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♞ The Knight's Tour

Can a knight visit every square on the chessboard exactly once?

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♞ The Challenge
Topics: graph theory, Hamiltonian paths, backtracking, problem-solving strategy
A chess knight moves in an L-shape: 2 squares in one direction and 1 square perpendicular (or vice versa).

The Knight's Tour: Starting from any square, can a knight move across the entire chessboard, visiting each of the 64 squares exactly once?

A closed tour returns to the starting square on the final move. An open tour ends anywhere.

A Knight's Tour exists for the standard 8×8 board — but not every starting square is equal, and smaller boards are worth exploring first.
5×5 board — try finding a tour! Click a square to place the knight.

Interactive: click squares in order to trace a knight's path. Reset button resets.

From any square (r, c), a knight can move to:
(r±1, c±2) and (r±2, c±1) — up to 8 possible moves from the centre of the board.

Corner squares have only 2 possible moves. Edge squares have 3–4. Central squares have up to 8.

Warnsdorff's Rule: At each step, move to the square that has the fewest onward moves. This heuristic finds a tour quickly on most boards.
📝 Questions
1
A knight is on square (1,1) of an 8×8 board (bottom-left corner). List all squares it can reach in one move. How many are there? 2 marks
2
How many moves does a knight in the exact centre of an 8×8 board have? (Use squares (4,4) or (5,5).) 2 marks
3
On a 4×4 board, is a complete knight's tour possible? Try to trace one, or explain using a colouring argument why it might be impossible. 4 marks
4
A Knight's Tour visits 64 squares, so it makes 63 moves. Each move changes the colour of the square the knight is on (light to dark or dark to light). How many light squares and dark squares does the knight visit in an open tour starting on a light square? 3 marks
5
For a closed tour (returning to start), the number of moves must be even (so the knight ends on the same colour it started on). An 8×8 board has 64 squares. Can a closed tour exist? Explain using the colouring argument. 3 marks
6
Extension: Warnsdorff's Rule says "move to the square with the fewest onward moves." Starting at (1,1) on an 8×8 board, trace the first 5 moves using this rule. Show the number of onward moves at each candidate square. 6 marks

Answer Key

♞ Knight's Tour

1.From (1,1): can reach (2,3) and (3,2) — 2 squares (corner has only 2 valid moves).
2.A knight in the centre can reach up to 8 squares.
3.A 4×4 board tour is impossible. Colouring argument: a 4×4 board has 8 light and 8 dark squares. A complete tour starting on light ends on dark (15 moves, alternating colours). This is achievable in principle, but with only 4 corner-constrained paths and the very limited moves available from corners, no complete tour exists. Accept either a failed attempt or the argument.
4.63 moves from a light square: the knight alternates colours on each move. Move 1 → dark, move 2 → light, … After 63 moves (odd), the knight ends on a dark square. It visits 32 light + 32 dark squares.
5.A closed tour needs 64 moves (to return to start). After 64 moves from a light square, the knight is back on a light square ✓. The colouring argument doesn't rule it out — and indeed, closed tours exist on the 8×8 board.
6.From (1,1), moves to (2,3) or (3,2). Count onward moves from each: (2,3) has 4 onward moves; (3,2) has 4 onward moves. Choose either. From the chosen square, continue selecting the square with fewest onward moves each step. Full path varies — award marks for correct application of the rule and counting onward moves accurately.

Answer Key

Knight's Tour

1.(2,3) and (3,2) — 2 squares.
2.8 squares.
3.Impossible on 4×4. Award marks for reasoning or attempted trace.
4.32 light, 32 dark squares.
5.64 moves returns to same colour — colouring argument doesn't rule it out. Closed tours exist.
6.Apply Warnsdorff: always pick the square with fewest onward moves. Award marks for correct move counts.
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