Consider a large chessboard. A limp rook is a chess piece that moves one step orthogonally, but it turns $90$ degrees after every move. The limp rook makes some moves, not crossing over its own path, and returning to the starting cell. It is known (see IMO Shortlist 2009 C6) that on an $n$ by $n$ grid, if $n$ is $3$ modulo $4$, the maximum number of steps the limp rook can take is $n^2-2n-3$.
A limp queen moves one step either orthogonally or diagonally, but it turns $45$ degrees instead (so changing its movement direction). How many cells can it visit in a cyclic path (not revisiting a square until the end) on an $n$ by $n$ grid if $n$ is $8$? $10$? How many for arbitrary $n$?