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

How long does a game of Snakes and Ladders last?

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

Research question

Using a Markov chain for a particular board, what is the probability that a game finishes within n turns, what are the expected and median game lengths, and which single snake most lengthens the game?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Snakes and Ladders has no decisions, so it is pure probability — and a perfect Markov chain. Powers of a transition matrix give the exact distribution of game length, which you can check by simulation and by playing.

The mathematics you'll need

Transition matrices (AI HL) for a 101-state chain
Matrix powers with technology; absorbing states
Cumulative distribution and median of game length
Expected value as a sum of tail probabilities
Simulation to verify

Where the data comes from

Record a board layout (a family board or one you design) and play or simulate games to test predictions.

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

A possible outline

Describe the board and rules (exact finish or not).
Build the transition matrix, explaining one row in full.
Compute P(finished by turn n) from matrix powers.
Find expected and median lengths; verify by simulation.
Remove each snake in turn to find the one with most effect; reflect on rule variants.

Pitfalls that cost marks

Building the matrix by hand for all 100 squares (use technology, explain one row).
Forgetting the rule for overshooting 100.
Confusing the median with the expected value.

Showing personal engagement

Use the board you grew up playing.
Design a board that makes games last about 30 turns.
Compare your model with games played with family.

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 (AI HL) for a 101-state chain; Matrix powers with technology; absorbing states
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 (AI HL) for a 101-state chain; Matrix powers with technology; absorbing states

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: building the matrix by hand for all 100 squares (use technology, explain one row); and forgetting the rule for overshooting 100. Say how each one affects your answer.

Use of mathematics (E)

HL: transition matrices (AI HL) for a 101-state chain; matrix powers with technology; absorbing states; cumulative distribution and median of game length; expected value as a sum of tail probabilities; simulation to verify, 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

Use the fundamental matrix (I − Q)⁻¹ (explain it) to get expected turns from every square.

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) explores a question like this one, 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 (AI HL) for a 101-state chain, matrix powers with technology; absorbing states, cumulative distribution and median of game length, expected value as a sum of tail probabilities and simulation to verify.

Where can I get data for this IA?

Record a board layout (a family board or one you design) and play or simulate games to test predictions. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How long does a game of Snakes and Ladders last?” filled in: the question, the maths and the data sources.

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