Vector Analysis: Chess

Question

I came across the following question on a mechanics assignment:

In chess, a knight makes L-shaped moves, two squares in one direction (horizontally or vertically) followed by one square in a perpendicular direction. Determine whether or not a knight can eventually reach every square on a 8 × 8 chess board. Hint: use vectors!

How would one go about it?

Answer 1

Take the knight to be in the top left position of a $3×3$ grid. This can be assumed because the edge is the only limiting factor and a configuration can be found by translation or rotation of a $3×3$ grid with the knight in the top left corner. We have

x 0 0    0 0 0    0 0 0    0 x 0
0 0 0 -> 0 0 x -> 0 0 0 -> 0 0 0
0 0 0    0 0 0    x 0 0    0 0 0

Thus we are able to move a knight arbitrarily on a chess board by repeating this pattern: moving to each square in a row, moving one column over then repeating the algorithm.