Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

What is the fewest number of exam sessions my school needs?

AI HLAA HLAI SL Solid Also in: Pure maths

Research question

Modelling subjects as vertices joined when a student takes both, what is the chromatic number of my school's exam graph, and how close does a greedy colouring algorithm get?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Exam officers solve a graph-colouring problem every season. Using your school's subject combinations (anonymised) makes a classic piece of discrete mathematics practical.

The mathematics you'll need

Graphs and graph colouring (explain)
Greedy algorithms and ordering
Lower bounds from cliques
Comparing with the real timetable

Where the data comes from

Anonymised subject-combination counts from your school's options (with permission) or your own class.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Build the conflict graph.
Find a lower bound using the largest clique.
Apply the greedy algorithm with different orders.
Compare with the actual exam timetable.
Reflect on other constraints (rooms, time of day).

Pitfalls that cost marks

Any identifiable student data.
Only one vertex order for the greedy method.
Confusing the chromatic number with the maximum degree.

Showing personal engagement

Use your own school's options.
Discuss with the exams officer.
Design a fairer timetable for students with many exams.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Graphs and graph colouring (explain); Greedy algorithms and ordering
AI SL	Good fit	Graphs and graph colouring (explain); Greedy algorithms and ordering
AI HL	Good fit	Graphs and graph colouring (explain); Greedy algorithms and ordering

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: any identifiable student data; and only one vertex order for the greedy method. Say how each one affects your answer.

Use of mathematics (E)

SL: graphs and graph colouring (explain); greedy algorithms and ordering; lower bounds from cliques; comparing with the real timetable, each explained in your own words and carried out correctly.

How networks & graphs explorations reach the top bands in general: Networks, graphs & algorithms IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Weight edges by the number of students affected and minimise back-to-back exams.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL networks & graphs exploration adds.

See a complete IA, marked

What is the quickest way to deliver a leaflet to every house on my estate? (AI HL) is a complete networks & graphs exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI HL, AA HL and AI SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Graphs and graph colouring (explain), greedy algorithms and ordering, lower bounds from cliques and comparing with the real timetable.

Where can I get data for this IA?

Anonymised subject-combination counts from your school's options (with permission) or your own class. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “What is the fewest number of exam sessions my school needs?” filled in: the question, the maths and the data sources.

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