How hard was my Cambridge interview? (ft. @TomRocksMaths)

To prove once and for all that Cambridge is better than O*ford 😤 (just kidding)

You might recognize Tom from his Navier-Stokes equations on the Numberphile channel, and he is a tutor at Oxford teaching students at St. Edmund Hall college in Oxford. I just thought it was funny to have a Cambridge student to interview a teacher at Oxford.

**I HAVE TO EMPHASISE THAT THIS IS ON A MUCH MORE DIFFICULT LEVEL THAN THE OTHER QUESTIONS I HAVE HEARD FROM MY FRIENDS. THIS IS AGREED BY MY FRIENDS, SO IT'S NOT JUST ME SAYING MINE ARE HARDER.**

I am a fourth-year Cambridge math student, and so I thought I would share with you the interview questions that I have gone through! I have signed a sort of NDA for not disclosing the exact interview questions, but it has now been 4 years since my last interview, so it is fine!

Actually, my interviewer knows about this channel, but I am not sure whether he watches it or not. I hope I don't get into trouble, because in the spirit of the confidentiality agreement I signed, I should be able to disclose the details of the interview questions now. It has been 4 years since my interview, so unless they are so uncreative to come up with new interview questions, I think this should be fine.

A bit of notes:

(1) Sorry for a lot of the technical glitches - we were very unlucky, because Tom has also used the same setup in his exam series without any problems.

(2) There is an organ playing in the background - they are practising for the evening service at the college! If you don’t like it, I’m sorry; if you like it, consider it background music.

(3) For Cambridge applicants: the interview format differs from college to college. If you are interviewing for St John’s, there are separate pure and applied interviews; for Trinity (which I strongly discourage any potential applicants to apply to, simply due to the competitive nature), your interview will be based on some questions you’ve attempted in a test prior to the interview. However, because of COVID, I am not sure if all these have changed.

(4) For the people who are here for the maths: for the second question, what I meant to say was that because there are 999 consecutive integers, when you shift by one place, then you are either (a) adding an even number and deleting an odd number, or (b) adding an odd number and deleting an even number.

In the case of (a), adding an even number does not change the number of primes, but deleting an odd number might or might not decrease the number of primes by 1, so the number of primes in the interval either changes by 0 or -1.

In the case of (b), deleting an even number again does not change the number of primes, but adding an odd number might or might not increase the number of primes by 1, so the number of primes in the interval either changes by 0 or +1.

Actually, even if it is 1000 integers, a similar argument applies, though I would say this parity argument is slightly more complicated. Anyway, the idea is that initially you have more than 10 primes, and then you have also constructed an interval with exactly 0 prime, so somewhere in between, there must be an interval with exactly 10 primes, because every time you are only either changing the number of primes by +1,0, or -1.

Video chapters:
00:00 Introduction
01:24 First question: graph sketching
08:54 Second question (Part I): 0 prime
19:59 Second question (Part II): 10 primes
29:47 Third question: physics
50:04 Final point: SAQ (now called My Cambridge)

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[mathemaniac]

I’m really out of my comfort zone with this video: revealing my face, and being unscripted, and I was really nervous (actually I am still second-guessing a lot of things I said in the video!). Hopefully this is still something that you will enjoy! I actually reached out to Tom first, just because it seems such a funny idea for a Cambridge student to interview an Oxford teacher, and thanks Tom for agreeing to help me out in his very busy schedule. There are also some notes about this video in the description, and please check it out!

Notes:

(1) Sorry for a lot of the technical glitches - we were very unlucky, because Tom has also used the same setup in his exam series without any problems.

(2) There is an organ playing in the background - they are practising for the evening service at the college! If you don’t like it, I’m sorry; if you like it, consider it background music.

(3) For Cambridge applicants: the interview format differs from college to college. If you are interviewing for St John’s, there are separate pure and applied interviews; for Trinity (which I strongly discourage any potential applicants to apply to, simply due to the competitive nature), your interview will be based on some questions you’ve attempted in a test prior to the interview. However, because of COVID, I am not sure if all these have changed.

