Most reasonably bright kids tire quickly of tic-tac-toe, but there is a version that is far more complicated: the (m,n,k)-game is played on a m x n rectangular board. Two players alternate in placing stones of opposite color; the first to get k stones in a row (vertically, horizontally, or diagonally) wins. Thus, tic tac toe is the (3,3,3)-game.
There are a gazillion scientific papers addressing the question of who wins the (m,n,k)-game for different values or groups of values of m,n and k. It is well known that the second player cannot win; this is called the strategy stealing argument
and is used all over the place in combinatorial game theory. (If the second player has a winning strategy, then the first player can make an arbitrary first move and then mirror the second player's strategy.) So the game is either a first player win or a draw.
There is one other fact that these papers will assume is known to the reader or will mention in passing: if the (m,n,k)-game is a (first player) win, then for every m' ≥ m and n' ≥ n, the (m',n',k)-game is also a win. Which is pretty obvious - whatever winning strategy the first player has on the smaller board also carries over to the larger board. Right?Wrong.
Where the above argument falls down is that if the first player blindly tries to carry over the original strategy to the larger board, then the second player may have a win, said win having been impossible on the smaller board because the winning move would not have been within bounds.
So is there some other way to transfer a winning strategy onto a larger board? I haven't been able to find any such way, and as for the people who claimed the extension to be obvious, they have no answer when presented with the counterargument.
Even worse, some authors (example
) also assume that if the (m,n,k)-game is a win, then so is the (m,n,k')-game for k' < k. As far as I can see, there is no known proof for this assertion either. (I will call the first statement conjecture 1 and this one conjecture 2.)
The whole situation shows the scientific peer review process at its worst. How did legions of presumably very smart researchers miss these obvious holes in the argument? Someone must have originally made the mistake, and everyone else assumed that the first guy did his homework. I'm sure some people noticed the hole, but were too timid to speak out, for fear that they might have missed something and look like a moron. It takes a child, an outsider, to point out that the emperor has no clothes.
Notice that I am not saying that either conjecture is false - I'm just saying that the commonly assumed "proofs" are flawed. In fact, I think it highly likely that both conjectures are true. It is highly improbable that every single first player winning strategy can be defeated in the manner outlined above, although it is theoretically possible.
Further, in the case of conjecture 1, we can prove a related but weaker statement: if the first player has a winning strategy in the (∞, ∞, k)-game with the property that there exists a mxn rectangle such that no matter how the second player plays, the first player never makes a move outside this rectangle, then the (m',n',k)-game is a win for all m' ≥ m and n' ≥ n.
Here the strategy transfer goes through with no problem.
Apart from conjectures 1 and 2, I believe there are two more questions that remain open:
Conjecture 3: if the (∞, ∞, k)-game is a draw, then is every (m,n,k)-game also necessarily a draw?
Conjecture 4: if the (∞, ∞, k)-game is a win, is the (m,n,k)-game is a win for all sufficiently large m and n?
I'm still researching the existing literature. Maybe I'll find out that everything I said is crap :)