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

Where should you build your first settlement in Catan?

AA SLAI SLAA HLAI HL Accessible Also in: Probability, Statistics

Research question

Using the probability distribution of the sum of two dice, which settlement spots give the highest expected number of resources per turn, and does expected value alone make a good strategy?

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Why it makes a good exploration

Catan players learn that 6 and 8 are good, but how good? Expected resources per turn, plus variance and diversity of resources, turns a board-game habit into a quantitative decision.

The mathematics you'll need

Probability distribution of the sum of two dice
Expected value and variance
Comparing spots by expected production
Simulation of games

Where the data comes from

Record a board setup; play and record production over several games.

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

A possible outline

Derive the two-dice distribution.
Compute expected production for each spot.
Consider variance and resource diversity.
Compare predictions with game records.
Reflect on trading and the robber.

Pitfalls that cost marks

Only expected value, no variance.
Too few recorded games.
Ignoring the robber rule.

Showing personal engagement

Use your family's board.
Test your strategy in games.
Design a fairer board and justify it.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Probability distribution of the sum of two dice; Expected value and variance
AA HL	Fits, but add an HL technique	Probability distribution of the sum of two dice; Expected value and variance
AI SL	Good fit	Probability distribution of the sum of two dice; Expected value and variance
AI HL	Fits, but add an HL technique	Probability distribution of the sum of two dice; Expected value and variance

Level: Accessible. A good first extended piece of maths, with room to go deeper. 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: only expected value, no variance; and too few recorded games. Say how each one affects your answer.

Use of mathematics (E)

SL: probability distribution of the sum of two dice; expected value and variance; comparing spots by expected production; simulation of games, each explained in your own words and carried out correctly.

How games & puzzles explorations reach the top bands in general: Games & puzzles IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Simulate 10,000 games' first 20 turns to compare placement strategies.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL games & puzzles exploration adds.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete games & puzzles exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Probability distribution of the sum of two dice, expected value and variance, comparing spots by expected production and simulation of games.

Where can I get data for this IA?

Record a board setup; play and record production over several games. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Where should you build your first settlement in Catan?” filled in: the question, the maths and the data sources.

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