STEP 3: What It Tests, How Hard It Is, and How to Prepare

Modern STEP consists of exactly two papers: STEP 2 (paper code 9470) and STEP 3 (paper code 9475), and candidates sit one or both depending on the requirements of the universities they have applied to. STEP 1 has not been offered since the June 2021 sitting, so any STEP 1 past papers you find online date from before that change; they can serve as warm-up material, but they no longer represent a live examination.

Officially, STEP 3 is based on A level Mathematics plus the full A level Further Mathematics course, whereas STEP 2 assumes only AS level Further Mathematics. There is an important subtlety here: the specification states that the pure content of STEP 3 builds on the prescribed pure content of A level Further Mathematics with modifications, with some topics removed and additional topics included. Completing your Further Maths course therefore does not automatically mean you have covered the STEP 3 syllabus. Work through the current specification (Version 1.3, for examinations from June 2026 onwards) line by line rather than assuming your school textbook maps neatly onto it.

The examination is administered by OCR, the exam board within the Cambridge family. It is a paper-based test lasting three hours, sat at an authorised centre with handwritten answers. Among the three official user universities (Cambridge, Warwick and Imperial College London), it is Cambridge that writes STEP 3 into its typical offer: Warwick requires one of TMUA or STEP and asks only for grade 2 in either STEP paper, while Imperial centres on TMUA and attaches STEP conditions only in particular circumstances. How seriously you need to take STEP 3 is therefore largely a function of whether Cambridge is on your UCAS form.

STEP 3 shares its structure exactly with STEP 2: a three-hour paper of 12 questions in three sections. Section A contains eight pure mathematics questions, Section B two mechanics questions, and Section C two probability and statistics questions. Every question carries the same maximum of 20 marks, and there is no restriction on how many you attempt from any section: answering six pure questions and nothing else is entirely within the rules.

The scoring rule deserves precise reading, because it is widely misquoted. Everything you attempt is marked, and your final score is the sum of your six best-answered questions, out of a maximum of 120. This is not "choose six at the start and commit". If a question stalls halfway, you can abandon it and start another; the marks already earned on the abandoned attempt are not wasted, because examiners reward good progress towards a solution even when the final answer is incorrect, and a correct answer earns full marks whatever method produced it.

Two further official points close the common loopholes. There are no bonus marks for completing a question: each is marked strictly against a 20-mark scheme. And attempting more than six questions is officially not advised: candidates who spread themselves across seven or eight attempts rarely score as well as those who concentrate on finishing six properly. Time, not access to questions, is the binding constraint.

STEP is unusually open about its numbers: the grade boundaries and score distribution are published every year, released alongside results by 08:00 UK time on results day (13 August in 2026), and the annual Explanation of Results documents are available online back to 2019. For the 2026 sitting of STEP 3, out of a maximum of 120:

Two pieces of arithmetic are worth internalising. Grade 1 sat at 56, which is less than half the paper and below the value of three perfect questions (60 marks). Grade S sat at 80, exactly the value of four perfect questions. In other words, three questions done thoroughly plus scattered part-marks elsewhere secures a grade 1; four near-perfect solutions with a little extra reaches an S. The official advice to concentrate on six questions, and within those to finish your strongest work properly, is not a platitude; it falls straight out of the numbers.

The boundaries are not fixed. They are set so that the number of successful offers closely matches the places available at the official user institutions (Cambridge, Warwick and Imperial College London), which means they move with the cohort's performance from year to year. Treat the 2026 figures as a reference frame, not a promise, and aim with a margin of safety rather than at the line itself.

Structurally the two papers are identical twins; every difference lives in the content. STEP 2 is based on A level Mathematics plus AS level Further Mathematics, while STEP 3 assumes the full A level Further Mathematics course. That additional further pure content gives STEP 3 questions a higher platform to launch from: a 20-mark question on STEP 3 will typically involve less familiar objects, more abstract definitions and longer chains of algebra, and it leans harder on your ability to stay rigorous inside a structure you have never met before.

Setting the two papers' published 2026 numbers side by side is instructive:

Notice the apparent paradox: STEP 3 has lower boundaries and a higher proportion of candidates reaching each grade. That does not mean STEP 3 is the easier paper. The more plausible reading is that the STEP 3 cohort is strongly self-selected, made up of candidates who have completed full Further Maths, most with Cambridge in their sights, so the field is stronger, while the harder content pulls raw scores down and the boundaries follow. For you as an individual the lesson is simple: never infer difficulty from boundary heights, and train for the two papers separately rather than treating STEP 3 as STEP 2 with a few extra topics bolted on.

