ConHex

ConHex Rules

• The board has 69 inner circles (clickable) and 41 surrounding regions.
• Players take turns claiming one empty circle by marking it with their colour.
• When a player has claimed at least half of the circles around a region, they take ownership of that region. The center pentagon (5 circles) needs 3; corner triangles (3 circles) need 2; the larger 6-circle regions need 3.
• Red connects the top and bottom sides through owned regions.
• Blue connects the left and right sides.
• Once a region is owned it stays that colour for the rest of the game.
• The first player to form an unbroken chain of owned regions between their two sides wins.

A bit of history

ConHex was invented by the Russian mathematician Michail Antonow in 2002. It belongs to the family of connection games — alongside Hex (Piet Hein, John Nash), Y, Havannah and TwixT — but it brings a key twist: instead of directly claiming territory cells, players claim the vertices (circles) shared between regions, and ownership of each region is decided by majority. This produces dynamic positions where a single circle can simultaneously contribute to several regions.

Mathematically the game keeps the same property as Hex: draws are impossible (a topological argument based on the dual planar graph) and the first player has a theoretical winning strategy by the strategy-stealing argument. Since the regions overlap fewer pieces than Hex tiles, ConHex rewards careful planning of where each circle's shared contribution lands more than raw connection power.