Neighbour-colour map

Assign three labels so every listed neighbouring pair differs, then justify the minimum with a triangle.

Adult planning suggestion: about 6–9 · Logic & rule checking · Shape & spatial thinking · Cooperation & communication

Players

Pairs

Time

12–18 minutes

Materials and quantities

One six-region graph, three labelled colour pencils and a neighbour audit table per pair.

Thirty pupils: fifteen pairs; checker and colourer swap after a complete audit.

Prepare

Use regions A–F with neighbour pairs AB, AC, BC, BD, CD, CE, DF and EF. A line means the two regions share the relevant boundary; a mere corner touch in an invented drawing does not add a pair. Use labels R, G and B along with colours so the worksheet remains usable in black and white. An adult can read the edge list aloud before the pair begins.

Play

Partners choose labels for regions, then check each of the eight listed pairs. They may revise a label, but must recheck all its incident edges afterward. Do not require every region to have a different colour; non-neighbours may share one. Record a check beside each pair rather than judging whether the whole map looks varied. The aim is a valid assignment, followed by a reason about the number of labels.

Worked example

One valid assignment is A=R, B=G, C=B, D=R, E=R and F=G. AB, AC and BC form a triangle. Its three vertices must all differ, so two labels cannot colour this graph. Since the displayed assignment uses only three, three is enough as well as necessary. This reasoning is about this supplied graph, not a proof of the four-colour theorem.

Finish and check

Finish when all eight pairs are checked and a partner can explain the triangle argument, or after the agreed twelve-minute search period. A different valid three-label assignment is equally acceptable. The answer sheet gives one model, not a prescribed colour preference. If an adult changes the neighbour list, the old minimality explanation may no longer apply and needs rechecking.

Adaptation and practical limits

For thirty pupils use fifteen pairs. One labels while the other reads and checks edges, then exchange roles on a fresh copy. Adult planning suggestion is roughly 6–9, depending on familiarity with pairwise constraints; the reference is mechanism background rather than a tested age claim. Hand-copy the six names and edges. No commercial map, special counters or printer is required.

Free alternatives: Use paper or clean reused cardboard and a pencil. Copy the supplied diagrams and tables by hand; no specialist kit or purchase is needed.

Printable materials

Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.