The official description of STEP's question character rewards repeated reading: questions may test the ability to apply mathematical knowledge in novel and unfamiliar ways, frequently draw on several specification topics at once, and typically require insight, ingenuity, persistence and the capacity to work through substantial sequences of algebraic manipulation. None of those four words describes fluent formula application; all four describe mathematical behaviour under uncertainty.

For STEP 3 that description carries extra weight, because the examiner has the whole Further Mathematics toolkit available and can braid together strands that your course taught in separate terms. The marking side is equally explicit: solutions must be clear, logical and legible with working fully set out, and marks may be lost if the examiner cannot follow your reasoning, even when a correct final answer appears at the end. Knowing how to solve a problem and being able to write the solution as a proof another mathematician can follow are two different skills, and the second has to be practised deliberately, on paper, against the clock.

Two further rules amplify the premium on fundamentals. Calculators are neither permitted nor required, and there has been no formulae booklet since 2019; the formulae you are expected to know are listed in the appendix to the specification. Accurate long-form hand computation and instant recall of standard results are trainable, and they need to become your default state well before exam day, or every step in the exam room costs you attention you cannot spare.

Almost every conditional offer for Mathematics at Cambridge (including the Mathematics with Physics route) includes STEP, and applicants are normally asked to take both papers. The published minimum offer level is A level AAA together with at least grade 1 in both STEP 2 and STEP 3. The grades actually required vary by College, so the authoritative source is the website of the College you are applying to, not a threshold circulating on forums.

For 2027 entry, eight Colleges (Downing, Emmanuel, Girton, Jesus, Newnham, Robinson, Sidney Sussex and Trinity Hall) operate a flexible offer scheme: the condition is met either by AAA with grade 1 in both STEP papers, or by AAA* with grade 1 in either paper, with both papers still to be sat.

On timing, STEP is a condition of the offer rather than a pre-application test: apply in October 2026 and you would expect to sit STEP in June 2027. One detail few applicants know: copies of candidates' answer booklets are sent directly to the Cambridge admissions team, precisely so that decisions about students who narrowly miss their offer can be made on the actual mathematics rather than the bare grade. Writing your solutions properly is therefore not only how you earn marks; it may also be what an admissions tutor reads when your result lands just short.

For contrast: Warwick accepts grade 2 in either STEP paper, whichever you prefer, and Imperial attaches STEP conditions in particular circumstances. Full-intensity STEP 3 preparation, though, has always had Cambridge applicants as its core audience.

The Faculty of Mathematics at Cambridge has developed a free official preparation programme, the STEP Support Programme, open to everyone and built from online modules with hints and full solutions. Its recommended sequence is explicit: Foundation modules starting in Year 12 (designed to be worked weekly over about six months), STEP 2 modules from September of Year 13, and STEP 3 modules from January of Year 13, moving on to timed full papers once the modules are complete. The programme has also published fully worked solutions to every STEP 2 and STEP 3 paper from 2019 onwards, written specifically so you can assess your own timed attempts.

Starting the STEP 3 modules in January is the pacing the programme itself recommends. Students on an accelerated Further Maths track, common in international schools in China, can reasonably start earlier, but the order itself (Foundation, then STEP 2 modules, then STEP 3 modules, then timed papers) is worth preserving.

Entry is made through an authorised centre; you cannot register yourself. For the 2026 series, registration opened on 1 March and closed on 4 May, with STEP 2 sat on 4 June and STEP 3 on 10 June: both papers within a single week, so your final rehearsals should simulate that double-header. Fees were £102.50 per paper in the UK and £142.25 (or $179.75 / €166) per paper elsewhere, with some centres adding an administration fee. One caution to end on: dates for the 2027 series have not been published, and OCR reserves the right to change dates and fees, so treat the 2026 calendar as an example rather than an annual pattern.

Students who have sat ESAT or TMUA on screen need to recalibrate: STEP is a handwritten, paper-based examination, and its rules are spelt out in the official instructions in unusual detail.

The common thread is that the answer booklet is your entire output channel. The efficient response is to practise the way you will be examined: no calculator, no formula sheet open in another tab, rough work managed alongside the fair copy on the same pages. Make examination-conditions writing your default months in advance, not a two-week conversion project at the end.

Given the rules, sound strategy is almost boringly simple: aim to do six questions properly. Read all twelve quickly at the start, begin with the question you understand best, switch without guilt when an attempt stalls (the part-marks stay banked), and write every step so that a stranger could follow it, because the mark scheme pays for visible progress towards a complete solution.

Five misconceptions come up constantly, and each has a clean official answer:

Each of these directly changes behaviour in the exam room, which is why they are worth correcting before your first timed paper rather than paying for them on the results slip.