This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a conjecture. Now, after several years of mathematical training at a university, I am still unable to settle the conjecture, so I present it to you.
The classical tic-tac-toe game is played on a $3\times3$ grid and two players take turns to put their mark somewhere in the grid. The first one to get three collinear marks wins. Collinear includes horizontal, vertical and diagonal lines. Experience shows that the game always ends in a draw if both players play wisely.
Let us write the grid size $3\times3$ as $3^2$. We can change the edge length by playing on any $a^2$ grid (where each player tries to get $a$ marks in a row on the $a\times a$ grid). We can also change dimension by playing on any $a^d$ grid, for example $3^3=3\times3\times3$. I want to understand something about this game for general $a$ and $d$. Let me repeat: The goal is to make $a$ collinear marks.
I assume both players play in an optimal way. It is quite easy to see that the first player wins on a $2^d$ grid for any $d\geq2$ but the game is a tie on $2^1$. The game is a tie also on $3^1$ and $3^2$, but my experience suggests that the first player wins on $3^3$ but the game ties on $4^d$ for $d\leq3$. It seems quite credible that if there is a winning strategy on $a^d$, there is one also on $a^{d'}$ for any $d'\geq d$, since more dimensions to move in gives more room for winning rows. This answer to a related question tells that for any $a$ there is $d$ so that there is a winning strategy on $a^d$.
This brings me to the conjecture:
There is a winning strategy for tic-tac-toe on an $a^d$ grid if and only if $d\geq a$.(Refuted by TonyK's answer below.)
Is there a characterization of the cases where a winning strategy exists? It turns out not to be as simple as I thought.
To fix notation, let $$ \delta(a)=\min\{d;\text{first player wins on }a^d\} $$ and $$ \alpha(d)=\max\{a;\text{first player wins on }a^d\}. $$ The main question is:
Is there an explicit expression for either of these functions? Or decent bounds? Partial answers are also welcome.
Note that the second player never wins, as was discussed in this earlier post.
A remark for the algebraically-minded: We can also allow the lines of marks to continue at the opposite face when they exit the grid; this amounts to giving the grid a torus-like structure. Now there are no special points, unlike in the usual case with boundaries. Collinear points on a toric grid of size $a^d$ corresponds to a line (maximal collinear set) in the module $(\mathbb Z/a\mathbb Z)^d$. (If $a$ is odd, then $a$ collinear points in the mentioned module add up to zero, but the converse does not always hold: the nine points in $(\mathbb Z/9\mathbb Z)^3$ with multiples of three as all coordinates add up to zero but are not collinear.) This approach might be more useful when $a$ is a prime and the module becomes a vector space. Anyway, if this version of the game seems more manageable, I'm happy with answers about it as well (although the conjecture as stated is not true in this setting; the first player wins on $3^2$).
