GUIDE TO ADMISSIONS IN MATHEMATICS

1 Cambridge Mathematics

The Cambridge undergraduate mathematics course, known as the Mathematical Tripos, is widely recog-

nised as one of the most rewarding - and correspondingly demanding - undergraduate mathematics

courses available. You will have to work hard, but will enjoy the opportunity to explore an exceptional

range of interesting and beautiful mathematics, and to interact with other enthusiastic and talented

mathematicians. Two other aspects of the course that our students greatly appreciate are its ﬂexibility

and the breadth of subjects offered.

2 Why Mathematics?

Here are some reasons often given for studying Mathematics at university.

• You ﬁnd mathematics interesting and you enjoy it. This is an

excellent reason.

• You are good at mathematics. This is a necessary, but not

sufﬁcient, condition (as mathematicians would say). You may

be ﬁnding the mathematics you are doing now quite straight-

forward, so that you hardly have to work at it. When you study

mathematics at higher levels it is not so straightforward, so

you have to be prepared to work hard at it. And remember

that this work will be a major part of your daily life.

• The job prospects are excellent. This is a true statement:

employers love mathematicians because mathematics is all

about the vital skill of problem solving, but it’s not on its own

sufﬁcient reason. You should choose to study mathematics

because you enjoy it! There are other ways of getting good

jobs than spending three or four years studying something

that you don’t enjoy.

“I ﬁrst developed an

enthusiasm for maths when

I was studying it at GCSE

level. I had always been very

good at maths but I’d always

just seen it as necessary

and functional. I hadn’t

realised the breadth of its

applications, and it had

certainly never occurred to

me how fun and rewarding

maths could be." Naomi,

Murray Edwards College

April 30, 2024

3 Why Cambridge Mathematics?

Here are some reasons for studying Mathematics at Cambridge.

• Cambridge is, according to all major surveys, one of the top

universities in the world.

• The Cambridge mathematics course is one of the very best

mathematics courses in the UK.

• The Cambridge mathematics course offers you lectures in

almost all areas of mathematics, from abstract logic to theo-

retical physics, quantum information to differential geometry,

mathematical biology to ﬁnancial mathematics, and allows

you to specialise in many different ﬁelds.

• The fourth year of our mathematics course (called Part III)

is world famous and a breeding ground for future leaders in

mathematical research.

• Cambridge Colleges offer a level of academic, pastoral and

ﬁnancial support that is unsurpassed by any UK university.

• Cambridge mathematicians are among the most sought-

after mathematics graduates in the UK, and go on to high-

level jobs in many diverse and very fulﬁlling careers.

• Cambridge is a beautiful, ancient and vibrant city.

“The two supervisions per

week, where you discuss

examples from the lectures in

pairs with an academic, are an

amazing chance to talk to

someone who is extremely

knowledgeable in that area."

Shona, Clare College

“Being able to speak to your

supervisor and say "I don’t

know this" is incredibly

reassuring, ... the system is in

place for us to succeed with

support." Clement, Jesus

College

“The breadth of courses on offer

gives you an in-depth

understanding of such a wide

variety of mathematical areas!"

Zain, King’s College

4 Our course

Introduction

Cambridge has enjoyed a reputation for excellence in Mathematics

since the time of Isaac Newton, over 300 years ago. Over the years,

some of the world’s leading mathematicians like Stephen Hawking

have taught in the Faculty, and even Nobel Prize winners (although

there is no Nobel Prize for mathematics), and many currently teach,

including Fields Medallists like Caucher Birkar, Tim Gowers and

Wendelin Werner (a Fields Medal is the mathematical equivalent of

a Nobel Prize) .

The Mathematics course in Cambridge is known as the Mathemat-

ical Tripos, comprising the 3 years of the undergraduate course

(Parts IA, IB and II) plus the optional one-year Masters course (Part

III). Alternatively, students can apply to Part III as a stand-alone

Masters course.

The course dates back to the time of Newton, whose pioneering

work in mathematics and physics was a strong inﬂuence for many

years. The name Tripos comes from the word for the three-legged

stool used by the ‘Ould Bachilour’ of the University who conducted

the University examinations in medieval times. The examination

then took the form of a debate or wrangle and concentrated on Grammar, Logic and Rhetoric. Although

the Mathematical Tripos has changed much over the centuries, some traditions remain: the students in

the ﬁrst class are still called Wranglers.

Features of the course

The main distinguishing features of the Cambridge Mathematics course are:

• It covers the whole range of mathematics: from number theory, logic,

geometry and group theory on the pure side, to ﬂuid dynamics, math-

ematical biology, quantum mechanics and cosmology on the applied

side, and includes subjects such as probability, statistics, numerical

analysis, ﬁnancial models and computing.

• It has an upside-down pyramid structure, with a set of compulsory

courses in the ﬁrst year, but a very wide choice in the third year and

fourth year. This means that you will not be tied down to a specialised

choice before experiencing university maths, but you will get a thor-

ough grounding which leaves all options open. Later, you will have

freedom to choose a specialization with true knowledge of your math-

ematical abilities and preferences.

• The examinations in the ﬁrst three years are non-modular in structure: it is not the case that each

examination paper is devoted to an individual lecture course. Instead, there are four three-hour

papers at the end of each year. In the ﬁrst year, two topics are examined on each paper and in the

second and third years the examination papers are cross-sectional, meaning that instead of each

lecture course having a dedicated examination paper, each examination paper has questions on

many lecture courses. The ﬂexibility that this allows is regarded as one of the great strengths of

the Tripos: this allows you to choose how many courses you wish to revise for the examination

and therefore to work at your own pace, which is important in mathematics. The examinations in

the fourth year are modular, but you have some ﬂexibility in the number of exams taken.

• Lecture courses in the ﬁrst three years are supplemented by supervisions. Supervision is the

Cambridge term used to describe teaching in a small group of students (usually two). The su-

pervisor, who is normally a member of the teaching staff or a post-doctoral researcher, sets work

for the students to prepare and then goes over it in the supervision. Usually the work takes the

form of examples sheets (sometimes called problem sheets) prepared by the lecturer to illustrate

the material covered in the lectures. A great strength of the supervision system is that it gives

students an opportunity to discuss their individual work and particular problems.

