# Strategy for Movable Tic Tac Toe

Movable Tic Tac Toe is similar to the classic game as it uses 3 x 3 matrix and has the same goal of winning, which is to align 3 X's or O's marks horizontally, vertically or diagonally. The difference is:

1. A player is limited to 3 X's or O's marks.
2. Once a player has all 3 marks placed on the grid, the player has to move a piece when it's his turn.

Play:

1. At the first phase, the players place their pieces one by one in a normal tic-tac-toe form.
2. At the second phase, after each player placed his 3 pieces, players can move any piece in the direction of the arrows presented. They can move up to two spaces in the same direction and can jump over their own and opponents pieces. Turning is allowed at the end of a turn but moving cannot happen afterward.

Q: What would be a strategy for this game?

Example game:

• You might want to ask: what is the name of this variant? This will allow to search for strategy for this variant. Commented Nov 23, 2021 at 8:47
• The game you are describing sounds much more like Three Men's Morris than Tic Tac Toe.
– Zags
Commented Nov 23, 2021 at 22:42
• In order to solve the game, we need to know which arrows each tile has. Commented Nov 24, 2021 at 10:54
• The arrows are just a two perpendicular lines. But it changes because players can rotate them 45 degrees. They can change between orthogonal and diagonal. Commented Nov 24, 2021 at 14:32
• @Cohensius, to clarify, the pieces themselves have the arrows and define the direction of the piece. The square they are on has no effect. Commented Nov 24, 2021 at 21:42

The game you are describing is more similar to Three Men's Morris than to Tic Tac Toe. Unlike Three Men's Morris, the extra movement options under the rules you cite mean the game is likely a draw under optimal play. I'm assuming that this game has something akin to Chess's draw due to Threefold Repition; if not, the optimal result of the game is that you play forever, in which case the only winning move is not to play.

The classic Three Men's Morris was solved in the 1200's as a first player victory. The Libro de los Juegos lists the following solutions (translation from here):

There are two possible victories, depending upon the second player's moves, the first being: 1. b2, a3; 2. b1, b3; 3. c3, a1; 4. b1-c1, any; 5. b2-c2. The second is 1. b2, b3; 2. a1, c3; 3. a3, a2; 4. a1-b1, any; 5. b1-c1.

Those solutions give a good starting point for this game. I'm going to gloss over a lot of the piece orientations during setup; in general, the optimal piece orientation is implied by the subsequent moves a player wants to threaten.

As player 1, you want to start in the middle with your piece orthogonal (B2+).

Case 1: If player 2 responds in the middle of a side (say B3), player 1 can force a win by playing 2. a1, c3; 3. a3, a2. This generates the following board, where player 1 can move a1-c1 to win:

``````  1 2 3
A X O X
B . X O
C . . O
``````

If player 2 instead plays C1 on turn 3, player 1 can move b2-a2 to win.

Given that, player 2 should respond by playing in a corner (say A3). Player 1 can then force either of the following boards:

Case 2 (1. b2, a3; 2. b1, b3; 3. c3, a1):

``````  1 2 3
A O . O
B X X O
C . . X
``````

This seems to be a draw. Unlike in Three Man's Morris, here Player 1 must move to a2 to prevent a loss (player 2 threatens b3-a2). b1-a2 is the best move as it threatens a player 1 win, but player 2 just responds with b3-c2. Player 1 would then move a2-b1, player 2 would move c2-b3, and the game would be a stalemate.

Case 3 (1. b2, a3; 2. a2, c2; 3. c3, a1):

``````  1 2 3
A O X O
B . X .
C . O X
``````

I can't see any way player 1 can force a win here. If player 1 moves c3-b3 to threaten a win, player 2 moves a1-b1 to block, and the players would then reverse those moves causing a stalemate. Anything else isn't moving towards a victory. I believe this case is a draw as well.

• In this variation, the direction of the arrows presented matter. Sometimes a move won't be possible in the cases above. Also, pieces can be moved directly across corners (i.e. a2-b1, b1-c2, c2-b3, or b3-a2). So in Case 1, after X does b1-c1, O could directly move b3-a2 and win the game depending on the arrows direction. Commented Nov 24, 2021 at 1:11
• @Jonah If the design of the board is relevant, please update the question to include a picture of the board.
– Zags
Commented Nov 24, 2021 at 1:12
• The board is a normal tic tac toe shape. It is just a 3x3 square. Whereas in Three Men's Morris, the board has a 3x3 of spaces, but the connections/paths are not the same. Hope this helps. Commented Nov 24, 2021 at 1:34
• I edited the question to show the board and pieces, but I am not sure this will help. In Three Men's Morris, an edge can lead to a corner and the center, but not another edge. In this tic tac toe variant, a piece oriented diagonally can jump from edge to edge directly. Commented Nov 24, 2021 at 18:20
• @Jonah You need to add substantially more detail to the question. Please include a full description of how wraparound movement works as well as how the diagonal and orthogonal pieces work (what are their initial states, how they change, and how they interact with the rest of the game). Adding several sample turns to your question could also help clarify the rules.
– Zags
Commented Nov 24, 2021 at 18:57