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

IA idea · Differential equations & dynamics

Can mathematics predict a battle? Lanchester's laws and Iwo Jima

AI HLAA HL Ambitious Also in: Statistics

Research question

How well do Lanchester's square-law differential equations, with parameters estimated as in Engel's 1954 study, reproduce the troop numbers in the Battle of Iwo Jima, and what does the square law imply about concentration of force?

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Why it makes a good exploration

Lanchester's equations explain why concentration of force matters, and in 1954 J. H. Engel tested them against real battle data. Re-doing the analysis is a rare chance to verify a classic piece of applied mathematics.

The mathematics you'll need

Coupled linear differential equations
Deriving the square law invariant aM² − bN² = constant
Euler's method with reinforcements
Parameter estimation and goodness of fit

Modelling the data, step by step

This idea compares models fitted to data. See it worked step by step, with a criterion tip at every step: Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Chi-squared goodness of fit. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Use the daily troop and casualty figures reported in Engel's paper and later reanalyses; cite them carefully.

Engel (1954), A Verification of Lanchester's Law — The classic paper fitting Lanchester's combat equations to daily troop numbers from the Battle of Iwo Jima.

A possible outline

Explain the model and its assumptions.
Derive the square law from the equations.
Fit parameters and simulate the battle.
Compare with the recorded numbers.
Reflect on what the model ignores and on using maths to study war.

Pitfalls that cost marks

Treating a historical simulation as a precise prediction.
Copying Engel's results instead of reproducing them.
Ignoring the reinforcement term.

Showing personal engagement

Discuss the ethics of military modelling thoughtfully.
Test the linear law as an alternative and compare.
Apply the square law to a strategy game you play.

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	Coupled linear differential equations; Deriving the square law invariant aM² − bN² = constant
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	Coupled linear differential equations; Deriving the square law invariant aM² − bN² = constant

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: treating a historical simulation as a precise prediction; and copying Engel's results instead of reproducing them. Say how each one affects your answer.

Use of mathematics (E)

HL: coupled linear differential equations; deriving the square law invariant aM² − bN² = constant; euler's method with reinforcements; parameter estimation and goodness of fit, each explained in your own words and carried out correctly.

How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Fit both the square and linear laws by least squares and decide which fits better.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL differential equations exploration adds.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations 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?

Coupled linear differential equations, deriving the square law invariant aM² − bN² = constant, euler's method with reinforcements and parameter estimation and goodness of fit.

Where can I get data for this IA?

Use the daily troop and casualty figures reported in Engel's paper and later reanalyses; cite them carefully. Checked sources and tools: Engel (1954), A Verification of Lanchester's Law.

Turn this idea into your IA

The planner opens with “Can mathematics predict a battle? Lanchester's laws and Iwo Jima” filled in: the question, the maths and the data sources.

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