# Why do we say Treblecross' octal game notation is .007?

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.
• Player 2's move should also take up three spaces, and cannot overlap player 1's move. Therefore in your second example player 2's move would not be a valid play in a .007 game because its buffer zone overlaps the buffer zone of player 1's move. Any move in treble cross that leads to an immediate loss is invalid in the equivalent .007 game. Note that the treblecross game on board size $n$ is equivalent to the .007 game starting with a pile of $n+2$ tokens, because in treblecross you can take the end squares that don't have a buffer zone on one side, while in .007 you always take 3 tokens.. Oct 21, 2021 at 15:26
• It seems more obvious when you say it like that. For some reason it hadn't occurred to me that both players would need an unclaimed space on either side. Thank you! Oct 21, 2021 at 17:14

For your variant, the actual octal notation would be 0.11337. For example, in the board X _ _ _ _ _ X, there is a single available space in the center, corresponding to a heap of size $$1$$. It is legal to play in that square, removing the heap.
It is true that it is valid to think of Treble-cross as octal 0.11337. It is also valid to think of it as octal 0.007, as long as you imagine a board of length $$n$$ is actually a heap of size $$n+2$$. Below are the values of the two octal games for $$n\in \{0,\dots,9\}$$, where you can clearly see they are equivalent except for a shift of $$2$$.
$$n$$ 0 1 2 3 4 5 6 7 8 9
0.11337$$(n)$$ 0 1 1 1 2 2 0 3 3 1
0.007$$(n)$$ 0 0 0 1 1 1 2 2 0 3