Hex

Hex is a two-player connection game played on a rhombus-shaped board of hexagonal cells. One player uses black stones and aims to connect the top and bottom sides of the board; the other uses white stones and aims to connect the left and right sides. Players take turns placing one stone on any empty cell. The first player to form an unbroken chain of their stones connecting their two sides wins. A key property of Hex is that draws are impossible — one player must always win.

Rules

1. The board is a rhombus (diamond shape) made of hexagonal cells, typically 11x11 but commonly played on sizes from 7x7 to 19x19.
2. Each pair of opposite sides is assigned to a player — one player connects top-to-bottom, the other connects left-to-right.
3. Players alternate turns, placing one stone on any empty cell.
4. The first player to complete an unbroken path of their stones from one of their sides to the other wins.
5. Stones are never moved or captured once placed.
6. There are no draws — the board topology guarantees exactly one player wins.

Swap Rule

To compensate for the first player's advantage, many tournaments use the swap rule: after the first player places their opening stone, the second player may choose to swap colors (claiming that stone and the first-move position) or continue as normal.

Strategies

• Center Control: The center cells are the most powerful because they can contribute to paths in all directions. An early center stone is strong.
• Bridge: Two stones separated by one empty cell, where you can connect through either of two hexagons, form a "bridge" — a virtual connection that your opponent cannot break.
• Ladder (Edge Template): A sequence of forcing moves along the edge that guarantees a connection. Recognizing and executing ladders is a fundamental skill.
• Template Patterns: Certain configurations of stones near the board edge guarantee a connection to that edge. Memorizing these templates is crucial for efficient play.
• Blocking Strategy: Sometimes the best defense is an active one — place stones that both block your opponent and advance your own connection.
• Reading the Board: Unlike Go or Chess, there are no captures in Hex. Every stone placed permanently affects the position, making board reading cumulative.

Hex was independently invented by two mathematicians. The Danish mathematician Piet Hein invented the game in 1942 and introduced it at the Niels Bohr Institute. Hein called the game "Polygon" and published it in the Danish newspaper Politiken, where it became very popular. He designed it as a game that elegantly embodied a mathematical theorem about connectivity.

Independently, the American mathematician John Nash invented the same game in 1948 while a graduate student at Princeton University. Nash proved that the first player has a winning strategy using an elegant non-constructive proof: if the second player had a winning strategy, the first player could "steal" it by making an irrelevant first move and then following the second player's strategy (since an extra stone can never hurt in Hex). This "strategy-stealing argument" proves a winning strategy exists without revealing what it is, and it remains one of the most beautiful proofs in combinatorial game theory.

The game was marketed as "Hex" by Parker Brothers in 1952, and this name has stuck. Despite Nash's proof that the first player can always win, no practical winning strategy is known for boards larger than about 9x9 — the game tree is too vast for brute-force computation on larger boards. This paradox (we know the first player wins, but we don't know how) makes Hex a fascinating subject in mathematics and computer science.

Hex has a devoted following in the abstract strategy game community. The game has been studied extensively in combinatorial game theory and artificial intelligence. Programs using Monte Carlo tree search and deep learning have achieved very strong play. Hex is closely related to many mathematical concepts, including the Brouwer fixed-point theorem and the Jordan curve theorem, and it remains an active area of research in mathematics.