(4) For the people who are here for the maths: for the second question, what I meant to say was that because there are 999 consecutive integers, when you shift by one place, then you are either (a) adding an even number and deleting an odd number, or (b) adding an odd number and deleting an even number.

In the case of (a), adding an even number does not change the number of primes, but deleting an odd number might or might not decrease the number of primes by 1, so the number of primes in the interval either changes by 0 or -1.

In the case of (b), deleting an even number again does not change the number of primes, but adding an odd number might or might not increase the number of primes by 1, so the number of primes in the interval either changes by 0 or +1.

Actually, even if it is 1000 integers, a similar argument applies, though I would say this parity argument is slightly more complicated. Anyway, the idea is that initially you have more than 10 primes, and then you have also constructed an interval with exactly 0 prime, so somewhere in between, there must be an interval with exactly 10 primes, because every time you are only either changing the number of primes by +1, 0, or -1.

[TomRocksMaths]

Thanks for having me - this was a lot of fun!

[Aaron-lpzt]

I have a lot of respect for Tom. It takes lots of courage to do these questions as a professor and risk your reputation.

[stephenmontgomery-smith]

When I went for my Cambridge interview back in 1981, I wasn't asked any questions in the Math part of the interview. My interviewer told me that he knew enough from my entrance exam results. Despite getting in, I was disappointed not to have the opportunity to discuss Mathematics at the interview. So now I watch these videos to see how well I would do.

The person who interviewed me was to later become my PhD advisor. He layer told me his typical interview question, which was to plot sin^2(x). He said invariably people plot it like |sin(x)| with sharp corners on the bottom. He points this out, and if they understand their mistake, he accepts them. If they see from their figure that it is (1-cos(2x))/2, he gives them a scholarship. I remember having thought about this long before the interview, and I would have answered it so quickly that I think he would have thought I was trained for this question.

[johnchessant]

Nice problems! Kinda jealous as an American because these math interviews seem significantly more fun than any part of the admissions processes we have here :D and in my opinion, watching how a student talks through a problem that's difficult but actually geared to what they've learned so far is a much better way of assessing whether they're suited for a top math program, especially compared to a multiple-choice standardized test where "you either know it or you don't"

[rapidfiregeekforhire]

Its so nice to finally have a face to the wonderful maths education I've been getting from this channel over the last gosh year or so. I started watching when you talked about the dream scandal. Here I come to find its with one of my favorite numberphile and computerphile youtubers. This is truly a great day, thank you for the high quality content you always produce!

[haniamritdas]

Thanks for going outside your comfort zone, Trevor! It can be intimidating to watch your highly polished and brilliant lessons -- but I have to say your insight is also very well expressed intuitively so I think they are also very accessible.

Anyway it is awesome to watch you two struggling a bit with questions out of the blue -- this being the way that maths is always learned in experience!

Also great to put your face to your voice!

[vincentchen]

It's fascinating that both Tom and Trevor considered the third question to be the hardest, when that was the only question that I felt was fairly straight forward. I doubt I would ever have been able to complete the prime number question, no matter how long I had to think about it!

[mtkimbrell]

@Tomrocksmaths I love that you take on these challenges. It shows that the great math minds don't always just wave their wand and "poof" out a proof. You actually do have to take the time like anyone would with a puzzle to figure out the solution. The secret to being good at math is to break it down and take the time for details. I was a horrible math student in primary school but then something clicked and I had the drive and patience to work through it. I went on to successfully complete about every Math class I could take with a B.S. in Electrical Engineering and M.S. in Engineering. Thank you for not making it look like a magic trick!

[adwz]

As someone currently undergrad student at Cambridge, I can tell u it is extremely nerve wracking sitting in that waiting room for so long, however as soon as the interview starts I promise u they will settle and you will feel like you are in flow state. Also, it is perfectly fine to not know an answer. In fact, most people who come out thinking they flunked the interview and aren’t going to get offers are actually the people who get them more frequently or not

[davidrobins]

The prime number one was really nice. I couldn't think how to approach it at all, but as soon as you said factorials, it all became clear. Just shows how one key insight or thought can be critical to solving a problem.