Lecture courses in the fourth year are supplemented by examples classes, where the set work in

examples sheets is discussed, and you can ask questions about material that you found difﬁcult

or complicated.

Lecture courses in Mathematics are organised by the Faculty of

Mathematics for students from all Colleges in the University.

Attendance at lectures is not compulsory but few students manage

to cover the material adequately by themselves even when good

textbooks are available. Each lecture lasts approximately 50 min-

utes and there are on average two lectures per day from Monday to

Saturday, in the mornings only. Lectures are given for eight weeks

in each of the Michaelmas and Lent terms and for four weeks in the

Easter term, ﬁnishing about ten days before the examinations. There

are no lectures in the Easter term in the third year.

Supervisions on the various courses are arranged by the Colleges rather than by the Faculty and stu-

dents receive on average two supervisions per week, each lasting about an hour, which usually take

place in the afternoon during weekdays. Examples classes in the fourth year last about one to two

hours, and their number depends on the courses you are taking.

Aims of the course

Our Mathematics course aims to:

• provide a challenging course in mathematics and its applications for a range of students that

includes the best in the country;

• provide a course that is suitable both for students aiming to pursue research and for students

going into other careers;

• provide an integrated system of teaching which can be tailored to the needs of individual students;

• develop in students the capacity for learning and for clear logical thinking;

• continue to attract and select students of outstanding quality;

• produce the high-calibre graduates in mathematics sought by employers in universities, the pro-

fessions and the public services, many of whom will become world leaders in their chosen ﬁelds;

• provide a Masters course (Part III) suitable for students wishing to embark on a research career

in the mathematical sciences.

Facilities and Resources

As a mathematics undergraduate at Cambridge you will have many resources to support your learning

and opportunities to broaden your experience.

• Library facilities are outstanding, which means you will not

need to buy any textbooks:

◦ Every College has a library which contains the standard

books recommended for each lecture course.

◦ The Betty and Gordon Moore Library, next to the Centre

for Mathematical Sciences, houses the main collection

of mathematical science books and journals.

◦ The University Library holds a copy of nearly every book

and journal published in Britain, and it also has very sub-

stantial stocks of other works.

• You are allowed (space permitting) to attend any lectures given in the University across all sub-

jects. There are many lectured in most years that are of particular interest to Mathematics stu-

dents:

◦ A non-examinable mechanics course aimed at ﬁrst-year students who have not taken much

mechanics.

◦ A non-examinable course on the History of Mathematics.

◦ A non-examinable course on Ethics in Mathematics.

◦ Prestigious annual lectures, such as the Rouse Ball Lecture, for which an eminent mathe-

matician is invited to Cambridge.

◦ A range of courses on computing offered by the University Computing Service:

• You can access interactive audiovisual and online resources by the Language Laboratories in

more than one hundred and sixty different languages, and receive individual advice on language

learning.

• University mathematics societies provide an invaluable source of en-

riching activities of all kinds, as well as information useful for your

studies. The Archimedeans is one of the oldest and most prestigious

student societies in Cambridge, open to all our mathematicians since

1935. The Emmy Noether Society, also open to all, was founded to

promote women studying mathematical sciences. Mathematical soci-

eties offer:

◦ Mathematical talks by mathematicians from Cambridge and from

the wider mathematical community.

◦ Social events throughout the year.

◦ Opportunities to contribute to mathematical publications.

◦ Opportunities to get involved in a leadership role.

◦ Ofﬁcial and unofﬁcial lecture notes.

• You have opportunities to get involved in other aspects of Cambridge Mathematics, and to repre-

sent students’ interests, by becoming a student representative on one of the Faculty committees:

◦ the Faculty Board: the governing body of the Faculty, which

has responsibility for the Mathematical Tripos,

◦ the Mathematics Undergraduate Admissions Committee,

◦ the Teaching Committee,

◦ the Curriculum Committee,

◦ the Part III Committee,

◦ the Equality, Diversity and Inclusion Committee.

• There are also many opportunities to become involved in outreach

and teaching, for example:

◦ STIMULUS is a community service programme which gives

Cambridge University students the opportunity to work with

pupils in local schools. As a STIMULUS student you can work

as a volunteer Teaching Assistant in a classroom, alongside

the class teacher.

◦ You can inspire young visitors with mathematical games and

other activities during the city’s annual Cambridge Festival

and other events for the public.

◦ You can become a Maths Ambassador.

The Mathematics course at Cambridge offers you an excellent experience all round. Don’t just take our

word for it.

The 2017

Unistats data from the National Student Survey speak

for themselves: as well as 94% overall student satisfaction, 95%

have said that ‘Staff are good at explaining things’, and 100%

consider that ‘The course is intellectually stimulating’.

Data from 2018 onwards are not available, because students boycotted the

survey in protest against Government plans related to University fees.

Structure of the course

This guide describes the course that is likely to be given to students starting in October 2024. Supple-

mentary material is available on the Faculty website at for

anyone wanting further details.

First Year (Part IA)

About the course

Results you learn

In the ﬁrst year only there are two options:

(a) Pure and Applied Mathematics;

(b) Mathematics with Physics.

Option (a) is designed for students intending to continue

with mathematics; option (b) is designed for students with

strong interests in both mathematics and physics, who want

to keep their options open until the end of the ﬁrst year.

About three-quarters of the ﬁrst year courses are common to

the two options. You can continue with Mathematics, rather

than physics, after taking option (b), and many students do,

but some vacation reading may be required.

There are 8 core lecture courses in the ﬁrst two terms, which

means you have two lectures a day, covering a wide range

of mathematics. Students take all courses, which serve as

a platform for later years.

There are courses in:

• abstract algebra, which is the study of mathematical

structures, such as sets, vector spaces and groups;

• analysis, which is the study of the foundations of calculus;

• number theory, in which equations involving integers are

investigated;

• differential equations, in which equations involving rates

of change are investigated;

• mathematical methods, which provide the basis for math-

ematical applications; for example, to theoretical physics;

• Newtonian dynamics and special relativity, in which the

laws of Newton and Einstein are formulated mathemati-

cally;

• probability, which is (probably) what you think it is.

At the end of the year, there are four three-hour exams.

• Here is a deﬁnition from the Analysis course.

