A Knight That Comes Home

Issue 10 Core Published 31 Aug 2026 By The Chalkboard Desk

The problem

A knight starts on a square of an ordinary chessboard, makes a sequence of legal knight moves, and returns to the square it started from. Show that the number of moves it made is even.

A knight move is two squares in one direction and one square perpendicular.
The board is the usual ; the knight never leaves it. The starting square is arbitrary.

Take a minute

Reveal hint 1

The obvious invariants — row, column, distance from the centre — all change in an uncontrolled way. Try colouring.

Reveal hint 2

What is the colour of the square after an odd number of knight moves?

Solution

AnswerThe path length must be even; every closed knight's tour has an even number of moves.

Derivation

Colour the board.
See the knight's effect.
Track the alternation.
Use the return condition.
Conclude.
Note the hidden structure.

Margin notes

Colour the board as usual, alternating light and dark squares; squares sharing an edge get opposite colours.
A knight move changes the row by or and the column by the complementary amount, so the sum (row column) changes by an odd number. Hence every knight move lands on a square of the opposite colour.
After moves the knight sits on a square whose colour differs from the start if and only if is odd.
Returning to the starting square means the final colour equals the starting colour, so is not odd.
must be even. On a standard board the smallest closed knight tour uses moves, and the familiar tours use .
The knight's graph on a chessboard is bipartite, split by colour; a closed walk in a bipartite graph always has even length. This is the same fact in different words.