[adamhilmi]

So close to interview season, it’s refreshing to see this!

[hainesensei]

As a 3rd year pure maths focused Cambridge student, I haven’t done any physics since A level physics really (I didn’t really manage to follow the dynamics and relativity course in first year). But the physics question still felt doable without many formulae (only knowing what things are):
Don’t consider the rotating frame at all, just consider the “x and y” components at the moment of shooting the gun that is its going 5m/s forwards and since the gun is moving at 5m/s sideways, it also has a perpendicular component of 5m/s (which corresponds precisely with the r and θ components, but thinking from a more A level physics perspective since rotating frames aren’t really covered there). This gets the 45° angle and the speed of 5√2 m/s. This avoids all the x’ = r’e_r + rθ’e_θ (I can’t write dot notation in YouTube comments).

[scollyer.tuition]

For Q1, you can note that after completing the square on both sides and simplifying, we obtain (y-0.5)^2=(x-0.5)^2, which is a translation (of 0.5 units) along both x and y axes of y^2 = x^2. And y^2 = x^2 trivially has solns y = +/- x.

[OneEyedJacker]

I was quite challenged by the prime number questions but understood them in terms of elimination of terms in factorials… the one involving the coreolis effect was much more straight forward. I’m a 71 year old mechanical engineer.

[justinroughley]

Thanks for video. This brought back memories of my Oxford interview in I think 1982. At that interview I was asked to show that integration was the inverse of differentiation which was a nice question that I hadn’t considered but that was within the scope of A-level knowledge.

[hsjkdsgd]

Cool collab! Bring more of such videos. They really help.

[simongu]

for question one, you can factor the euqation and you'll get (y-(x+1))(y-x)=0 and vice versa, then you'll get the two lines symmetric

[ibcavid]

In the second question I think this should be the way to make proof more neat: by shifting interval by 1, number of primes can either +1 -1 or don't change. (0, 1000) has more than 10, (1000!, 1000! + 1000) has 0. So there is an interval where we would get exactly 10.

[TheNothingNihilates]

Love Q2! [spoilers]

So brave of you guys to do this unprepared, especially Tom. Even after the explanation, I'm still not convinced Tom got it. I was also thrown by part 2 being a yes/no existence question, not a constructive one (is there a nice construction for the interval?) But once that is clear, it is relatively easy. But the explanation in the vid is not super clear.

Part I. This is kinda like the flip side of Euclid's proof that there are infinitely many primes: n! + 1 has a prime factor greater than n, therefore there is a prime greater than n for all n ( ~ Euclid), .

For this question, n! + 2 has factor 2, n! + 3 has factor 3 ... n! + n has factor n => there are no primes among the n - 1 consecutive integers from n! + 2 thru n! + n. Set n = 1000 and you are done.

Part II. The explanation was basically right but not clear. Consider the 999 consecutive integers in the interval [2, 1000]. The interval contains more than 10 primes. We know from Part I that the interval [1000! + 2, 1000! + 1000] contains 999 consecutive integers but no primes. Now, start sliding the lower interval upwards by integer increments: [3, 1001], [4, 1002] ... With each shift, the number of primes in the sliding interval will do one of the following:
1/ stay the same (prime comes in at the top and prime drops out the bottom, or composite comes in at the top and composite goes out the bottom), [because 1001 is not prime, the former in fact never happens for windows of odd size, like 999]
2/ reduce by one (prime drops out the bottom, and composite comes in at the top), or
3/ increase by one (prime comes in at the top, composite drops out at the bottom).
Thus, the count of primes in this shifting window of 999 consecutive integers begins above 10, and (from Part I) eventually drops to 0. Since it changes only by increments or decrements of 1, it must at some point be exactly equal to 10. (NOTE: there is no need to assume that it is a decreasing function, even though the overall trend is downward. Also, 999 being odd makes no difference to the proof.)