It says that, roughly, you can draw a continuous

function f without taking the pencil off the paper:

Given ε > 0, ∃ δ such that

|x − a| < δ ⇒ |f(x) − f (a)| < ε .

• Here is an equation from Vector Calculus.

It says that the amount that stuff expands in a ﬁxed

volume is equal to the amount of stuff crossing the

boundary of the volume:

∇ · FdV =

∂V

F · dS .

• This result from Probability says that random

things tend to be Normally distributed if there are

enough of them:

lim

n→∞

P (

√

n(S

− µ)/σ ≤ z) = Φ(z) .

• Here is an equation from Group Theory. It says,

for example, that if you shufﬂe a pack of cards

(same shufﬂe) 80,658,175,170,943,878,571,660,

636,856, 403,766,975,289,505,440,883,277,824,

000,000,000,000 times, the pack returns to its

original state (try it!):

|G|

= e .

• The relativistic rocket equation, from Dynamics

and Special Relativity

V = c tanh





tells us how fast a rocket goes if it expels a mass

− m

of fuel at speed v

“Among all courses I have taken so far, I enjoyed the courses

on group theory the most. I knew nothing about groups be-

fore I went to Cambridge, and so it seemed to be very hard

to understand when it was ﬁrst introduced to me during the

IA Groups lectures. However, once I got used to the basics, I

started to appreciate the beautiful structures of groups."

Isabella, Murray Edwards College

Second Year (Part IB)

About the course

Results you learn

In the second year, there are 15 lecture courses, and a

Computational Projects course. Students decide how many

courses to take: unusually (maybe uniquely) there is no

ﬁxed number that students must take to exam.

The course becomes broader and deeper. On the pure side,

the foundations of calculus are examined further and new

algebraic systems are developed. On the applied side, there

are courses on some of the most important developments in

19th and 20th century physics.

There are more courses in:

• abstract algebra;

• analysis;

• mathematical methods.

There are new courses, including:

• geometry of curved spaces;

• quantum mechanics;

• ﬂuid dynamics;

• electromagnetism;

• statistics;

• optimisation.

Reports are submitted in the second and third term for

the Computational Projects course. At the end of the year,

there are four three-hour exams.

• The Schrödinger equation

−

∇

φ + V φ = i~

∂φ

∂t

expresses the conservation of energy in quantum

mechanical systems.

• Maxwell’s equations are the fundamental

equations of electromagnetism; solutions tell us,

for example, how light propagates.

∂

= µ

[ab,c]

= 0 .

• The basic equation of complex analysis, due

to Cauchy (as are most other equations in the

subject), is

f(z)dz = 0 ,

which is an integral round a closed path in the

complex plane.

• The Cayley-Hamilton theorem for a matrix A

asserts that any matrix satisﬁes its own

characteristic equation:

P (λ) ≡ det(λI − A) =⇒ P (A) = 0 .

• In statistics, the Rao-Blackwell theorem is a

statement about expected loss:

E(L(δ

(X))) 6 E(L(δ(X))) .

“I loved Markov Chains, it was short, it was sweet, it made

perfect sense and Professor Grimmett was hilarious."

Clement, Jesus College

“I really enjoyed Statistics in IB. Today, mathematicians are valued in almost every sector,

because data interpretation has become so important - and so complex - that statisticians are

needed to build a bridge between data and the real world. I felt that the Statistics course laid

the foundation for this, and I hope I’ll be able to use these skills working for the United Nations

or some government agency in the future."

Maël, Homerton College

Third Year (Part II)

About the course

Results you learn

In the third year, there are more than 35 lecture courses,

and a Computational Projects course. As in the second

year, students decide how many courses to take: usually

three, four or ﬁve a term. Again, there is no ﬁxed number for

examination purposes.

The courses include some whose content may be guessed

at from the titles, such as:

• Number Theory,

• Coding and Cryptography,

• Mathematical Biology,

• Cosmology,

• Logic and Set Theory,

• Principles of Statistics,

• Waves

and some whose content remains obscure unless you know

about these things:

• Galois Theory (advanced group theory in which it is

proved that there is no general formula for the solutions

of a quintic equation);

• Algebraic Topology (in which properties of similar

shapes - such as doughnuts and teacups - are classiﬁed);

• Asymptotic Methods (how functions behave at large

values of their arguments);

• General Relativity (a theory of gravity);

• Stochastic Financial Models (how to predict

unpredictable markets).

• Mathematics of Machine Learning (the mathematics

needed to build classiﬁcation algorithms, as used e.g. to

aid medical diagnosis or for search engines).

Reports are submitted in the third term for the Computational

Projects course. At the end of the year, there are four three-

hour exams.

• θ = 2 arcsin

is the angle of the wake made by

a ship or a duck, which is derived in the Waves

course.

• The Einstein equations

−

8πG

are solved in General Relativity.

• The Prime Number Theorem, discussed in the

Number Theory course:

π(x) ∼

log x

approximates the number of prime numbers less

than a given number x.

• In Coding and Cryptography, the RSA, which is

one of the ﬁrst public-key cryptosystems, is derived:

c ≡ m

mod n m ≡ c

mod n .

• The Riemann hypothesis

ζ(z) = 0 =⇒ Rz =

(or z = −2m)

gets a mention, but not a proof, in Further Complex

Methods.

• Black and Scholes received a Nobel prize for their

celebrated equation

∂V

∂t

∂

∂S

+ rS

∂V

∂S

− rV = 0 ,

which is derived by our third-year students in

Stochastic Financial Models.

• In Quantum Information and Computation,

|ϕi =

√

[a|0i(|00i + |11i) + b|1i(|10i + |01i)]

is shown to be the ﬁrst step needed to achieve

quantum teleportation.

“ “If you observe a secondary rainbow in the sky, then that’s

the solution of an Airy equation!". I stole this quote from

Professor Manton. If you ﬁnd it interesting and would like to

know more, come to the course on Asymptotic Methods."

Yujun, Trinity Hall

Fourth Year (Part III) - optional, leading to MMath

Part III is the jewel in the crown of our course. It goes back to 1769, when it was known as ‘The Smith’s Prize

examination’, and is recognised as a world-leading taught Masters course in mathematics and one of the best

ways of preparing for graduate work in mathematics or theoretical physics.

