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

What is the cheapest way to connect villages with fibre broadband?

AI HLAI SL Solid Also in: Geometry & Voronoi, Finance

Research question

What is the minimum total length of cable needed to connect [a group of villages] using a minimum spanning tree, and how much would adding one extra junction point (a Steiner point) save?

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Why it makes a good exploration

Utilities use minimum spanning trees to plan networks. Using real villages near you, and then discovering that adding a new point can beat the MST, is a strong, original extension.

The mathematics you'll need

Weighted graphs and minimum spanning trees
Kruskal's and Prim's algorithms (show one in full)
Distance calculations from coordinates
Steiner points and 120° angles (explain)

Where the data comes from

Village coordinates from OpenStreetMap; road distances if cables follow roads.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Real data for this idea

Primary source: OpenStreetMap. Read off coordinates (right-click, “Show address”) or use Export for a small area. Licence: Open Database Licence: credit “© OpenStreetMap contributors”; data you build from it must be shared alike.

OpenStreetMap is under the Open Database Licence: credit “© OpenStreetMap contributors”. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain the problem and assumptions (cable along roads or straight lines).
Build the distance table.
Find the MST with one algorithm, checking with the other.
Test a Steiner point and compute the saving.
Reflect on costs, terrain and existing infrastructure.

Pitfalls that cost marks

Repeating both algorithms in full (show one).
Mixing straight-line and road distances.
No real-world check of the result.

Showing personal engagement

Choose villages near your home.
Research the real broadband rollout in the area.
Estimate the cost using published per-km figures.

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	Good fit	Weighted graphs and minimum spanning trees; Kruskal's and Prim's algorithms (show one in full)
AI HL	Good fit	Weighted graphs and minimum spanning trees; Kruskal's and Prim's algorithms (show one in full)

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: repeating both algorithms in full (show one); and mixing straight-line and road distances. Say how each one affects your answer.

Use of mathematics (E)

SL: weighted graphs and minimum spanning trees; kruskal's and Prim's algorithms (show one in full); distance calculations from coordinates; steiner points and 120° angles (explain), 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

Find the optimal Steiner point position by minimising the total distance with calculus or technology.

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 Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Weighted graphs and minimum spanning trees, kruskal's and Prim's algorithms (show one in full), distance calculations from coordinates and steiner points and 120° angles (explain).

Where can I get data for this IA?

Village coordinates from OpenStreetMap; road distances if cables follow roads. Checked sources and tools: OpenStreetMap and GeoGebra.

Turn this idea into your IA

The planner opens with “What is the cheapest way to connect villages with fibre broadband?” filled in: the question, the maths and the data sources.

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