I've recently been reading a lot about game theory and octal games, and in the few sources on it I can find, people seem to agree that Treblecross' value is .007.
For reference, the game of Treblecross is as follows. Start with a 1-by-n board of empty squares. Players take turns placing X's into the empty squares. The first player to make a sequence of three consecutive X's wins the game. So, for example, if we start with a 1-by-6 board, a game may play out as thus:
☐☐☐✕☐☐ ✕☐☐✕☐☐ ✕☐✕✕☐☐ ✕☐✕✕✕☐
And Player 2 wins.
The reasoning I've seen for this game being the .007 game is that since no reasonable player would ever place an X next to an X already on the board, every move effectively takes up three spaces - the X itself, and the two spaces immediately adjacent to it. When a player has no more "reasonable" moves left, they immediately lose, since no matter where they place their X, the other player will instantly be able to make three-in-a-row.
I do not disagree with the notion that a game on a 1-by-n board where you place 1-by-3 blocks is equivalent to the .007 game. However, I do not understand why Treblecross is cited as being equivalent to this game. Am I misunderstanding in thinking that every X would actually take up five spaces? It is clear that when placing down an X on the board, there is a buffer zone of one on either side of the X, but shouldn't that buffer zone extend for two spaces on either side? For example, consider the following game.
☐☐☐✕☐☐ ☐✕☐✕☐☐ ☐✕✕✕☐☐
Player 2 stayed outside Player 1's one-space buffer zone, and then Player 1 still capitalizes on their move by placing an X immediately in between them. Thus, placing the X has the effect occupying five spaces, not three, correct?
Furthermore, if this is the case, wouldn't its code be .00337? Because
- A player cannot move in such a way that blocks off access to only one or two spaces.
- A player can only claim three spaces by placing an X in the first or last space, and cannot split the board into two sections using this method. This is also permitted to be the player's winning move.
- A player can only claim four spaces by placing an X in the second or penultimate space, and cannot split the board into two sections using this method. This is also permitted to be the player's winning move.
- A player can claim five spaces and split the board in the process. This is also permitted to be the player's winning move.