Cambridge Maths Interview - Krollo

Maths Interview Report - Corpus Christi College, Cambridge - December 2015

On the morning of my interview, I woke up reasonably late, sorted my stuff out, and had an awkward breakfast with the other maths applicants. We all made a fairly concerted effort not to discuss much maths given the circumstances - indeed, the hall at Corpus that morning was one of the few rooms in my life where I increased average social ability when I walked in. In any case, the time soon came to pick up the pre-reading material. We assembled in a draughty office, and were then led down by the Director of Studies (DoS) to some sort of conference room set out not unlike an exam hall, though smaller and with more reproductions of Impressionist works on the walls. (I expect this may have been a step down for some of my fellow applicants given their backgrounds; no doubt they were uncomfortable without an original Sisley at least.)

We were then given the pre-reading material itself. It consisted of two excerpts from maths books, each about ten or so pages long; it covered three major topics in all: [redacted]. It mainly consisted of worked examples. We had nearly three hours to read through this; my initial read-through took about half an hour, and by then all of us were starting to realise that this would be a very long three hours indeed. I went back to the start and read through it again, in the absence of anything more productive to do. I noticed one chap had got a completely unrelated book out of his bag, which he proceeded to read for the remainder of the time; this unnerved me somewhat, expecting him to be a top-flight Olympiad man or something. Someone else was taking remarkably copious notes. I’m not really sure what he was taking notes on, but he seemed very determined about it.

I then decided that it would be at least slightly productive to try and predict the questions that were going to come up on the test, and solve them in advance. This occupied me for about an hour, and it turned out to be magnificently pointless in almost every respect. Some time later the DoS returned, bearing examination papers. He handed them out ten minutes in advance, in case we wanted to clarify anything before the test began. Beyond his handwriting, which was remarkably clear for a mathematician, all seemed more or less in order, so I set to solving as many of the questions as I could in my head. The first thing that struck me was that the questions fell into two halves - three questions that seemed very similar to ones in the booklet, and three questions where the links were rather more tangential (no pun intended).

Perhaps oddly, I started with two questions of the latter category, mainly because they looked suspiciously like problems I had seen previously. I was quite lucky in this respect, because the questions in the test leaned quite heavily towards an area of mathematics which I had read around. These two I polished off within the first quarter of an hour or so. I then tried two from the former category, which were reasonably accessible - in one case it got to the stage where I was almost copying out one of the proofs from the booklet. (We were allowed the booklet in the test.)

The last two were rather more tricky, and both were on a subject I was rather less comfortable with. Suddenly I saw how one of the questions in the test linked to an example in the book, which gave me a fifth solution that I was reasonably proud of. The last question was tough, but I made some progress at least.

We then went en masse to lunch, where we dissected both the test and the tuna. I was glad to confirm my five full solutions with others, but more unexpectedly glad to compare my own performance. Of the six of us, I had managed five full solutions, another guy (who I knew beforehand, through a friend) had managed four, there was one on three, one on two (and some partial attempts), and one on but a solitary full solution (plus partial attempts to every other question). The sixth maintained a steely silence on the topic, suggesting that he either got all six correct, or none. (Incidentally, the chap who I had seen ignore the pre-reading material for most of the three hours had two full solutions - there was not a small amount of schadenfreude going on.)

Soon it was time to head off to my first interview proper, which was at the top of a disconcertingly long staircase. Soon enough, one of my two interviewers, whom I recognised from a bit of preparatory internet stalking, invited me in. After a few pleasantries, about how I found the test and how I found the college, he told me about how in this interview we would do solely maths, which was largely what I expected (the slightly odd implications about the second interview didn’t occur to me until some time later).

Without going into too much detail - for I have signed a non-disclosure agreement with various threatening words on it - the first question started out with a fairly familiar proof, before branching out into a tad unexpected journey into combinatorics, but which I was largely happy with. I needed little prompting here and I was quite happy with my performance. The next question introduced a new concept regarding polynomials, which I believe I understood quite quickly. Firstly they asked me a basic question to check I understood the definition of it, then they started me off with something a little tougher. I was about to start a fairly laborious method, which I eschewed at their behest, and immediately realised why their method was superior. I made a few useful observations, and then managed to put the solution together without too much of a problem, after a couple of hints. They then asked me a third question on this subject, which I gather not everyone was asked, which was fairly difficult. They almost immediately gave me a subsidiary problem to help me along the way, which I got through. I suggested a couple of ideas that started to circle around the original solution, at which they gave me one more clue. Suddenly, in that way mathematics problems do, the idea came to me in one fell swoop, and they saw I had the right idea so moved on quite quickly (without wanting me to do the algebraic grunt work).

