, they may extend it to a somewhat more general question, for example:

• “How would you go about finding the distance between the points , travelling along the surface of the unit cube”

Top tip: the interviewers often like to generalise results as a way of extending questions; if you get given a sheet of questions beforehand by your college, it might be useful to think: “How might I generalise this result?”

In this instance the interviewer might be looking for you to say something like: “We know a lot less about the points in this case. We would have to find all four possibilities and take the minimum”.

### What is the question testing?

The tutors not only want to test your technical knowledge and skills in more algebraic problems but also the spatial reasoning used in questions relating to geometry. This question is more than just applying a formula, and the interviewers will want to see how you respond to this. Most A-level students should be familiar with finding the Euclidean distance between two points in space, ie. “, and this might be your initial response to the question, but this question should stretch this and see how you think about other shortest paths on surfaces that are ‘flat’ or Euclidean.

### Related topics from university:

This question is in the topic of geometry, a course that you will meet in your first term at Oxford. The idea of shortest paths around a solid or in space is called a ‘geodesic’.

### Question 2 [Definitions/Modulus]

Question: How would you define the maximum of two real numbers, x and y?

Student: The maximum of x and y is the biggest of the two numbers

Interviewer: Ok, yes, but that doesn’t sound very formal.

Student: [pause to think] The maximum is x if x is bigger than y, and it is y otherwise.

Interviewer: Can you write that down for me?

Top Tip: From experience, the interviewer will try and get you to formalise your thoughts and move away from more vague language.

Student:

when

and

when

Interviewer: Are you sure?

Top Tip: Often the interviewer will ask you whether you’re sure about an answer, regardless of whether it’s right or wrong. They aren’t trying to trip you up; they want to analyse every possibility.

Student: [pausing to think]

Interviewer: Have you considered the case where x = y; what happens then?

Student: Well, in that case, they are the same number, so either of them could be the maximum.

Interviewer: Can you write that down for me?

Student:

when

and

when

Interviewer: Good. How might you write this down in just one formula?

Student: [pausing] I’m not sure. I don’t really know how we could do it without cases?

Interviewer: Ok. Maybe think about the average of the two numbers and the difference of the two numbers? How do they relate to each other?

Top tip: Don’t worry about needing hints from interviewers. They are trying to push you to new material!

[Silence; this is natural during difficult steps of the questions]

Student: Ah. I see that one of the numbers is the average plus half of the difference and the other is the average minus half of the difference.

Interviewer: Ok, so can you now have a go at writing the maximum function down?

Student: I think I should be writing something like:

but I know that’s not quite right because the difference should always be positive?

Top tip: Even if you know you’ve not quite got it, tell them what you’re thinking!

Interviewer: Do you know a way of making it always positive?

Student: Oh yes, I can take the modulus, so I think the formula is

Interviewer: Yes that’s it.

### Further Hints:

• If you are struggling more in the interview, they may give more comprehensive hints. For example: “Can you write down for me and in terms of and ?” This technique is often used in mathematical proofs.

### Extending the question:

• Again, depending on how quickly you follow the question and derive the formula for max, there are many follow up questions that could be asked.
• A nice next step could be: “If that is your formula for max, can write down your formula for min?”, prompting you to reflect on their responses from earlier and write
• They might ask: “Can you write a formula for the maximum and minimum of three numbers?”, which might prompt you to note that the, from which we can use the formula derived above.
• A more challenging extension onto this might be what is the middle number in x, y, z: can you write down a formula for this? This could be approached a few ways and you might notice that: .

### What is the question testing?

This question is testing how well you can formalise your own intuitive ideas of maximum and minimum, and using what you already know about modulus.

A lot of the focus of pure maths in university is rigour and formality. You can only use words which you have defined mathematically and properly. This question aims to challenge you in this way and move away from less mathematical words like “bigger”.

### Related topics from university:

Ideas of maximum and minimum and order inequalities are looked at in much of the analysis course in your first term at university.

### Question 3 [Prime Numbers/Möbius Function]

Question: The Möbius function, μ(n), is defined as follows for natural numbers:

m(n) = 1 if n is square free and has an even number of prime factors. m(n) = -1 if n is square free and has an odd number of prime factors. m(n) = 0 if n is not square free. Calculate m(35), m(66) and m(35 x 66).

Student: I’m not sure what it means for n to be square free; can you clarify what you mean by this?

Top tip: Don’t be afraid to ask for the interviewer to clarify a word. They don’t expect you to know everything.

Interviewer: Of course. A number is square-free if it doesn’t have any squared prime numbers as factors, for example 6 is square-free because its only factors are 2 and 3 and neither are repeated.

Student: Ok, I think I understand that. [pause to think about the calculation] So 35=5×7 and 5 and 7 are both prime so m(35) must equal 1. And 66=6×11=2x3x11, and 2, 3, and 11 are all prime so m(66) must equal -1.

