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

Question

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.

Answer 1

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