The third big topic was something that again I was quite comfortable with, but I think by then the mental stress of the previous problem was getting to me and I made a fair few silly errors. I first decided to sketch a graph, which they seemed happy with; I got most of the right ideas and put them together quite quickly (with a few prompts to correct stupid mistakes). The second part involved estimating something with this graph, and again I got the right idea more or less straight away. I made a silly error again, which the interviewers finally got me to correct. Time seemed to be becoming a problem, so again they curtailed the algebraic misery. I was not particularly happy with this question as I left the room, but the first two I felt had gone quite well. Overall I felt this interview was decent, but by no means one of my best performances.

I had an hour to wait until my next interview, so I went to the common room where the other interviewees were and wondered what to do. I was wondering whether to try and shore up what I remembered of my answers from the test, but the details had slipped away by that point and I decided to with the flow. The discovery of a Nintendo 64 made my decision certain. While I was trying to beat a Romanian girl on Rainbow Road, I noticed the other maths applicants were being conspicuously productive. One had an A4 folder with profiles of all his interviewers and their recent work, while another was doing a STEP paper (though not tremendously successfully from what little I saw). Soon enough it was time to head across to the next interview, which was in a somewhat obscure location by all accounts. I decided to get one of the students to show me where to go for the avoidance of doubt.

About five minutes later, the previous candidate walked past me. “Good bloody luck in there. Trust me, you’ll need it.” His thousand-yard stare made the context clear. I was thus a little stirred by the time, another five minutes later, when the interviewer ushered me in. He seemed a bit off, not offering to shake my hand and not even saying hello. While I was a bit worried at first, in retrospect it seems clear to be merely an example of the unique mental condition of the professional mathematician. The setting was rather grand, and I gathered it to be the chap’s living room. For some reason the only thing that I recall specifically is a volume of Solzhenitsyn on the shelf.

He started by explaining to me the format the interview would take. There was another chap sitting on the other side of the room, who, it was explained, would be observing. (It turned out that he was mainly quizzing people on the questions I had completed successfully.) The interview opened with some questions about my application in general, starting with a question about whether a problem with my economics teaching had been resolved. He also asked about what Olympiads I had done, and I was happy to be able to refer to my recent success in the Physics Olympiad which he seemed to register. Finally he made reference to the STEP classes at Leeds University I was allegedly attending, so I gave him a brief run-through of the structure while making reference to my result in STEP 1 last year, which it was clear he had also registered.

He then pulled from out of nowhere my paper from the test earlier, which I had been anticipating. He immediately asked about that one dodgy question, as I had expected him to, so I wasn’t entirely at a loss about how to begin. He was much less helpful as an interviewer than the previous interviewers and I wasn’t sure whether I was making progress, until he eventually made - about five minutes later - a very pointed remark which pushed me in the right direction. He then asked me to clarify a minor point from one of my other solutions, which he seemed much more happy with.

Somehow, I noticed, half the interview had gone and I hadn’t felt that I’d made a good impression. Next he gave me a suspiciously easy graph sketch to start with - it was literally just a quadratic from C1, but in the stress of the situation it took me irritatingly long to get it down on paper. He then asked me to draw some related functions. The first took me a while to see since I was fixating on solving it algebraically, but it was clear from his interruptions that he wanted me to take a rather more qualitative view; once I had loosened up a bit it flowed nicely. He then followed this up with another related function, and at this point I had got the gist of the question.

He then followed this up with another graph sketch, involving the floor function - not something seen often in A-Level, but I had come across it a great deal in my readings around the subject. I was thus very confident with this graph, and after explaining a couple of features I more or less drew it straight onto the paper. I expected this to be going somewhere, but no - he just replied “You should hear back from us by the first week of January” (which turned out to be a week too optimistic) and sent me on my way without so much as a goodbye.

I went back to hang around the common room for a bit, to see how other people were finding things. I thought, beyond the test, my performance in both the interviews was unremarkable; I was, however, pleased not to have been asked any more awkward questions in the second interview. One person was apparently asked in a rather accusatorial way “Why on earth did you get 87 in FP1?” while others’ solutions in the test were apparently quite literally laughed at. If nothing else, this chap did not seem quite the natural interviewer. But now it was all over - I had done all I could. I played another few games of MarioKart and went to find something to eat, reasonably satisfied but not utterly joyous about how the day had gone.

On January 12, 2016, I received an offer to study Mathematics at Corpus Christi College, Cambridge.