The University Gatekeepers (Hotseat: MAT 2013)

Contents

1 Background
2 Selected Problems in Depth
3 Synthesis

Background

I didn’t have to take the Mathematics Admission Test when I applied to Imperial (though the Cambridge STEP was a part of the conditions of my offer, probably because I didn’t take FM). This was added as a part of the admission process in 2013, though seemingly only for students aiming to read Mathematics. Oxford has been running this test for fairly long, and they use it both for Mathematics as well as Computer Science. Unlike STEP, candidates don’t get a choice of questions, but instead are prescribed questions based on the programme they’re applying to.

The paper has seven questions and candidates need to answer five. All candidates have to answer question 1, which is actually ten multiple-choice questions worth 4 marks each (no penalties for guessing). There are then six 15-mark questions that follow. Candidates in Computer Science generally have two questions that the broader Mathematics population also takes – in 2013 this featured an algebra question on functional equations, and a discrete mathematics/combinatorics question. There are two further questions, one of which tends to incorporate a significant element of mathematical logic (2013’s problem revolved around the notion of common knowledge, much like the infamous Cheryl’s birthday question), and another that deals with formal languages and reasoning about similar structures (the 2013 problem focused on an inductively generated language).

I did the Computer Science stream of the paper, and managed to put together a score of 95 (40 on the first problem; 14 on Q2, 15 on Q5, 15 on Q6 and 11 on Q7) owing to a lucky (or, depending on how you see it, educated) guess on one of the multiple-choice problems. The “Mathematics-only” questions Q3 and Q4 didn’t seem any harder, though; I cleared both of them without much difficulty (whereas Q2 was somewhat nastier, in general). I lost the points on Q7 owing to a careless error; specifically, that if an expression $Bw$ is reversed, the resultant expression is certainly not $wB$ (well, unless $w$ is palindromic!)

The specific paper I did is here.

Selected Problems in Depth

Problem 1: The Gauntlet

I don’t find multiple choice questions to play very well with mathematics, because it’s very often possible to reverse engineer a solution or otherwise arrive at the correct answer via an unexpected solution path. For example, consider problem G:

$p_n(x) = (x - 1) + (x - 2) + \ldots + (x - n)$

What is the remainder when $p_n(x)$ is divided by $p_{n-1}(x)$ ?

Presumably, the intended solution path was to express $p_n(x)$ in a more convenient form (noting the arithmetic progression in the constant terms) and then actually do the division. However, given the options, it sufficed to simply test it for $p_3(x)$ and $p_2(x)$ , noting that the result of the division was negative, and that only one of the options was negative.

Nonetheless, there were certainly ample opportunities for trick questions (this paper had E, where there was a trap with an algebraic expression easily being bounded by degree 7, though it wasn’t the case that it actually was degree 7).

I got through 9 of the problems fairly smoothly, though struggled with H (failing to realise that one of the equations was actually that of a semicircle, and plowing into aggressive trigonometry-based integration). In the end I ran out of time towards the end of the integration, but I guessed the only option that featured the subtraction of the area under one of the lines, which turned out to be correct.

Problem 2: Functional Equations

This was generally fairly straightforward till the end, where the task was to find some function $f$ that satisfied $f(t) - f(1-t) = (2t - 1)^3$ . In hindsight it seemed fairly obvious to conclude that a good way of solving this would be to treat $f$ as a cubic polynomial, though I struggled for a while substituting random values of $t$ in to the functional equation to see if anything dropped out. I then went through some messy reasoning along the lines of “the difference between two points increases cubically as the (x-)distance between said points increases linearly, so the derivative is probably quadratic, thus the function is probably cubic”. I then established $f(t) = At^3 + Bt^2 + Ct + D$ and was on my way after a modest algebraic bash.

Problem 6: Reasoning about Knowledge

This question introduces Alice, Bob and Charlie; Charlie writes a (different) number on Alice and Bob’s foreheads. Thus Alice and Bob can see the other’s number, and Charlie can see both. The question begins with a little warm-up to prompt candidates into thinking about epistemic reasoning (if Alice knows her number right off the bat, it must be 2 – because she can see Bob’s 1; for any other number she wouldn’t be able to know). Things heat up a little later on:

Charlie: Your numbers are between 1 and 10 inclusive.
Alice: I don’t know my number. Is my number a square number?
Charlie: If I told you, you would know your number.
Bob: I don’t know my number.

What is Alice’s number?

We start by noting that the squares are 1, 4 and 9. Since Alice can see Bob’s number, she already knows that her number is one of two possibilities, spaced two apart. For Charlie’s statement to make sense, exactly one of the possibilities has to be a square. This tells us that Bob’s number is 2, 3, 5 or 8. (It couldn’t be 10, because if it was Alice would know that her number is 9.)

Next, Bob can see Alice’s number, but doesn’t know his. Since he can see Alice’s number, Alice’s number $A$ must be one for which both $A + 1$ and $A - 1$ are candidates. This gives us the answer: her number has to be 4.

This one, in my opinion, stopped at a fairly early level; things could develop a fair bit more. However, assuming proportional time weighting (and bearing in mind that I did write my thesis on epistemic logic) the 22.5 minutes candidates were given for this is reasonably challenging.

Synthesis

I found STEP to be somewhat more demanding, perhaps because it had questions that focused more on depth rather than breadth, and also because I took the computer science stream of the paper while STEP just focuses on mathematics.

The average score of all candidates on questions 1 through 5 was 44.8, with 60.6 for successful applicants. Nonetheless, I subsequently worked through problems 3 and 4 and actually found them considerably easier than 2. Interestingly, there was a candidate who scored between 85 and 89 but was nonetheless rejected; also, no one scored 100, which seems surprising (this would certainly have been possible if not for the careless slip I made on problem 2 – or maybe the marking for some aspects such as the rigor demanded in proofs was stricter than my own marking).

This was quite enjoyable, actually. I’m not sure how I would have fared had I taken this exam in the past! I’d think that my knowledge of logic, as well as the more computer-science oriented topics (discrete mathematics and formal languages) would have improved, though knowledge of say calculus or analysis probably would be slightly weaker than when I finished high school.