Four Colour Theorem for Kids: Colour a Map, Only Four

Give a child four crayons and a map they drew themselves, with the rule that no two regions sharing a border can match. They will do it. Then tell them to invent a map that four crayons cannot handle, and let them try. They will fail, and so has everybody else. The four colour theorem for kids is the rare activity where the puzzle is easy, the challenge is impossible, and the reason it is impossible took 124 years and a computer to establish. It is also the cheapest way to start an argument about what a proof actually is.

Key Takeaways

• You’ll draw and colour four of your own maps with exactly four crayons, build one that genuinely needs all four, and then try to break the rule.
• It teaches graph colouring, and the modern question of whether a proof a human cannot read still counts as a proof.
• Best from 8. Younger children enjoy the colouring; the “try to need five” challenge needs enough persistence to fail productively.
• One map is 2-colourable and looks like it should not be. Drawing it is the best surprise in the activity.
• Draw the maps with a pencil first and colour last. Colouring as you go means redoing it.

What You’ll Need

Item	How much	Where to get it	If you don’t have it
Plain paper	6 sheets	Printer	Back of old homework
Crayons or coloured pencils	Exactly 4 colours, plus a pencil	Pencil case	Four different shading patterns: dots, lines, crosses, blank
A printed map of your country’s regions	1	Print a blank outline map, or copy one by hand	Any atlas page traced on thin paper
Pencil and eraser	1 each	Anywhere	Pen, but you will regret it
Ruler	1, optional	School supplies	Freehand is fine and often better

Time: drawing 15 min, colouring 25 min, the challenge 15 min Cost: nothing Mess level: low

Do the Experiment: The Four Colour Theorem for Kids

Step 1 – Draw a scribble map and colour it with two crayons

Without lifting the pen, draw one long loop that crosses itself eight or nine times and comes back to where it started. You now have a map. Try to colour it with only two colours so no two regions sharing an edge match. It always works, which nobody expects. Put that map aside; you will come back to it.

Step 2 – Draw an ordinary map with ten regions

Now draw a map the normal way: an outline split into about ten regions with lines meeting at junctions, including some points where three regions meet. Keep the regions big enough to colour.

Step 3 – Colour it with four, and state the rule precisely

The rule is that regions sharing a border must differ. Regions meeting at a single corner point are allowed to match, the way Utah and New Mexico do. Say this out loud before they start, because it is the source of almost every argument that follows.

Step 4 – Try the same map with three crayons

Sometimes three is enough and sometimes it is not. Let them push on it for five minutes. If three works, challenge them to add one region that breaks it.

Step 5 – Build a map that provably needs four

Draw a large triangle, then a small triangle in the middle, then connect the small triangle’s corners to the big one’s corners, giving four regions: three around the outside and one in the centre. Every region borders every other. Four regions all touching each other means four colours, with no argument available.

Step 6 – Turn a map into dots and lines

Take the ten-region map and put a dot in the middle of each region. Join two dots whenever their regions share a border. Now the problem has changed shape: colour the dots so joined dots differ. This is the same puzzle with the geography deleted, and it is how mathematicians actually work on it.

Step 7 – Colour a real map of your country

Use the printed outline and four crayons. This takes longer than the invented maps because real borders are awkward, and there is usually one region that forces a redo of three others. That backtracking is a genuine algorithmic experience.

Step 8 – Try to invent a map needing five colours

Give them ten minutes and real encouragement. They will produce maps that look impossible and then turn out to need four. Then tell them the punchline: people tried this for 124 years, one published proof was wrong for a decade before anyone noticed, and the eventual proof needed a computer.

What you should see: The scribble map colours with two every time, and it looks like it should not. The ten-region map takes two or three attempts with four crayons, usually with one backtrack. Three crayons fail on most maps with more than six regions. The real country map is the longest part, 15 to 20 minutes, with at least one moment of having to recolour a whole cluster. Nobody will produce a five-colour map, and the honest thing to say is not “well done, it’s impossible” but “nobody knew that for certain until 1976”.

What’s Actually Going On

The problem is graph colouring. Delete the geography, keep a dot for each region and a line for each shared border, and the question becomes: what is the smallest number of colours that always suffices so that joined dots differ? For any map drawn on a flat sheet or a sphere, the answer is four.

Two colours are enough for the scribble map because of a specific property: every crossing point has exactly four lines leaving it, which makes the regions alternate like a chessboard. Three is not enough in general, and the triangle-in-a-triangle map from step 5 shows why in one picture: four regions each bordering all three others cannot be done with three colours, and you cannot argue with it. The hard part was never showing that four are sometimes needed. It was showing that four are always enough.

