A closed knight's tour on a five-by-five board

Question

A knight's tour is a sequence of knight moves that visits every square of a board exactly once. It is closed if the last square is a knight's move away from the first, so the knight could keep going round forever. 1. Can a 5 by 5 board have a closed knight's tour? 2. What can you say about which squares an open tour of the 5 by 5 board must start and end on?

Accepted Answer

No closed tour exists on a 5 by 5 board. Open tours exist, and they must start and end on squares of the majority colour. Counting colours. A 5 by 5 board has 25 squares: 13 of one colour (the corners' colour) and 12 of the other. A knight alternates colours on every move. Closed tours. A closed tour is a cycle through all 25 squares. Around the cycle the squares alternate in colour, so the cycle must contain equal numbers of each colour. Thirteen and twelve are not equal. No closed tour. The same argument for any board. A closed knight's tour needs the board to have an even number of squares…

A closed knight's tour on a five-by-five board

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