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Part III (MMath/MASt)


Exam timetable

The timetable for the 2023 examinations is available here.

Exam rubrics

Exam rubrics for the 2023 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:


As part of the Master of Mathematics / Master of Advanced Study, candidates may choose to offer an essay. This will count for 3 units, or about a sixth of the course. There is no prescribed length for an essay, but the general opinion seems to be that 5,000 to 8,000 words is a natural length. The essay does not have to be original in content.

Each year members of the Faculty propose suitable topics; links to those for the most recent submissions are provided below (subsequent years are expected to be broadly similar, although not identical). Students are also free to propose their own topic (subject to confirmation by the Faculty Board of Mathematics). Note that if an essay is written on a particular topic in a given year then that exact topic cannot be set in the next year.

Please also note that there will be changes in 2023-24 to how the essay option is organised. Details will be published in the new academic year.

A list of extra topics is generally offered during Lent Term, on the same basis and terms as any on the initial list (listed as "Additional Essay Topics").

The primary requirement on the presentation of Part III essays is that they are legible. Hand-written essays are acceptable (if legible), but you may prefer to use the text formatting software which is available on the University PWF network.

Essay descriptors

The Faculty Board believes that the essay is a key component of Part III, and does not necessarily expect the mark distribution for essays to be the same as that for written examinations. Indeed, in recent years 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.

In light of these beliefs, 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 prescriptive nor comprehensive, but rather general guidance consistent with long-standing practice within the Faculty.

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’ (see Appendix III 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 α-, while the minimum performance deserving of a pass is associated with β-.

An Essay of α-Grade Standard (α-, α, α+)

Typical characteristics expected of an essay of α-grade standard 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 well-written essay of appropriate length (5000-8000 words) with
    • few grammatical or presentational issues;
    • a clear introduction demonstrating an appreciation of the context of the central topic of the essay;
    • a coherent presentation of that central topic;
    • a final section which draws the essay to a clear and comprehensible end, summarising well the key points while suggesting possible future work.

An essay of α-grade standard would be consistent with the quality expected of an introductory chapter of a PhD thesis from a leading mathematics department. 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 would be one possible characteristic of an essay of α-grade standard. Furthermore, it would be expected that an essay containing 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 be justified by an explicit additional 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 all should demonstrate understanding and synthesis of mathematical concepts at the level expected for a pass mark in a Part III lecture course.

Typical characteristics expected of an essay of β+ standard include:

  • Demonstration of good mastery of most of the underlying mathematical content of the essay.
  • A largely well-structured essay of appropriate length (5000-8000 words) with
    • some minor, grammatical 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;
    • a final section which draws the entire essay to a comprehensible end, summarising the key points.

Such essays would not typically exhibit extensive reading beyond the suggested material in the essay description, or original content.

Typical minimum characteristics of an essay of β- (pass) standard include:

  • Demonstration of understanding of some of the underlying mathematical content of the essay.
  • An essay exhibiting some non-trivial flaws in presentation through, for example
    • an inappropriate length;
    • repetition or lack of clarity;
    • lack of a coherent structure;
    • the absence of either an introduction or conclusion.
  • An essay consistent with the quality expected of an upper-second-class final-year project from a leading mathematics department.

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.