About the course

Results you learn

The course is exciting and varied as no other mathematics course. Part III offers around

80 different courses (you would normally choose between six and eight) and often

more than 100 possible topics for the optional essay in which students have to review

recent research in an area of their choice. Courses on offer span the whole range of

Mathematics and its applications, Theoretical Physics and Probability and Statistics,

and aim to introduce students to the latest developments in the ﬁeld, in preparation for

research. Part III provides an essential link in maintaining a buzz of mathematical excite-

ment all the way up from ﬁrst-year undergraduates to research students and academic staff.

Currently around 90 Cambridge mathematics undergraduates stay on to do Part III. They

are joined by around 160 students from other Cambridge departments, other universities in

the UK, and the rest of the world. With students from many different backgrounds, you will

have the opportunity to experience high-level mathematics within a truly rich environment.

Topics at the cutting edge of mathematical research are taught by some of the world’s best

mathematicians, sometimes the very people who introduced them or who have made the

greatest strides in research in the ﬁeld. Some recent examples among the many courses

offered include:

• Quantum Computation (qubits and other tools to go beyond the capability of any classi-

cal computer);

• Algebraic Topology (using tools from abstract algebra to assign algebraic invariants to

topological spaces);

• Geometric Group Theory (study of algebraic and algorithmic properties of inﬁnite groups

via their actions on spaces);

• Algebraic Number Theory (which lies at the foundation of research such as Fermat’s

last theorem);

• String Theory (which describes elementary particles as excitations of a quantised string);

• Analysis of Partial Differential Equations (an introduction to the modern rigorous math-

ematical study of the fundamental equations in nature);

• Advanced Probability (introducing rigorous analysis of stochastic processes, such as

Brownian motion, ubiquitous in applications of probability theory);

• Category Theory (which studies mathematical structures and the mappings between

them, unifying ideas from different areas of mathematics);

•

Advanced Financial Models;

• Black Holes;

•

Fluid Dynamics of Climate;

• Statistics in Medicine.

There are many truly

marvellous equations

in Part Ill of the

Mathematical Tripos,

but the margin of this

booklet is too

narrow to contain

them.

At the beginning of the third term, after the Easter break, you decide which courses you wish to take to exam.

At the end of the year, there are exams in each of these: some are three-hour, some two-hour.

“The breadth of Part III is truly something special. I was able to take courses on all aspects of

geometry and analysis, ranging from Elliptic PDEs to Characteristic Classes and K-Theory. Then

writing an essay gave me the chance to really delve deep into the area I found most exciting."

Paul, St. Catharine’s College

5 Admissions Criteria

Which A-levels?

A-levels are referred to here because the majority of our applicants take A-levels. Nevertheless, note

that

• other qualiﬁcations at roughly the level of A-levels provide excellent preparation and are equally

acceptable ( e.g. International Baccalaureate or Scottish Advanced Highers);

• if you are taking the IB new Mathematics syllabus, you should take IB Higher Level ’Analysis and

Approaches’;

• many applicants are accepted every year with a variety of international qualiﬁcations.

You can obtain information about other qualiﬁcations from or from indi-

vidual Colleges or from our web site

The best advice is to do as much mathematics as possible. The current normal minimum require-

ment for our course is A-level Further Mathematics (or an equivalent qualiﬁcation). Note that if your

school does not offer teaching for Further Mathematics, you may be able to get help from the Advanced

Mathematics Support Programme ( ).

If a choice of mathematics topics in Further Mathematics is available to you (and we recognise

that for most of you there will be little or no choice of which topics you study at school), it is best (from

the point of view of our course) to take as much pure mathematics and mechanics as possible, in

preference, say, to statistics.

Our course contains a signiﬁcant component of Theoretical Physics in the ﬁrst and second years; in

the third year there is even more but you can avoid it completely if you want to. Nevertheless, you

should not worry if you are not taking A-level Physics because we teach Theoretical Physics courses

from scratch. You should also not worry if you have not enjoyed Physics much so far, because we teach

Theoretical Physics courses from a mathematical point of view. However, some of the material in the

A-level Physics course does provide useful background for our course. For those students who have not

had the opportunity to study much mechanics, we offer a short non-examinable course, ‘Introduction to

Mechanics’, intended to provide catch-up material.

As for other A-level or AS-level subjects, you should just choose the subjects you enjoy most.

STEP

All Cambridge Colleges normally include Sixth Term Examination Papers (STEP) grades in their condi-

tional offers, a number of other universities, for example Warwick, Imperial College, UCL and Durham,

also use STEP as part of some of their offers, and many other universities recommend that their math-

ematics applicants practise on past papers as preparation for university-style mathematics.

You can sit STEP examinations in centres in the UK and abroad (which can often be your school).

The reasons Colleges like to make offers involving STEP are:

1. STEP is an excellent predictor of success in the Mathematical Tripos, partly because the ques-

tions are less standard and less structured than, for example, A-level questions, which helps to

distinguish between ability (or potential) and good teaching.

2. Preparation for STEP also serves as useful preparation for our course.

3. The STEP marks and the scripts themselves are available for inspection by College staff. This

means that it is possible to make allowances for a near miss and to make judgements on the

actual work rather than on just the marks or grades.

4. STEP is the same examination for all applicants (whatever qualiﬁcations they may have studied

for). In a year when some examinations may still be teacher-assessed and awarding levels are

uncertain, STEP provides a fair comparison across the board.

You may ﬁnd STEP a bit daunting at ﬁrst, especially if your school does not offer any help with it, but

you should not be worried. Many students who did well in STEP did not have any help. Here are two

important pieces of advice (and see Appendix A for more), and a fact that may surprise you:

• Do not worry if your school is not able to provide help

with STEP.

There is plenty of material with which you can help yourself

freely available online, and many students who have done

well in STEP didn’t have any help from their school. The

best preparation for STEP is to work through past papers. To

this end, the University of Cambridge provides many free re-

sources and other support, including an online STEP Support

Programme, all available through . Much

useful advice and speciﬁc hints are available to guide you if

you get stuck.

• Do not worry if the STEP questions seem difﬁcult.

