Exam timetable
Exam rubrics
Exam rubrics for the 2026 examinations are available here.
Past exam papers
All papers are copyright by the University of Cambridge and may not be reproduced without permission.
The papers are stored as PDF files. The PDF files can be viewed and printed using the Adobe Acrobat viewer. This program is available free of charge from the Adobe web site.
Papers are available for past years from 2001, with the exception of 2020. Click on the links below to select the year you wish to view:
- 2026
- 2025
- 2024
- 2023
- 2022
- 2021
- 2019
- 2018
- 2017
- 2016
- 2015
- 2014
- 2013
- 2012
- 2011
- 2010
- 2009
- 2008
- 2007
- 2006
- 2005
- 2004
- 2003
- 2002
- 2001
Essay
The essay is a key component of Part III and counts for 3 units of examination credit, equivalent to a 24-lecture course. Note that, since there is a maximum of 16 units of credit that can be obtained from examination papers for lecture courses, it is essential to submit an essay in order to achieve a total amount of examination credit in the range 17-19 units. (See Sections 9 and 10 of the Part III Handbook.)
Experience has shown that the great majority of Part III students find that working on their essay is an enjoyable change from learning from lectures and is valuable training for research as well as a range of other careers.
A list of essay titles approved by the Part III Examiners will be announced by the end of the fourth week of the Michaelmas term and will appear together with more detailed descriptions in the Essay Booklet:
This booklet also gives details of the process of allocating essay titles to students, based on their preferences, and includes general guidelines and instructions about writing the essay, including advice concerning plagiarism.
Additional essay titles may be approved by the Part III Examiners and will be added to the booklet not later than 1 March.
Essays and grade descriptors
Just as with written examination papers, the Assessor awards a numerical mark out of a maximum of 100 to each essay and in addition assigns a ‘quality mark’ or grade (see Appendix IV of the Part III Handbook). The Faculty Board has specified that the minimum performance deserving of a distinction on a paper or an essay is associated with a grade of α-, while the minimum performance deserving of a pass is associated with a grade of β-.
The Faculty Board does not necessarily expect the mark distribution for essays to be the same as that for written examinations. Indeed, experience has shown that for many students the essay mark has been amongst their highest marks across all examination papers, both because of the typical amount of effort they have devoted to the essay and the different skill set being tested (compared to a time-limited written examination). The Faculty Board wishes that the hard work and talent thus exhibited should be properly rewarded.
To set clear expectations concerning the length of an essay, the Faculty Board has agreed that all candidates should use a standard LaTeX template, and that the length of an essay should normally be between 20 and 30 pages and should only exceptionally be more than 35 pages. The Board recognises that the length of an essay is not a reliable indicator of the quality of exposition and at most only weakly reflects the quantity of work involved; it is also anxious to prevent the essay absorbing too much of the candidate’s time. The Board is therefore content if a topic is set for which an excellent essay requires about 20 pages and would normally be unhappy if a topic were set for which an excellent essay required more than 35 pages.
In light of remarks above, as well as the comments of both internal and external Examiners over the years, the Faculty Board considers the following descriptors of the broad grade ranges for an essay to be appropriate. The Board trusts that these guidelines prove useful in guiding the judgement of the inevitably large number of Assessors marking essays, and thereby strengthen the mechanisms by which all essays are assessed uniformly. They are intended to be neither overly prescriptive nor fully comprehensive, but rather general guidance consistent with long-standing practice within the Faculty.
An Essay of α-Grade Standard (α-, α, α+)
Typical characteristics for the award of an α grade include:
- Demonstration of clear mastery of the underlying mathematical content of the essay.
- Demonstration of thorough understanding and cogent synthesis of advanced mathematical concepts.
- A well-structured and clearly-written essay of appropriate length with
- few expository or presentational flaws;
- a clear introduction demonstrating an appreciation of the context of the central topic of the essay;
- a coherent presentation of that central topic;
- clear and comprehensible conclusions, either in the body of the essay or in a final section summarising well the key points and suggesting possible future work.
An α grade would be consistent with the quality and level of understanding expected of an introductory chapter of a PhD thesis from a leading mathematics department. A possible characteristic of such an essay might be a more elegant presentation and synthesis than that presented in the underlying papers, perhaps in the form of a shorter or more efficient proof of some mathematical result. It would be expected that an essay containing original, publishable results would be of α+ standard, but, for the avoidance of doubt, publishable results are not necessary for an essay to be of α+ standard. A mark in the α+ range should, however, be justified by an explicit statement from the Assessor highlighting precisely which aspects of the essay are of particularly distinguished quality.
An Essay of β-Grade Standard (β-, β, β+)
Essays of β-grade standard encompass a wide range, but in all cases candidates should demonstrate understanding and an ability to synthesise mathematical concepts at the level expected for a pass mark in a Part III lecture course.
Typical characteristics expected for the award of a β+ grade include:
- Demonstration of good mastery of most of the underlying mathematical content of the essay.
- A largely well-structured essay of appropriate length with
- some minor expository or presentational issues;
- an introduction demonstrating an appreciation of at least some context of the central topic of the essay;
- a reasonable presentation of that central topic;
- comprehensible conclusions, either in the body of the essay or in a final section summarising the key points.
Candidates would not typically show evidence of extensive reading beyond the suggested material in the essay description, or original content.
Typical positive characteristics for the award of a β- grade (pass standard) include:
- Demonstration of understanding of some of the underlying mathematical content of the essay;
- An overall standard consistent with the quality expected of an upper-second-class final-year project from a leading mathematics department;
while negative characteristics might include some non-trivial flaws in exposition, for example:
- An inappropriate length;
- Repetition or lack of clarity;
- Lack of a coherent structure;
- The absence of either an introduction to provide context or of clearly identified conclusions.
For the avoidance of doubt, a key aspect of the essay is that the important mathematical content is presented clearly in (at least close to) the suggested length. An excessively long essay is likely to be of (at best) pass standard.
Examination allowances
Examination allowances may be granted by the EAMC (Examination Access and Mitigation Committee) when a student is prevented from fully completing their assessment due to illness or other grave cause, and when the difficulties are unanticipated and arise close in time to the examination.
The general University guidance on exam allowances is available here:
2025-26 Exam Allowances: Guidance for Students and Colleges
In addition, there are important special arrangements and exemptions that have been agreed for Part III Mathematics (in consultation with the EAMC):
The Faculty guidance notes should be consulted carefully, in conjunction with the University guidance, when submitting an application for an exam allowance in Part III.