Strong connection

From HexWiki
Jump to: navigation, search

A strong connection or virtual connection between two (or more) pieces consists of a set of cells that are either empty or occupied by pieces of the same color (the "carrier" of the virtual connection), such that whatever moves the opponent makes in the carrier, it is always possible for the player to respond in the carrier in such a way that the pieces remain connected.

Virtual connections are not always equivalent to actual connections. In order to keep their connection, the player has to answer the opponent's threats. The opponent can use this to their advantage, for example by stealing territory. Playing in the carrier of an opponent's virtual connection is called intruding on the virtual connection.

Every chain is trivially a virtual connection. Virtual connections are often formed by connecting several chains via templates and double threats. A set of virtually connected pieces is also called a group.

Examples

The simplest and most common example of a virtual connection is the bridge.

AB

If Blue plays in one of the empty hexes, Red can respond in the other one. Therefore, Red can guarantee that pieces A and B stay connected.

More complex virtual connections can be formed by a combination of templates and double threats. For example, consider the following position:

As shown in the next diagram, Blue's stones A and B are connected to the right edge by edge template II and a ziggurat, respectively. C is connected to either A or B by double threat at x and y. C is connected to D, E, and F by a crescent, bridge, and trapezoid, and finally F is connected to the left edge by another ziggurat. Therefore, Blue has a virtual connection between their two edges, and has won the game. The carrier of the virtual connection is shown in gray.

AFxCEyBD

Red can either resign, or continue playing in the carrier and forcing Blue to defend the virtual connection. If Red plays outside of the carrier, Blue could just ignore Red's move and move anywhere or even pass.