STEP is supposed to be difﬁcult: it is aimed at the top few

percent of all A-level candidates. It is therefore important to

adjust your sights when tackling a STEP paper. The questions

are much longer and more demanding than A-level questions

(they are intended to take about 30 minutes, rather than the

10 or so minutes for an A-level question). They therefore look

daunting; but you should not be daunted.

• Every year, about a third of our places are ﬁlled by appli-

cants who have missed their STEP grades

STEP is an important part of our conditional offer and it en-

ables us to compare applicants directly. However, Colleges

use all available information together, existing and predicted

grades, school reference, personal statement, performance at

interview, and the actual STEP scripts, taking individual con-

text into account, to form a picture of each applicant. In this

way we are able to make allowances for many applicants who

miss their STEP grades.

“STEP can seem impossible,

but with enough preparation

it becomes do-able"

Matthew, King’s College

“The main challenge for me

was the STEP exams after

I had my conditional offer. I

spent the summer waiting for

results convinced that I

hadn’t got in. The marking is

more generous than you

may expect so I met my

offer, and the experience left

me far better prepared for

the pressure of the Tripos

exams." Josh, King’s College

“As a foreign student, my

school didn’t offer support for

STEP - the book that got me

through was Stephen Siklos’

Advanced Problems in

Mathematics, freely available

online. When I started

revision, I wasn’t even able

to answer most questions on

STEP I. So don’t panic, and

practise regularly! " Alex,

Clare College

“Don’t let anybody tell you STEP is something ‘you can either do or you can’t’. It might seem

impossible at ﬁrst but it’s like anything else and the more you practise the better you get."

Katie, Murray Edwards College

Finally, if you are from a non-selective UK state school that offers no help with STEP preparation, and

you hold a conditional offer to read mathematics, you may qualify for STEP workshops provided by

Cambridge University. Eligible students will be sent an email with details after they have received their

offer.

Gap Year

Only a small minority of our mathematics students take a gap year. Although in many subjects the extra

maturity gained from a gap year is a great asset, in mathematics this has to be balanced against the

danger of going ‘off the boil’. If you do take a gap year, then you should plan to keep up your mathe-

matics in some way if possible, and you should certainly get back into good practice (for example, by

working through past STEP papers) before you start the course. Some Colleges are more encouraging

than others to those wishing to defer entry, and Colleges realise that mature applicants will have had

‘gap years’ for a variety of reasons during their lives before applying to university: see section 7.

6 Admissions Process

College Offers

Admissions are handled entirely by individual Colleges. Most applicants name a College on their appli-

cation form but you may instead make an open application, in which case you will be allocated a College

on the basis of the number of mathematics applications per available place in each College.

All Colleges look for talented mathematicians who have a deep interest for the subject. Colleges assess

applicants using a combination of many different criteria, allowing them to show strength in a range of

areas. They achieve this by each using a slightly different style of assessment, which includes interviews

with specialists in both pure and applied mathematics, and mathematical problems at time of interview.

As in previous years, we continue to use STEP as part of our conditional offer. We believe that STEP

provides excellent preparation for university mathematics here and elsewhere.

Typical offers across Colleges are broadly the same, normally A*A*A at A-level plus conditions based

on STEP papers 2 and 3 (with often at least a grade 1 required in both STEP papers). However, in

order to take into account the background of individual applicants, Colleges are willing to be ﬂexible

in both assessing candidates and making offers. In particular, in the case of applicants from groups

that are currently under-represented at Cambridge

, or those who have had to overcome signiﬁcant

educational disruption and/or socio-economic disadvantage, some Colleges

may make an A-Level

applicant a ‘ﬂexible offer’: this is an offer which will be met if applicants achieve either A*A*A with at

least grade 1 in STEP 2 and 3, or A*A*A* with at least grade 1 in just one of the two STEP papers taken.

If you are made a conditional offer and you do not quite fulﬁl the conditions, you may still be accepted

by your chosen College; otherwise, you may be pooled and your application will then be considered by

other Colleges.

All Colleges encourage applications from well-qualiﬁed applicants from groups that are currently under-

represented and/or disadvantaged.

In any case, the common features of the admissions process are:

• All Colleges are prepared to be ﬂexible to meet the needs of individual applicants.

• All Colleges like to interview all realistic applicants.

• All Colleges require some information beyond references and A-level grades (or the equivalent

qualiﬁcation if you are not taking A-levels). All conditional offers for Maths will require STEP to be

taken, and no offer is made without the applicant having been interviewed.

• All Colleges assess applicants by considering all available information as a whole (for example

a single bad grade or weak reference will not in isolation mean you do not get an offer). Inter-

views are intended to complement and explore the data provided by exam grades, application

statements and references.

The two mature Colleges (Hughes Hall and St Edmund’s), which admit only students who will be 21 or

over on the 1st of October of the year they start, have particular expertise in assessing non-standard

qualiﬁcations and different paths to higher education, and tend to be more ﬂexible. Their admissions

procedures reﬂect this, for example by accepting candidates for interview at an additional round in

March. However, they still aim to admit only candidates for whom the course is suitable, and require

evidence of a high level of mathematical ability.

Details of currently under-represented groups are detailed in the University Access and Participation Plan available at

Downing, Emmanuel, Fitzwilliam, Girton, Lucy Cavendish, Murray Edwards, Newnham, Robinson, Sidney Sussex, Trinity

Hall.

The interview

Interviews form an important part of our selection procedure.

Don’t worry, and especially do not listen to the hype about Oxbridge interviews that circulate on some

social media! There are no trick questions. The main purpose of the interview is to see how you think

about a mathematical problem.

Useful things to know about the interview process:

• Interviews take place in early to mid-December. If you’re invited for

interview, your College will send you a letter around mid-November.

• You’ll normally be given two interviews, sometimes three especially

if you’re applying for Maths with Physics.

• An interview will typically last for about 20 to 40 minutes. You’ll be

told in advance how long your interview will be.

• Interviews are conducted in an informal atmosphere. Just wear

something you’re comfortable with - we’re only interested in your

mathematical potential!

“I was scared, but the

interviewers made me

feel more conﬁdent."

Matthew, King’s College

“There was no nonsense

about what books I’d read

or whether I’d got my

bronze Duke of

Edinburgh, we got

straight to the maths."

