# What I learnt from a STEP Speedrun

I’ve been doing some work on STEP recently - maths exams used mainly for entrance at Cambridge and Warwick, who want some way to differentiate between very good A-level candidates.

When I was in Year 13, I had an interview - in fact, two interviews - at Cambridge; at one of them, I overheard an admissions tutor at one of the colleges saying (and I quote) “a trained monkey could get two As at A-level.” It’s possible that that was the point at which I started thinking “perhaps Cambridge isn’t for me, people like me don’t belong here”; in my class, there were several decent mathematicians who certainly weren’t going to get two As, but were just as certainly not trained monkeys.

But I digress. I eventually received (and declined, because St Andrews felt much more like home) an offer from Cambridge, which included getting two grade 2s in the 1996 STEP II and STEP III.

I wondered: how would I have got on with those papers? How would I do today? So I ran through the STEP II. Here are some of my thoughts.

Edit: @christianp pointed out that I hadn’t linked to the paper. Here it is!

### 1. How I did

Obviously, I’ve got a lot more maths under my belt now than I had when I was 18. I raced through the bulk of the pure section, giving solid answers to six questions in about 90 minutes. There are likely mistakes in there, but I’m happy I would get at least a grade 2 from those six questions.

Interestingly, I found the mechanics and statistics significantly harder, especially in terms of communicating my answers. Question 9 (the centre of gravity of a weeble) and 11 (a spring in a lift), I’m happy I got the answer right – but I feel a bit dubious about whether I’d have got full marks as my explanations felt a bit fudged.

### 2. Familiarity

One of the characteristics of the STEP is that many of the questions introduce you to something nice - a property of the Fibonacci sequence, or of roots of unity, or of prime factors. The trouble is, I’ve been around the block a bit and sometimes I’m *aware* of the nuggets they’re trying to push me towards discovering - and inadvertently use those nice properties on the way to discovering them, which is rather circular.

In particular, it makes questions like 6 (on factors) much easier for now-Colin than they would have been for then-Colin, because I’ve done all the discovering ahead of time.

### 3. Sticky bits

I found both of the statistics questions difficult (and I would class my answers to them as the kind of ‘fragmentary’ solution that would pick up few, if any, marks).

On question 10, the henhouse of doom, I felt like I was most of the way there, but not quite at the answer they asked for - I made an error early on that I’m not certain I fixed properly.

### 4. Thinking time

My favourite bit of the paper was question 5, linking the fifth roots of unity to trigonometry. As I say, this is something I’ve played with a lot, and I was following a path that used more advanced knowledge than I had available to me until I put the question aside for a few minutes.

At that point, I realised I was missing a more elementary trick, and once I figured that out, I could complete the question without the heavy artillery.

### 5. Careless errors

In a couple of places - especially question 4 (a vile integral) - I made arithmetic errors that led me to a solution that didn’t match what I was trying to show. In that case, I found the error and managed to correct my way through the question, but it looked very messy and could easily have introduced new errors had I not been so lucky.

I think, next time, I would rewrite my answer from the point of error (it’s not like I was short on time) and - only then - cross out my erroneous work.

### 6. Efficiency

In the first question, which asked for the $x^6$ coefficient of $(1-2x+3x^2 - 4x^3 + 5x^4)^3$, I multiplied the brackets out by hand. I’m good at that. It doesn’t take long. But in retrospect, I could have worked carefully with general polynomial multiplication to find all of the terms with an $x^6$ component. I don’t know if that’s a *better* method, but it certainly feels less of a mess.

### 7. How would then-Colin have done?

That’s a good question, and not an easy one to answer (I don’t really know *what* my strengths were back then). I like to think I’d have done well with the expansion question, the system of equations in 2, the roots of unity question (5), and the fixed square (7). The weeble question (9) also feels like it would have been bread-and-butter to me, making a total of five questions I think I’d have liked. There are others I’m sure I’d have had a stab at in a pinch.

I think then-Colin would have done OK with paper II. (Paper III is a different kettle of fish, though – there are only three or four that seem then-Colin-friendly).

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How would you have done on these papers? How about in the year you applied to university? Did you sit the STEP then? How did you get on?

* Edited 2018-01-14 to include a link to the paper. Thanks, Christian!