The history is one of the best stories in mathematics, and Georges Gonthier’s 2008 account in the Notices of the AMS lays it out crisply. Francis Guthrie posed the question in 1852. It embarrassed his brother, his brother’s professor Augustus De Morgan, and everyone De Morgan showed it to. Alfred Kempe published a proof in 1879, and it took the mathematical community about ten years to notice that it was wrong. Birkhoff found the first useful piece of machinery in 1913, Heesch added a counting technique in 1969, and in 1976 Kenneth Appel and Wolfgang Haken finally proved it, with a published version appearing in the Illinois Journal of Mathematics in 1977.

Here is the part worth telling a child, because it is about the nature of proof rather than about maps. Appel and Haken’s proof worked by reducing the infinite problem to a finite list of map fragments and having a computer check every one. No human could read the whole thing. Robertson, Sanders, Seymour and Thomas produced a cleaner version in the 1990s that cut the list to 633 configurations, but 137 of those have a boundary ring of 14 regions and each requires checking about 20 million cases. Still not readable by a person.

Gonthier put the objection better than anyone: running code several times with the same input is weak evidence, because it is “blind to the most common cause of ‘computer’ error: programmer error.” So in 2005 he and his colleagues did something different. They formalised the entire proof, the human argument and the computation together, inside the proof assistant Coq, so that a separate and much simpler program could verify every single logical step. That is now the state of the four colour theorem: proved, checked by machine, and genuinely difficult for any one person to hold in their head.

One last honest caveat, because children find it immediately. The theorem applies to maps on a flat surface or a sphere where each region is one connected piece. If a country has two separated parts that must share a colour, four is no longer enough, and you can draw a counterexample in two minutes. And on a doughnut-shaped surface the answer is seven, not four. The theorem is about a specific situation, which is true of most theorems and almost never said out loud.

How to Explain It to Your Kid

Ages 5–8: neighbours can’t match

Forget the theorem. Give them a map with six big regions, four crayons and one rule: touching regions must be different. That is a genuinely absorbing puzzle at this age and it trains the planning-ahead they will need for everything else. Then show them the corner rule, two regions meeting at one point, and let them decide whether that counts. Deciding on a rule together is good mathematics.

Ages 9–12: build the one that needs four

The key move is step 5. Have them draw the triangle inside a triangle and then prove to you that three colours cannot work, by trying and explaining why each attempt dies. That is their first proof of impossibility, and it is short enough to hold. Then do the dots-and-lines translation and point out that the map’s shapes stopped mattering.

Ages 13+: what counts as a proof?

Give them the history and the argument. The theorem is true, nobody can read the proof, and a different program has verified every step of a formal version. Ask whether they believe it, and why. Then set the investigation: write a colouring algorithm in words that works on any map, test it on five of their own maps, and find a map where their algorithm has to backtrack. They will discover greedy colouring, that it sometimes uses more colours than necessary, and that a method which always finds the minimum is a much harder problem.

The question to ask: “I have drawn a map where every single region touches every other region. How many regions can my map have, at most, before that becomes impossible to draw on a flat sheet?”

Why Teaching This Early Pays Off

Most of what children are taught about mathematics is settled, old and anonymous. This is settled, recent and has names attached, including the name of a man whose wrong proof stood for ten years. That is a more honest picture of how knowledge actually accumulates.

The concrete payoff is graph colouring itself, which is not a curiosity. Timetabling a school so no teacher is double-booked, assigning radio frequencies so neighbouring transmitters do not clash, allocating registers in a compiler and scheduling exams are all colouring problems. A child who has backtracked through a map by hand has done the same work a scheduling algorithm does.

The deeper payoff is the question about proof, which is becoming urgent rather than academic. Computer-assisted and machine-checked proofs are now normal in parts of mathematics, and arguments about whether an automated system can be trusted are live. Jeannette Wing’s account of computational thinking is about exactly this: formulating a problem so that a machine can work on it, and then reasoning about whether the machine’s answer means anything.

As always, the modest version is the true one. An afternoon with crayons does not improve anyone’s grades, and the research on hands-on activity does not promise that. What it offers is one concrete memory of trying to break a theorem and failing, which is a far better preparation for being told things than agreeing with them.

Before You Start: What Parents Should Know

Best age	8 and up. The colouring puzzle works from 5, but the challenge in step 8 needs a child who can fail at something for ten minutes and find that interesting rather than defeating.
Supervision	Low. Your real job is the rule in step 3 and the honesty in step 8: do not tell them it is impossible until they have properly tried.
Real hazards	None at all. Paper and crayons.
Setup	A table and good light. You need four clearly distinguishable colours, which rules out a pack where two blues look alike under a lamp.
Frustration point	The real country map, where one bad early choice forces recolouring several regions. Say in advance that backtracking is expected and is not a mistake.
Don’t	Don’t colour as you draw, and don’t let the corner rule stay vague. Both produce arguments that look like the theorem failing.

