Why Four Colours Are Enough
The four-colour theorem works because a careful swapping argument on coloured chains can always remove the one configuration that seems to demand a fifth colour.
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Why are four colours always enough to colour any map so that no two neighbouring regions share a colour?
Mapmakers once believed some maps needed five colours — then a quiet theorem proved four always suffice.
It sounds impossible: with infinite map shapes, why can't four colours ever fail?
Tangled map examples, a near-miss fifth-region scenario, and a step-by-step colouring routine that the learner drives.
Four colours always suffice because any troublesome region can be 'shrunk' into a point without breaking the colouring — a Kempe-chain argument in action.
More complicated maps should need more colours, so surely some map exists that truly needs a fifth one.
- Full computer-assisted proof details of the four-colour theorem
- Non-planar maps on tori or higher-genus surfaces
- History of false proofs by Kempe and others
- 01The Map-Colouring PuzzleslideQuestion
Open with the driving question and show ordinary maps of countries to set the scene.
- Map colouring asks that neighbouring regions differ in colour
- Three colours sometimes fail on simple maps (think: a triangle of three regions)
- Does every map really fit inside four colours?
- 02Commit to a GuessquizPrediction
Ask the learner to predict whether any map ever truly needs a fifth colour before any proof is shown.
- Make one independent choice between 'always four' and 'sometimes five'
- 03Try to Build a Map That FailsinteractiveEvidence
Let the learner add regions to a blank canvas and attempt to force a fifth colour, watching how quickly the chain of neighbours pushes back.
- Drag to drop new regions along existing borders
- Watch the live four-colour suggestion update
- Notice that any fifth region can usually be recoloured by swapping two existing colours
- 04Kempe Chains: The Swapping TrickslideExplanation
Explain Kempe-chain recolouring: how swapping two colours along a connected chain can resolve a stuck region without creating new conflicts.
- A Kempe chain is the connected set of regions coloured with two specific colours
- Swapping those two colours along the chain is legal everywhere on the chain
- This swap can free up a colour for the stuck region
- 05Run the Swap YourselfinteractiveEvidence
Give the learner a deliberately tricky near-five-colour map and let them perform a Kempe swap to fix it.
- Identify a region that appears stuck with a fifth colour
- Toggle colours along its Kempe chain
- Confirm the swap leaves the rest of the map valid
- 06Where Four Colours BreakslideBoundary
Show that the rule depends on the map being planar — drawn on a flat surface without crossings — and hint at how the count changes on a torus.
- Planarity (no border crossings) is essential
- On a donut-shaped torus, seven colours can be needed
- The theorem says nothing about non-planar surfaces
- 07From Maps to Schedules and BeyondslideTransfer
Transfer the idea: colouring conflicts appears in scheduling, register allocation in compilers, and Sudoku, where the same chain-of-neighbours logic applies.
- Schedule conflicts form a 'map' of competing tasks
- Compilers colour variables so simultaneously-live ones differ
- The Kempe-chain idea generalises to many graph-colouring tasks
- 08Why Four Is Always EnoughslideResolution
Return to the driving question and deliver the resolution: any apparent demand for a fifth colour dissolves under a Kempe-chain swap.
- No planar map forces a fifth colour
- Kempe-chain swaps fix the only configuration that looks stuck
- The theorem, proven with computer help, depends on planarity and this swapping argument
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