Colourings

A colouring problem asks you to assign colours to objects — cells of a grid, faces of a cube, vertices, edges — so that certain local conditions are met simultaneously.

Types of Constraint

Neighbour-count rules. Each object of one colour must border exactly $k$ objects of another colour. For example: "each black cell must share a side with exactly one white cell."

Window rules. Every small sub-grid of a fixed size must contain all colours. For example: "every $1 \times 3$ horizontal strip must use all three colours."

Count rules. The total number of cells of a given colour is fixed, sometimes combined with a parity condition. For example: "exactly 6 cells are black, and each row contains an even number of white cells."

Spatial exclusion rules. No two cells of the same colour may be king-neighbours (sharing a side or corner).

How to Approach These Problems

1. Count the constraint edges. If every white cell must border exactly one black cell, then the total number of white–black adjacencies equals the number of white cells. Equating this to the count from the black side often pins down the ratio of colours.

2. Use parity. Row/column parity constraints are linear over $\mathbb{F}_2$. Think of each row as a binary vector and the constraint as a system of equations.

3. Try small cases. A $4 \times 4$ grid has only 16 cells; systematic trial with a few key cells often reveals the pattern.

4. For cubes. Draw the net for face problems. For vertex and edge problems, use the known adjacency graph: each face touches 4 others, each vertex touches 3, each edge touches 4.

Problems in This Guide

• COL-001A/B/C — $4 \times 4$ Board: Each Cell Borders a Fixed Number of Opposites
• COL-002 — $4 \times 4$ Board: Four Colours in Every Rectangle and Square
• COL-003 — $6 \times 6$ Board: Three Colours in Every Three-Cell Strip
• COL-004 — $4 \times 4$ Board: Six Black Cells with Row and Column Parity
• COL-005A/B/C — Cube Faces, Vertices, and Edges with Two Colours
• COL-006A/B — $8 \times 8$ Good Colouring (at most 16 black cells)
• COL-007 — $7 \times 7$ Board: More Red Than Blue, Each Red Borders Three Blue
• COL-008 — Count the Good $4 \times 4$ Colourings