## Task 34/78

On a regular chessboard, it is well known that a knight can be moved in such a way that it visits each square exactly once and returns to the starting square. This is possible in various ways.

In how many different ways is this possible on a (hypothetical) chessboard with $9 \cdot 9=81$ squares?
If the knight is to visit all squares and return to the starting square, he must make exactly 81 moves. With each move, he changes the color of the square, so that after an odd number of moves, he always lands on the color that is not the same as the color of the starting square.

Thus, it is already shown that he cannot be on the starting square after 81 moves. The number of possible different ways is therefore zero.