Chessboard Colouring

Colouring is one of the cleanest ways to turn a picture into a proof.

The basic move is simple:

1. Colour the board in a repeating pattern.
2. Work out what each allowed move or tile does to the colours.
3. Compare that with what the whole board requires.

Sometimes the colouring is the ordinary black-white chessboard. Sometimes a stronger pattern is needed: a 3D checkerboard, a triangular-grid orientation count, or a weighted repeating pattern.

The Core Question

Most problems in this guide ask:

Does the proposed path, movement, or tiling preserve a colour balance that the final board does not have?

If the answer is yes, the problem is impossible for a structural reason, not because we failed to find a clever construction.

Problems In This Guide

• CCB-001 R2-D2 Hamiltonian Path
• CCB-002 Mutilated Chessboard Dominoes
• CCB-003 Stormtrooper Shuffle
• CCB-004 Two Knight Scouts
• CCB-005 T Tetrominoes on a 10 by 10 Board
• CCB-006 Alternating Local Extrema
• CCB-007 Missing Centre Cube
• CCB-008 Star Destroyer Triangular Tiling
• CCB-009 1 by 4 Rectangles on a 10 by 10 Board

Return to the Mathematics in Chess overview for related chessboard problems.