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

Which Monopoly squares are landed on most often?

AI HLAA HL Ambitious Also in: Networks & graphs, Probability, Matrices & Markov

Research question

Using a Markov chain that includes the Jail rules and the Go-to-Jail square, what is the long-run probability of landing on each square, and which property group gives the best return on investment?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Players argue about which sets are best. A Markov chain gives long-run landing frequencies — Jail and the squares after it stand out — and combining them with rents gives an evidence-based strategy.

The mathematics you'll need

Transition matrices with sums of two dice
Steady-state vectors (eigenvectors or high powers)
Expected rent per turn and payback period
Simplifying assumptions and their effect

Where the data comes from

Use the board's published rules and rent values for the edition you own.

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

A possible outline

State the rules modelled and simplifications (cards, doubles).
Build the transition matrix.
Find the steady state.
Combine with rents and costs.
Reflect on the simplifications and test with simulated games.

Pitfalls that cost marks

Ignoring Jail (the main effect).
Including every card without a plan — start simple.
Steady state without interpretation.

Showing personal engagement

Use your own edition of the board.
Test your strategy against family members.
Add the chance and community-chest cards and see what changes.

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	Transition matrices with sums of two dice; Steady-state vectors (eigenvectors or high powers)
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Transition matrices with sums of two dice; Steady-state vectors (eigenvectors or high powers)

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: ignoring Jail (the main effect); and including every card without a plan — start simple. Say how each one affects your answer.

Use of mathematics (E)

HL: transition matrices with sums of two dice; steady-state vectors (eigenvectors or high powers); expected rent per turn and payback period; simplifying assumptions and their effect, 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

Model the three-doubles rule and the choice to stay in Jail late in the game.

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

What mathematics does this IA use?

Transition matrices with sums of two dice, steady-state vectors (eigenvectors or high powers), expected rent per turn and payback period and simplifying assumptions and their effect.

Where can I get data for this IA?

Use the board's published rules and rent values for the edition you own. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Which Monopoly squares are landed on most often?” filled in: the question, the maths and the data sources.

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