Re-entrant Queen's Tour

Theorem
Consider a chessboard $\CC$ of size $n \times n$ such that $n > 3$.

Then there exists a re-entrant queen's tour on $\CC$ of $2 n - 2$ moves.

For $n < 6$ it is necessary for the queen to move outside the boundary of the chessboard in order for this to happen.

Proof
First it is shown that at least $2 n - 2$ moves are needed.

Let there be $R$ rows and $S$ columns which have none of the given moves on them.

The $R \times S$-square segment of this chessboard has a $2 R + 2 S - 4$ squares on its edge if $R$ and $S$ are both greater than $1$.

Each diagonal moves covers at most $2$ of these boundary square.

Thus in the set of moves there are at least:
 * $R + S - 2$ diagonal moves
 * $n - R$ horizontal moves
 * $n - S$ vertical moves.

Thus there are at least $2 n - 2$ moves needed to cover the entire chessboard.

It remains to demonstrate that no more than $2 n - 2$ moves are needed.