Problem
COM-B2-M05-P009 Knight Moves and Colours
#9
★★★☆☆ Level 3 of 5
A knight is to visit all squares of a \(5\times5\) board exactly once. Prove that such a path cannot be closed, meaning the last move cannot return the knight to the starting square.
A knight changes the colour of the square on every move.
Colour the board like a chessboard. Every knight move takes the knight to a square of the opposite colour.
If a path through all \(25\) squares were closed, it would form a cycle of odd length \(25\) with colours alternating. But a closed alternating cycle must have even length. Contradiction.
The problem is useful because the colouring invariant is applied to the graph of moves.