When It Doesn’t Work

• They insist four is not enough. Check the corner rule: they are probably treating two regions that meet at a single point as neighbours. Under the standard rule those may share a colour.
• The map is impossible because a region is in two pieces. This is a real exception, not a mistake. If one country appears twice on the map and both parts must match, four colours can genuinely fail. Point it out and keep the map.
• The country map takes forever. Start from the region with the most neighbours rather than from a corner. Colouring the most constrained region first cuts the backtracking dramatically, and that is a real algorithmic strategy.
• Three colours worked and they want to know why they bothered. Ask them to add one region to the map that breaks it. It is usually possible in one line, and finding it is better than any explanation.
• A young child finds the whole thing just colouring. That is fine, and it is still worth doing. Add the goal of using the fewest colours they can and it becomes a puzzle again.

Take It Further

Invent a greedy algorithm and find where it wastes colours. Write a rule in words, such as “go region by region and use the lowest-numbered colour that does not clash”. Test it on five maps and count the colours used. Then find a map and an ordering where it uses five. Greedy colouring is a real algorithm and its weakness is exactly this.

Colour a map on a doughnut. Draw a rectangle, declare that the left edge joins the right and the top joins the bottom, and colour a map under those rules. Now four colours are not always enough; the answer for a torus is seven. This is the cleanest demonstration that a theorem depends on the surface it lives on.

What to Watch For Over the Next 3 Months

• Week 4: They check real maps in books or on walls and point out that four colours are enough, usually with some triumph.
• Month 2: The word “neighbour” or “clash” shows up in a scheduling argument, about seating, teams or who is sitting next to whom.
• Month 3: The question worth encouraging is whether a computer can be trusted to prove something. There is no settled answer, which makes it a very good thing to be arguing about at thirteen.

Frequently Asked Questions

Do regions meeting at a corner have to be different colours?

No, not under the standard statement of the theorem. Only regions sharing a stretch of border must differ. Regions touching at a single point may share a colour, which is why the real map of the United States works despite the Four Corners. Agree this before you start colouring.

Why did the four colour theorem take so long to prove?

Because showing four colours are always enough means ruling out every possible map, and the number of possible configurations is vast. Kempe’s 1879 attempt was accepted and then found to be wrong about a decade later. A complete proof arrived only in 1976, with a computer checking a finite list of cases.

Is a proof a computer checked really a proof?

It is accepted as one, but it changed what mathematicians argue about. Appel and Haken’s version could not be read by a human. A 1990s revision reduced the case list to 633 configurations, many still needing millions of machine checks, and in 2005 the whole proof was formalised in the Coq system so a separate program could verify each logical step.

Can you always colour a map with three?

No. The simplest counterexample is a triangle with a smaller triangle inside it joined to all three corners, giving four regions that each border the other three. Four regions mutually touching require four colours, and a child can construct it in a minute.

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About the author

Ricky Flores is the founder of HiWave Makers and an electrical engineer with 15+ years of experience building consumer technology at Apple, Samsung, and Texas Instruments. He writes about how kids learn to build, think, and create in a tech-saturated world. Read more at hiwavemakers.com.

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Sources

1. Gonthier, G. (2008). “Formal Proof — The Four-Color Theorem.” Notices of the American Mathematical Society, 55(11), 1382–1393. https://www.ams.org/notices/200811/tx081101382p.pdf
2. Appel, K., & Haken, W. (1977). “Every planar map is four colorable. Part I: Discharging.” Illinois Journal of Mathematics, 21(3), 429–490. https://doi.org/10.1215/ijm/1256049011
3. Robertson, N., Sanders, D., Seymour, P., & Thomas, R. (1997). “The Four-Colour Theorem.” Journal of Combinatorial Theory, Series B, 70(1), 2–44. https://doi.org/10.1006/jctb.1997.1750
4. NRICH, University of Cambridge. “Four Colours.” https://nrich.maths.org/problems/four-colours
5. Wing, J. M. (2006). “Computational thinking.” Communications of the ACM, 49(3), 33–35. https://doi.org/10.1145/1118178.1118215
6. National Research Council. (2012). A Framework for K-12 Science Education. National Academies Press. https://doi.org/10.17226/13165

Related reading: the seven bridges puzzle that invented networks, which shapes tile a floor and why, and how mathematicians verify a proof with Lean.