Interviewer: Before you work it out, do you have any instinct on what m(35 x 66) might be?

Student: Well 35 has an even number of prime factors, and 66 has an odd number of prime factors, so together there’ll be an odd number of prime factors hence I think m(35 x 66 = -1)

Interviewer: That’s correct. Can you state the connection between m(35), m(66) and m(35 x 66) more exactly?

Student: [pausing] It seems like m(35)m(66) = m(35 x 66) …. Yes, I think that’s the rule.

Interviewer: Ok. Does this rule always work for two numbers x, y say?

Student: Yes, I don’t see why it shouldn’t. The total number of prime factors of xy is equal to the sum of the numbers of prime factors of x and y. So, if the number of prime factors of each of them are both odd or both even, the factors of xy will be even, and m(xy) = 1, whereas if one has an odd number of factors and one an even number, there will be an odd total, and m(xy) = -1. Although, saying that out loud, I’m now thinking that it won’t work if x and y share factors, because then xy will not be square free….

Top tip: If you think you’ve taken a wrong turn don’t be afraid to correct yourself, and the sooner you do so the better! It shows you’re able to question yourself and listen to the advice of the interviewer.

Interviewer: Continue. Can you give me an example?

Student: Well, m(6) = 1 but m(6 x 6) = 0 ≠ m(6)m(6). I think that this relation will always hold when x and y don’t share factors; that is, they are coprime.

Interviewer: Can you explain why you think it will always hold in cases like this?

Student: Well, if x has a squared prime factor, so will xy, and then m(xy) = 0 = 0 x m(y) = m(x)m(y). If neither have a squared prime factor, and since they are coprime, their factors are different, so the total number of unique factors of xy is equal to the sum of the numbers of unique factors of x and y. So by the reasoning above, it must hold that m(xy) = m(x)m(y).

Interviewer: Good. This property is called multiplicativity. Can you think of any other examples of a multiplicative function?

Student: I guess just a simple function like f(x) = x will be multiplicative, because f(xy) = xy = f(x)f(y). Actually, thinking about it, any function like f(x) = xn will work.

Interviewer: That’s right.

#### Further Hints:

• If you have not noticed that the rule m(xy) = m(x)m(y), the interviewer might prompt you by saying: “What is m(x)2 going to be for any x?” or, “Can you think of a simple counterexample to this?”
• Thinking of counter examples and proving things should go hand in hand. Where a proof fails is where you can construct a counterexample, and why you can’t think of a counterexample should help you see your method of attack for proof
• When asked to think of an example of another multiplicative function, it needn’t be complicated. The interviewer may prompt you in this way by asking something like, “What are some easy functions you know – are they multiplicative?”, in response to which you might name some simple polynomial, constant, or exponential functions.

#### Extending the Question:

• This could naturally be followed up by asking you to reason why, “If xy is square free, then x is square free”. You could reason this by saying, “xy being square-free means that whatever x and y are, they must be coprime” and “m(xy) ≠ 0 implies that m(x) ≠ 0 and m(y) ≠ 0”.
• Alternatively, the interviewer could give you a new function that you will probably not have seen before and ask you to prove or reason why it is multiplicative. For example, they might give you the unit function, u(n) for positive integers, defined as 1 when n = 0 and 0 otherwise, and ask you to show that it is multiplicative. This could be done by splitting it into the cases of x = y = 1, x ≠ 1 ≠ y, and x = 1, y ≠ 1. Proving properties of function/objects is commonplace in pure mathematics.
• If you have shown competency in the question, the interviewer could ask you about other multiplicative functions in the realm of number theory, such as Euler’s totient function, phi(n) , which counts how many numbers less than n are coprime to n, or the divisor function, sigmak(n), which sums the kth powers of all the divisors n.

#### What is the question testing?

This question is aiming to test your skills in number theory and numerical reasoning. You are purposefully exposed to a function which you will likely never have seen before, with the hope that the interviewers can see how you respond to the new material. The ideal student will, even though in unfamiliar territory, be able to understand and apply the definition in order to calculate and prove later results. Working from definitions is a vital skill to have, as the pace at which you learn maths at Oxford is fast, and so it is essential that you are able to confidently apply mathematical definitions.

#### Related topics from university:

The Möbius function itself doesn’t come up in first year mathematics, but the reasoning used in this question about factors, comprimes etc. is relevant to the groups and group actions course in the second and third terms at Oxford.

### Conclusion

If you found this style of dissecting interview questions useful, see the STEPMaths website for lots more examples. We’ve also got an example of a statistics-style interview question, if you want more examples of maths interview questions. Our Maths interview preparation courses might also interest you, run by our expert specialist maths tutors.

Remember: the tutors aren’t trying to catch you out, and they don’t want to see you fail. They want to see your best, and they want to understand the way you think. If you can prepare well, and are able to talk through your thought process, you’ll be well on your way to success.

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