Connection Games: Hex, Lines of Action and Tak
Winning here is judged topologically: not by counting, but by whether the shape closed.
A game that is also a theorem
Hex was invented by the Danish polymath Piet Hein in 1942, and independently reinvented by John Nash at Princeton in 1948 — for a while it was simply called Nash. The rules could hardly be simpler: place stones in turn, and win by linking your two opposite edges with an unbroken chain.
But it carries a property too elegant to look like a game rule: when the board fills, exactly one player must be connected, so a draw is impossible. That is not a design decision but a provable topological fact. Because draws cannot happen, Nash used a strategy-stealing argument to prove the first player wins — although no explicit winning strategy is known for large boards. A game that doubles as a mathematical object is extremely rare.
Two other ways to connect
Lines of Action (1969, designed by Claude Soucie and introduced by Sid Sackson in A Gamut of Games) inverts the idea: edges are irrelevant, and you must gather every remaining piece of your own into a single connected group, counting all eight directions. Its movement rule is beautifully economical — a piece moves exactly as far as the total number of pieces on its line. Gathering therefore changes your own mobility as you do it: the more crowded the line, the further you travel.
Tak (2016, James Ernest and Patrick Rothfuss) connects a road: place flats and capstones, move stacks, and link either the left and right edges or the top and bottom. Its origin is unusual — the game first appeared as a fictional pastime in Rothfuss's novel The Wise Man's Fear, and readers wanted to play it, so a designer was brought in to make it real.
The 3 games in this family

Hex
Europe · Early modern
Create an unbroken path between your two sides of a board that can never end in a draw.

Lines of Action
The Americas · Contemporary
Move by the number of pieces on a line and connect your entire army first.

Tak
The Americas · Contemporary
Stack pieces, carry piles, and complete a road across the board.