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

How many flights does it take to get anywhere?

AI HLAI SL Ambitious Also in: Geometry & Voronoi

Research question

In the network of [a set of airports], what is the maximum number of flights needed to travel between any two (the diameter), and which airports would most reduce it if given one new route?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Airline networks have hubs, which keep journeys short. Building a network from OpenFlights data and asking where a new route would help most is a real network-design problem.

The mathematics you'll need

Directed graphs and adjacency matrices
Powers of the adjacency matrix to count routes
Shortest paths and diameter
Effect of adding an edge

Where the data comes from

Routes and airports from OpenFlights; restrict to a region or airline to keep it manageable.

OpenFlights data — Airports (with latitude/longitude) and airline routes as CSV.

Real data for this idea

Primary source: OpenFlights data: Airports and routes. airports.dat gives name, IATA code, latitude and longitude; routes.dat gives each route's source and destination airport. Licence: Open Database Licence. The routes file is a 2014 snapshot.

The OpenFlights routes file is a 2014 snapshot: treat it as historical and say so. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Choose and justify the subnetwork.
Build the adjacency matrix.
Use matrix powers to find shortest path lengths.
Find the diameter and critical pairs.
Test new routes and reflect on the data's age (a snapshot).

Pitfalls that cost marks

Using the whole world network (too large).
Ignoring that the data is an old snapshot.
Directed vs undirected routes unclear.

Showing personal engagement

Include your home airport.
Plan a real trip using your model.
Compare hub-and-spoke and point-to-point airlines.

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	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Fits — ambitious at SL	Directed graphs and adjacency matrices; Powers of the adjacency matrix to count routes
AI HL	Good fit	Directed graphs and adjacency matrices; Powers of the adjacency matrix to count routes

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: using the whole world network (too large); and ignoring that the data is an old snapshot. Say how each one affects your answer.

Use of mathematics (E)

SL: directed graphs and adjacency matrices; powers of the adjacency matrix to count routes; shortest paths and diameter; effect of adding an edge, 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 distance and compare the shortest-by-flights and shortest-by-distance routes.

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 and AI SL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Directed graphs and adjacency matrices, powers of the adjacency matrix to count routes, shortest paths and diameter and effect of adding an edge.

Where can I get data for this IA?

Routes and airports from OpenFlights; restrict to a region or airline to keep it manageable. Checked sources and tools: OpenFlights data.

Turn this idea into your IA

The planner opens with “How many flights does it take to get anywhere?” filled in: the question, the maths and the data sources.

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