IA idea · Optimisation & linear programming

Square or hexagonal? Packing tins into a box

AA SLAA HLAI SL Solid Also in: Geometry & Voronoi, Pure maths

Research question

For a given number of cylindrical tins, which arrangement (square grid, hexagonal rows or a mix) needs the smallest box, and at what number of tins does hexagonal packing start to win?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Everyone 'knows' hexagonal packing is more efficient, but in a small box it often isn't. Finding the crossover is a satisfying mix of geometry, algebra and case-by-case thinking.

The mathematics you'll need

Packing density: π/4 for square and π/(2√3) for hexagonal (derived)
Box dimensions as functions of the number of rows and columns
Trigonometry and Pythagoras for offset rows
Comparing discrete cases; finding a crossover

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Measure real tins and boxes from a supermarket, and test your predicted arrangements with coins or bottle tops.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive the two densities.
Write the box area for n tins in each arrangement.
Compare for n = 1 to 60 and find the crossover.
Test predictions physically with coins.
Reflect on why real cases use square packing (strength, handling, labelling).

Pitfalls that cost marks

Quoting the densities without deriving them.
Comparing only large n, where the answer is obvious.
Ignoring the box's wall thickness.

Showing personal engagement

Use the tins your family buys.
Test with coins and photograph the arrangements.
Ask why your supermarket's boxes are packed the way they are.

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	Packing density: π/4 for square and π/(2√3) for hexagonal (derived); Box dimensions as functions of the number of rows and columns
AA HL	Good fit	Packing density: π/4 for square and π/(2√3) for hexagonal (derived); Box dimensions as functions of the number of rows and columns
AI SL	Good fit	Packing density: π/4 for square and π/(2√3) for hexagonal (derived); Box dimensions as functions of the number of rows and columns
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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

Personal engagement (C)

Optimise a decision that is really yours or your school's (a timetable, a budget, a delivery), gather the real constraints yourself, and say which ones you chose to ignore and why.

Reflection (D)

Compare the mathematical optimum with what people actually do, and test how sensitive the optimum is: which constraint, if relaxed a little, would change the answer most? For this idea, start with: quoting the densities without deriving them — say how it affects your answer.

Use of mathematics (E)

SL: An objective function and constraints set up from the context, solved correctly (graphically for two variables, or with differentiation), the optimum checked and interpreted, and any new method such as linear programming explained in your own words.

HL: Optimisation with two or more variables, a justified numerical search, a proof that the optimum lies at a vertex, or a sensitivity analysis with calculus, used because the problem needs it.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Extend to three dimensions (stacking oranges) or to tins of two different sizes.

Extending it for HL

Add a second variable or a non-linear constraint, use a numerical search where calculus alone is not enough, and analyse how the optimum moves as a parameter changes.

See a complete IA, marked

Our annotated exemplar When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature (AI SL) asks a different question, but shows how a complete optimisation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).