Nick, Christ’s College

The best ways of preparing for interview are:

• Practise lots of maths problems, including material from the STEP

Support Programme foundation modules at

, but also maths quizzes

and fun problems from websites such as

• Practise sketching functions.

• Practise solving problems saying aloud to a friend or parent what

you’re doing (so you’ll be used to saying aloud what you’re thinking

during the interview).

• When looking at mathematical statements and problems, practise

asking yourself questions such as: “What if ...?" (for example what

if, instead of all natural numbers in this problem we look at only

even numbers?), or “Can this be extended ...?" (for example, some-

thing valid for a particular function, which happens to be an even

function, can it be extended to all even functions? Yes/no - why?).

“I panicked a lot at the

start of my ﬁrst interview

but the interviewers were

really nice and prompted

me in the right direction.

The main thing to

remember in the

interviews is to think out

loud, so they can see

your thought processes

even if you have no idea

how to solve the

question."

Ellen, King’s College

Which College?

Your choice of College is quite separate from your decision to study mathematics at Cambridge, and is

in many ways secondary with respect to this: often your choice will be based on factors such as the size

or situation of the College, sporting or musical facilities, and other personal preferences. The University

Undergraduate Prospectus includes a section about the Colleges (

), which contains a substantial amount of useful information and will help you choose a

College.

If you are not made an offer by your chosen or allocated College, your application may be made available

to all the other Colleges in the Winter Pool. Every year many applicants for Maths are ’pooled’ and a

substantial proportion made offers by other Colleges

More information is provided in the table on the next page. For further details, you should get in touch

with individual Colleges directly (enquiries are welcome) or consult their web pages: a convenient

central access point is the Faculty page ().

7 Admissions Data

The following table gives some information which you may ﬁnd useful. Last year, about 1600 students

applied for the roughly 250 places allocated to Mathematics; about 500 conditional offers were made,

about 150 of them to pooled applicants.

COLLEGE No. of places Applications Attitude to Using Interview

per year per place gap year ﬂexible offer Format

Christ’s 8 – N NO tbc

Churchill 15 ↑ N NO O

Clare 10-12 – DU NO tbc

Corpus Christi 7 – N NO O

Downing 4 – N YES O

Emmanuel 11 – DU YES P

Fitzwilliam 8-9 – N YES tbc

Girton 10 ↓↓ DU YES O

Gonville & Caius 10 – DU NO PUK/OO

Homerton 10 ↓↓ EI NO PUK/OO

Jesus 8 – N NO tbc

King’s 10 ↑↑↑ DU NO P

Lucy Cavendish 6 N/A** N YES O

Magdalene 5 – D NO tbc

Murray Edwards 6 ↓↓ N YES tbc

Newnham 6 ↓ N YES tbc

Pembroke 8 – N NO tbc

Peterhouse 8 ↓ DU NO tbc

Queens’ 14 – E NO O

Robinson 6 – EI YES O

St Catharine’s 8 – DU NO O

St John’s 14 – N NO O

Selwyn 6 – N NO PUK/OO

Sidney Sussex 6-8 – N YES O

Trinity 40 – N NO PUK/OO

Trinity Hall 6 – N YES tbc

Hughes Hall* N/A N/A N/A NO O

St Edmund’s* N/A N/A N/A NO O

Note that that the number of places per year in this table is the target intended for next year, and

applications per place is an average based on recent history.

* Hughes Hall and St Edmund’s only accept ’mature’ students, i.e. students who will be 21 or over. The

number of mature applicants in mathematics in any given year is small, so entries in this table would not

convey useful information. By deﬁnition mature students have had ‘gap years’ for a variety of reasons

at some point in their lives before applying to university.

** Lucy Cavendish College accepted only mature students until 2020. Following a major change in

admissions policy, the College is now accepting applicants from the standard university age, so average

applications per place based on recent history would not be representative.

Key:

Number of applicants per place Number of applicants per place for Mathematics compared with the

average (of about 5.8) for all Colleges: much higher ( ↑↑↑); higher (↑↑); slightly higher (↑); about

the same (−); slightly lower (↓); lower (↓↓).

Attitude to gap year Attitude to deferred places (i.e. a gap year): Discourage, Discourage Unless you

have something particularly worthwhile/relevant to do, Neutral, Encourage If you have something

particularly worthwhile/relevant to do; Encourage.

Flexible offer see page 12.

Interview format P: in-person, O: online, PUK/OO: in-person for UK-based students/online for overseas-

based students.

8 Careers

What Cambridge offers

Mathematics is at the heart of a wide range of careers and underpins many others. A mathematics

degree opens doors to careers in areas as diverse as ﬁnance, medical technology, teaching, software

development and many more. Employers greatly value the strong analytical and problem-solving skills

that mathematics graduates have. You will be taught by lecturers whose academic research collabora-

tions and real-life industry experience inform their teaching and directly beneﬁt students. The Mathe-

matics Faculty has a wide interdisciplinary network of industrial, business, governmental and academic

partners. This broad range of connections will enrich your learning as well as your career prospects.

Cambridge mathematics offers you opportunities to broaden your experience. These include:

• Summer Undergraduate Research Opportunities (for 2nd and 3rd year students)

• Post Master Placements (for 4th year students)

• Teaching maths to young students at the Sutton Trust Summer School (for all students)

• Volunteering with the STIMULUS community service programme as a Teaching Assistant in a

classroom (for all students).

Some of these opportunities are competitive.

“My aim was always to work at the interface between biology and

maths. Given that I specialised in pure maths, this was a signiﬁcant

change, and the Post Master Placement helped enormously to

make this transition as smooth as possible. I received lab training

and designed and carried out experiments, gaining valuable lab

experience."

George, Peterhouse, Post Master Placement at the Sainsbury

Laboratory for plant science

“I spent twelve weeks at the Bermuda Institute of Ocean Sciences

through a fully funded internship. My stay included participation in

one of the cruises collecting data offshore, which was a very

interesting experience. I was joined by a large number of interns

mostly from the US and Canada. The people were very friendly and

I enjoyed my stay very much."

Alex, Trinity College, Cambridge Cawthorne internship for

undergraduates - available every other year

Above all, the challenging nature of the work you will do here is the best preparation for any career: you

will develop the ability to think on your feet, be creative, make connections between different topics and

persevere until you crack difﬁcult problems.

