Can a knight return home in an odd number of moves?

Question

A knight moves in an L-shape: two squares in one direction and one square at right angles. Starting from some square, it makes a sequence of moves and ends up back on its starting square. 1. Can the number of moves be odd? 2. Can a knight travel from the corner a1 to the opposite corner h8 in exactly 5 moves? In exactly 7?

Accepted Answer

No, a knight always needs an even number of moves to return. And a1 to h8 can be done in 6 moves but never in 5 or 7. The colour argument. A knight's move goes two squares one way and one square the other, so it changes the row by an odd amount and the column by an even amount, or vice versa; either way the sum of row and column changes by an odd number. On a chessboard, squares whose row plus column is even are one colour and those with an odd sum are the other. So every knight move changes the colour of the square it stands on. After an odd number of moves the knight is on the opposite colo…

Can a knight return home in an odd number of moves?

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