The Kingly Tour may have involved more work and exhibited less beauty
than a good puzzle should. So let's allow N E W S moves only, like
a Rook, but limit moves to one square at a time, like a hobbled Rook.
A hobbled Rook tours an nxn chessboard without visiting a square twice. Your
opponent begins by placing the hobbled Rook in any corner. Thereafter, you and
she alternate making single-square N E S W moves. A player loses when there
is no previously unoccupied square available for a move.
For what n do you have a winning strategy?
Is there an n for which you have a winning strategy in the modified case
where your opponent can make any move on her turn, while you are
constrained to make single-square N E S W moves only?