Mathematics in Chess

Chessboard puzzles are a compact way to learn how local rules create global structure.

You are not expected to play strong chess here. You are expected to reason about:

• parity and colouring,
• graphs of legal moves,
• extremal upper bounds and constructions,
• local degree rules that force global patterns.

This page remains the umbrella guide for the whole collection, and the material is split into focused library guides.

Click any heading below to open the full guide.

Focused Guides

1. Knight Returns

Problems about returning a knight to its starting square after exactly $k$ moves.

• Main ideas: colouring invariants, closed walks in graphs, repeating even cycles.
• Includes short and long exact-return questions for a knight.

2. Maximum Non-Attacking Pieces

Extremal placement problems for kings, rooks, bishops, knights, and queens.

• Main ideas: upper bounds, constructions, row/column counting, diagonals, checkerboards.
• Includes maximum non-attacking kings, rooks, bishops, knights, and queens.

3. Exact Opponent Attacks

Two-colour placement problems where each chosen piece attacks exactly a fixed number of opponent pieces.

• Main ideas: regular bipartite attack graphs, degree sums, cycle structure, blocking, and constructive patterns.
• Includes degree-constrained knight constructions and black/white knight and queen attack puzzles.

4. Other Chessboard Problems

Structural construction problems that do not fit the other main families.

• Main ideas: row/column count constraints.
• Includes equal-file/distinct-rank constructions.

5. Knight's Tour

A classic Hamiltonian-path and Hamiltonian-cycle puzzle on the knight-move graph.

• Main ideas: graph modeling, Hamiltonian paths/cycles, Warnsdorff-style heuristics.
• Includes tours on board sizes from 5x5 up to the classical 8x8 board.

6. Flying Rook Cycles

Hamiltonian cycles on the flying-rook move graph: visit every square exactly once and return to start.

• Main ideas: custom move graph (rook moves of length at least 2), Hamiltonian cycles.
• Includes flying-rook cycle problems on 4x4 up to 8x8 boards.

Shared Problem-Solving Habits

Across all these guides, the same workflow keeps appearing:

1. Find the governing invariant or counting constraint.
2. Translate the board into a graph or a table of row/column totals.
3. Prove an upper bound or impossibility first.
4. Build a construction that meets the bound or satisfies the rule exactly.
5. Use the interactive as a fast checker, then write the proof cleanly.

Use the practice list on this page for the mixed collection, or jump into one of the focused guides when you want a single method family.