STEP preparation · undergraduate mathematics

Beyond STEP

Past STEP questions as a doorway to university mathematics

STEP is the hardest mathematics examination most school students ever sit. It is also one of the best: its questions are short, unfamiliar and deep, and the best of them are small pieces of university mathematics in disguise.

This book takes the most interesting STEP questions, from the first papers in 1987 (and the specimen papers that preceded them) to the present, and uses each one as a doorway. We solve the question carefully, asking the questions a good student asks along the way, and then step through the door: where the idea comes from, what it turns into at university, and what to read next.

It is written for strong 17–18-year-olds preparing for STEP who want more than practice papers: to start thinking like a mathematics undergraduate, and to find out what is waiting for them.

How each section works

The questionAn unedited STEP question. Try it first, for at least half an hour.
The hookWhy this question is more interesting than it looks.
Working it throughA guided solution in stages, with questions to ask yourself along the way.
What’s really going onThe university mathematics hiding behind the question.
The thinking skillThe habit of mind the question trains, which you will use again and again.
Try this nextFollow-up problems, and reading suitable for a first-year undergraduate.

Every section has a printable worksheet (the question, the hints in order, and the follow-up exercises) and every chapter can be downloaded as a typeset PDF.

Chapters

1Number Theory and Proof Primes, irrationality and Diophantine equations: the most proof-heavy chapter, moving from the infinitude of primes through irrationality to the structure of number systems. 9 sections · read now → 2Sequences, Series and Convergence What a limit really is, then famous infinite sums and products, then how fast things converge. 9 sections · read now → 3Functions, Inequalities and Rigour Opens with error-spotting, then the classical inequalities, then functions defined by their properties. 9 sections · read now → 4Calculus Done Properly From the definition of the integral, through integrals that prove things, to geometry and the calculus of variations. 9 sections · read now → 5Complex Numbers Arguments and roots of unity, then complex-number geometry, then complex functions as maps. Placed before Chapter 6 because roots of unity drive the algebra there. 8 sections · read now → 6Algebra and Structure Roots of unity and polynomials lead towards Galois theory; the chapter closes with three group-theory questions. 8 sections · read now → 7Linear Algebra and Geometry Matrices as objects in their own right: eigenvectors, determinants, rotations and invariants, ending with Euler's formula. 9 sections · read now → 8Differential Equations and Dynamical Systems Existence and qualitative behaviour first, then models, then chaos in discrete maps. 8 sections · read now → 9Counting and Chance Pure combinatorics and generating functions, followed by classical counting problems from the probability papers. 13 sections · read now → 10Conservation Laws, Collisions and Gravity Momentum and energy as constraints, variable mass, and Newtonian gravitation. 8 sections · read now → 11Oscillations, Rotation and Moving Frames Oscillations and rotating or accelerating frames, closing with a section of surprising trajectories. 14 sections · read now → 12Probability Five themes: thinking clearly, symmetry, expectation and games, random processes, and limits. 25 sections · read now → 13Statistics and Randomness in the World Estimation, inference and statistical literacy. 7 sections · read now →