“Studying mathematics at Cambridge has helped me in a number of ways. Of course, the speciﬁc

subject matter from some courses can be useful. In addition, being able to solve problems and

understand logical arguments is an important skill. But also, the experience of having to deal with

difﬁcult work, not always with a clear path forward laid out, has been an important grounding for real-

world work. It has made me relish tackling situations where the best course forward is not obvious

and a combination of creativity and hard work is called for." Tim, Trinity College

What some of our former students say

Understanding diseases using machine learning

‘I joined GSK in 2016, with a Masters in Computational Biology to comple-

ment the Mathematics BA. Now, nearly 5 years on, I work in a research-

driven machine learning team dedicated to understanding, modelling, and

predicting the complex behaviour of different diseases. Designing good

models requires familiarity with the underlying mathematics, and I regularly

rely on knowledge from a range of my old courses, from Differential Equa-

tions to Statistics to Graph Theory. My maths degree gave me the tools I

need to keep on top of the fast-paced world of machine learning."

Finnian Firth, Emmanuel College, Machine Learning Engineer at Glaxo-

SmithKline

Managing investment portfolios

“Maths taught me how to think. After ﬁnishing my maths degree, I went to

Goldman Sachs trading desk, then joined Bayview as a portfolio manager.

The more I advanced my ﬁnance career, the more I learned to appreciate

the skills that the maths degree at Cambridge equipped me with. Not only

do I apply the maths modelling knowledge daily, but I also leverage my

analytical and critical thinking whenever I navigate uncharted waters. The

rigorous logic training gave me the tool to learn new knowledge, the ability

to ﬁnd novel solutions, and the conﬁdence to take on new challenges."

Zhu Gong, Lucy Cavendish College, Head of European ABS at Bayview

Asset Management

Doing a PhD in Algebraic Number Theory

“The Cambridge maths course is challenging, exciting and rewarding – it

teaches you to learn and more importantly enjoy a lot of beautiful maths,

and has given me the perfect platform to start a PhD and career in research.

During my time there I met people from a huge variety of backgrounds and

interests, as well as friends who I will keep for life!"

Muhammad Manji, Trinity College, PhD student in Algebraic Number

Theory at the University of Warwick

Developing software to analyse ﬁnancial risk

‘After graduating in 2020, I now work as a data scientist in a Risk Technol-

ogy team at Morgan Stanley. My job varies from coding to analysing trends

and I use the problem solving skills from my maths degree every day."

Anna Neely, Murray Edwards College, Technology Associate at Morgan

Stanley

Making AI work for us

“After graduating in maths from Cambridge, I did a PhD in Physics at Queen

Mary University of London, then I switched into AI, working in industry, be-

fore going back into academia, with postdocs in Montreal and Oxford. Most

recently I joined DeepMind, where I now work as a Research Scientist. My

research is focused on how to correctly specify what we want an AI system

to do. It involves a mixture of maths, computer science, statistics and phi-

losophy. My maths degree is the foundation of all of my research - I think it

can provide students with the ability, outlook and conﬁdence to tackle chal-

lenging mathematical problems across the sciences"

Zac Kentonn, Emmanuel College, Research Scientist at DeepMind

Developing machine learning models for text

“Two years after graduating from Part III, I now develop machine learning

models for text at AstraZeneca. This is a really exciting time, as I am a part

of introducing this new, disruptive technology into a careful and regulated

industry. I notice the value of my maths degree everyday: we were trained

to have a sensitivity for hidden assumptions, the creativity to propose new

ones, and the skill to work out the outcome, which is useful for designing

machine learning systems, coding, and even big project meetings. Also,

being a maths graduate means the symbols in machine learning papers

are nothing to fear!"

Khyla Kadeena-Miller, Trinity College, Data Scientist at AstraZeneca

Communicating science news to a non-specialist audience

“Going into journalism after doing a maths degree always felt like an odd

move. I knew I’d be competing with lots of people with what seemed to me

to be far more appropriate degrees. I was dead wrong; the media is well

aware that there are too few science-literate people in the business, and we

are in demand. I didn’t start by covering science. In the course of my career

I’ve been sent to the top of Mont Blanc and to two different swingers’ clubs.

I’ve also been arrested in three different countries, only once for espionage.

Eventually, though, science called me back. I’ve spent ten years happily

covering the most exciting subject in journalism - including, in 2017, being

sent back to my old department to interview Stephen Hawking."

Tom Whipple, Churchill College, Science Editor at the Times

Working as an actuarial consultant

“After graduating from Newnham College, I am now working as an actuar-

ial consultant and studying towards the actuarial professional qualiﬁcation.

At work, mathematics comes everywhere in modelling and calculations for

client deliverables. A Maths degree has allowed me to obtain good problem

solving skills which greatly help my effectiveness and efﬁciency working as

a consultant."

Ruby Zhao, Newnham College, Actuarial Associate at PwC

Building open-source tooling for the data ecosystem

“I help build and implement open-source libraries and tools for the data

ecosystem, helping data engineers, scientists and researchers around the

world democratise cutting-edge data analytics. I’ve also helped our teams

use these tools for various projects, of which my favourite was building a

self-learning driving agent to help a Formula E racing team win. The an-

alytical and problem-solving skills I picked up during my maths degree at

Cambridge come into play every day, but some of the more speciﬁc courses

I’ve taken, like Bayesian inference are often directly applicable to my work."

Zain Patel, King’s College, Software Engineer at QuantumBlack

Developing and evaluating climate models

“Thirty years after going up to Cambridge to read Maths, I am now a science

fellow at the Met Ofﬁce leading a group of scientists working on modelling

the global ocean and the shelf seas. I am also a coordinating lead author of

the Ocean, Cryosphere and Sea Level Change chapter for the IPCC Sixth

Assessment Report. My work has included managing people and projects,

publishing research papers and brieﬁng government departments, but my

maths degree underpins every aspect of my work."

Helene Hewitt, Fitzwilliam College, Science Fellow at the Met Ofﬁce

Appendix A STEP

This section is intended to give you more information about the Sixth Term Examination Papers (STEP),

and the resources available to help you prepare for it, in addition to what, as already mentioned, is

available at .

