The Hobbled Rook Tour

[bonanova]

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?

[Prime]

Same as Kingly Tour.

For even "n" you must win.

After her move an odd number of squares have been used. She can only split an even number of squares board into even and odd territories. Whereafter you can step into the odd territory and maintain your advantage. Conversely, you must avoid splitting the board into two odd territories.
It is still possible to lose with even "n", but you must make a mistake for that.

On the diagram 6) C4 was a blunder. Now she wins by 7) D4. Curiously, 7) B4 loses, even though it's a step into an odd space. Because after 8) B3, she has the only move 9) A3 giving you a choice of two odd territories.

Instead of 6) C4 the correct move would be 6) B3 winning in 4 moves at the most.

(Unlike customary chess notation, I count a move for each side as one move here.)

A general strategy would be:

When you have a space of an odd number of squares -- go inside and pack it tightly without leaving any holes until you see a winning combination.

[flamebirde]

Any as in any true rook move or like as in your last puzzle?

[bonanova]

Same as Kingly Tour.

For even "n" you must win.

After her move an odd number of squares have been used. She can only split an even number of squares board into even and odd territories. Whereafter you can step into the odd territory and maintain your advantage. Conversely, you must avoid splitting the board into two odd territories.

It is still possible to lose with even "n", but you must make a mistake for that.

Kingmoves4.gif

On the diagram 6) C4 was a blunder. Now she wins by 7) D4. Curiously, 7) B4 loses, even though it's a step into an odd space. Because after 8) B3, she has the only move 9) A3 giving you a choice of two odd territories.

Instead of 6) C4 the correct move would be 6) B3 winning in 4 moves at the most.

(Unlike customary chess notation, I count a move for each side as one move here.)

A general strategy would be:

When you have a space of an odd number of squares -- go inside and pack it tightly without leaving any holes until you see a winning combination.

You have it. Here is a succinct approach.

The board is initially tiled with dominoes.

She selects a domino with her move. You take the other half.

This way, she can move on any square on each of her turns.