OCR, which administers STEP, has a STEP website at

and have a Customer Support Centre who can be emailed at

STEPmaths@ocr.org.uk;

or you can call 01223 553366.

STEP papers are taken in June. They ﬁt in the time-line for applications as follows (you should check

the exact dates yourself).

• 15 October: deadline for UCAS applications.

• December: interviews (you will be invited for interview unless we believe that our course is not

suitable for you).

• January: conditional offer letters sent.

• Early May: deadline for STEP registration

• June: STEP examinations. You sit the papers speciﬁed in your conditional offer (see below);

you can sit a paper or papers not speciﬁed in the conditional offer (if, for example, required or

recommended by another university).

• Mid-August: STEP results (at the same time as A-level results).

There are two mathematics papers, paper 2 and 3 (paper 1 was discontinued in 2020; past papers

are still available, though, and they are useful initial preparation for STEP 2 and STEP 3). Each paper

has 12 questions: 8 pure, 2 mechanics and 2 probability, and you are assessed on 6 questions (the 6

questions best answered). There are ﬁve grades: S, 1, 2, 3 and U.

Your Cambridge offer will normally be based on grades in STEP 2 and 3.

The syllabus for STEP 2 is based on A-level Mathematics plus AS Further Mathematics.

The syllabus for STEP 3 is based on a ‘typical’ Further Mathematics A-level syllabus, with the pure

mathematics content based on the Further Mathematics core syllabus. Full syllabus speciﬁcations can

be found on the OCR website above.

If STEP clashes with one of your other exams, then you should contact the exam ofﬁcer at your school/-

college, who should advise about alternative arrangements.

If you live in the UK, you should be able to sit the STEP examinations at your school.

If you live abroad, it may still possible for you to sit STEP at your own school, provided your examinations

ofﬁcer is happy to administer the test. This may involve setting up the school as an international exam-

ination centre. If your school have not previously run STEP or are not sure whether they are approved

to offer STEP, they should contact the OCR STEP Customer Support Centre.

Alternatively, you can sit the examination at a British Council ofﬁce, but the British Council may apply a

signiﬁcant additional fee; or the STEP helpline may be able to advise you of a nearby school in which

candidates are taking STEP papers, and you can also use their online search at

to ﬁnd a

centre, in the UK or abroad, where you can sit your STEP exams.

You can ﬁnd answers to most other questions you may have about STEP at

Section 5 carried two important pieces of advice:

• Do not worry if your school is not able to provide much help with STEP.

The University of Cambridge provides a wealth of mathematical resources designed to develop

your problem-solving skills, mathematical conﬁdence and mathematical thinking, and some specif-

ically designed to help you prepare for STEP:

 An online STEP support programme, at , to help potential

university applicants develop their advanced problem-solving skills and prepare for sitting

STEP Mathematics examinations. This includes:

∗ foundation modules to gradually build up your conﬁdence with STEP questions at

∗ topic notes to support your preparation for A-level at

∗ videos showing worked solutions to past STEP questions at

 An NRICH site intended to help students to prepare for studying mathematics at university:

This is an accessible and structured introduction to

advanced problem solving, which will help build conﬁdence, ﬂuency and speed. An excellent

starting point.

 STEP questions with solutions at Underground Mathematics, available at

. Underground Mathematics offers free resources to

support the teaching of A-level mathematics, as well as selected past STEP questions with

fully worked solutions and explanations.

Further free resources:

 Advanced Problems in Mathematics: Preparing for University is a combined and

much improved version by Stephen Siklos of his two previous booklets on STEP problems:

Advanced Problems in Core Mathematics and Advanced Problems in Mathematics. It is

free to download from . It has past

papers, hints, full solutions, and much useful advice.

 The OCR STEP website has past papers from 2014 (with solutions), available from

 The MEI website at has full solutions to the papers for

1996 to 2019, to guide you if you get stuck.

You can get tuition support and much more when studying Further Mathematics, whether in a

school/college or by yourself, from the Advanced Mathematics Support programme:

• Do not worry if the STEP questions seem difﬁcult.

As mentioned previously, STEP is supposed to be difﬁcult and you need to adjust your sights

when tackling a STEP paper. It is also worth repeating: the questions are much longer and more

demanding than A-level questions and you are only expected to answer a few of them.

You may be interested to know the exact borderlines in terms of marks. They vary from year to

year, since the marks are not scaled to ﬁt pre-stated borderlines (such as UMS marks at A-level).

Here are some examples (questions are marked out of 20); more information can be found on the

Admissions Testing Service STEP website.

2017 S/1 1/2 2/3

Paper 2 101 80 69

Paper 3 95 69 57

2019 S/1 1/2 2/3

Paper 2 90 68 55

Paper 3 77 57 48

2021 S/1 1/2 2/3

Paper 2 92 67 54

Paper 3 89 67 54

As you see, the grade borderlines can vary signiﬁcantly from year to year, depending on how hard

the paper turns out to be. However, the standard required for the different grades does not vary.

This guide is intended for students who are considering applying to Cambridge to study the undergrad-

uate Mathematics, or Mathematics with Physics, course starting in October 2024.

The information contained here is only a rough guide. Further general information about admissions

can be found in the University Undergraduate Admissions Prospectus obtainable online at

or from

Cambridge Admissions Ofﬁce, Student Services Centre,

New Museums Site, Cambridge, CB2 3PT

(telephone (+44) (0) 1223 333 308, e-mail: [email protected]),

or from individual Colleges.

Further information about the mathematics course can be found in the leaﬂet Guide to the Mathematical

Tripos (undergraduate course in mathematics) obtainable from

or from

Undergraduate Admissions, Undergraduate Ofﬁce, The Faculty of Mathematics,

Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA

(telephone: (+44) (0) 1223 766879; e-mail: [email protected]).

All the documentation is available at

The pages of the individual Colleges can also be accessed from this site.

We hope that you have found this information useful, but let us know if you have any questions

which are left unanswered.

Our contact:

Email: [email protected]

Phone: +44(0)1223 766879

Undergraduate Admissions,

Faculty of Mathematics,

Centre for Mathematical Sciences,

Wilberforce Road,

Cambridge CB3 0WA,

United Kingdom

April 30, 2024