# Hex Go

## by David Kater

David teaches math at City College and has authored a book with his father.

Instructional Objective The learner will demonstrate logical thinking by strategic placement of stones in such a way as to capture the maximum possible territory.

Time Required A game may last from one to several hours.

Learner Description The game is most appropriate for adult learners.

Origins
This game is based on the game of Go that was developed in ancient China and adopted in Japan as the national pastime. Hex Go is played on a hexagonal grid rather than the traditional rectangular grid. The rules of play are identical to those of Go.

Object of the Game
To build boundaries of stones that surround the largest number of vacant intersections. Each vacant intersection represents one point. The player with the highest point count wins the game.

Rules One player receives Black stones, the other White stones. Black always makes the first play, and starts by placing one stone on any vacant intersection. White then plays a stone, and play alternates from that point. Once a stone is played, it is never moved unless it is captured.

The following scenarios describe the basic concepts of Hex Go:

Units of stones Units of stones are formed when two or more stones of the same color are at adjacent intersections connected by line segments. Each unit of stones is treated as a whole throughout game play. It becomes safe as a unit, or it is captured as a unit.

Two units of connected stones.

Liberty Liberty denotes any intersection that is vacant and adjacent to a stone or unit of stones. A stone of either color may be played on any liberty; however, the removal of a liberty by an opponent's stone, constitutes a threat. When a stone or a unit of stones is left with only one liberty, the stone is in danger of capture on the next move.

Liberties are illustrated by triangles. Notice that only adjacent intersections are identified as liberties.

Capture Capture occurs when a stone or unit of stones has no remaining liberty point. Such a stone or unit of stones is not allowed to remain on the board. A player is not allowed to place his stone into a position of capture unless he captures the opponent's stone(s). A stone with only one remaining liberty is in check (in danger of capture).

In area 1, when White plays at A-11, he captures Black's stone at A-10. In area 2, when Black plays at E-4, he captures White's three stones. In area 3, when White plays at H-9, he captures Black's stone at H-8. In area 4, when White plays at I-14, he captures Black's four stones. In area 5, White can not play into Q-14 because it has no liberty point. In area 6, White can play at O-5 to capture Black's three stones. A stone may be played into a momentary position of capture if it has a liberty at the end of the turn.

Repetitive capture In order to resolve repetitive capture situations, a stone which captures cannot then be captured on the next play by the opponent. The opponent must make at least one play elsewhere on the board first.

In area 1, after Black plays at A-8 to capture White's stone at A-9, White must play one turn elsewhere before attempting to recapture A-8 by playing at A-9. In area 2, after White plays at I-6 to capture Black's stone at H-6, Black must play one turn elsewhere before attempting to recapture I-6 by playing at H-6. Area 3 is similar to area 2. In area 4, after Black plays at G-13 to capture White's two stones, White must play one turn elsewhere before attempting to recapture G-13 by playing at H-14.

Eye formation An eye is one or more connected vacant intersections completely surrounded by stones of the same color. Note that an eye is not necessarily safe from capture.

In area 1, A-8 is a vacant eye. In area 2, the white stones are positioned as in area 1, but they are surrounded by black stones (i.e., in check). If Black plays at A-14, White's stones will be captured. In area 3, an eye of two connected intersections is surrounded by White stones. In area 4, Black's stones surround White stones arranged as in area 3. If Black plays at N-13, the White stones will be captured.

Safe territories A unit of stones with two separate eyes results in a safe territory because the opponent could never play into either eye without playing into a capture situation.

In area 1, Black would have to play at both A-2 and C-1 simultaneously to capture the White stones, and that is not allowed. In area 2, Black has established a safe territory (two separate eyes at A-14 and A-16). Areas 3, 4, and 5 each contain a safe unit of stones that cannot be captured. Note that minimal safe territories are radically different shapes than in traditional Go, but they generally require about the same numbers of stones.

Strategy Traditional heuristics for winning also apply to Hex Go:

* Attempt to secure territory in the corners first, then along the edges, then work toward the center.

* Resist the temptation to form complete safe territories at the beginning of the game. Rather, spread your stones across the board to form foundations for safe territories later.

* Be willing to lose territory or stones if you can secure a greater advantage elsewhere.

The geometry of the Hex Go board has some subtle but important effects on strategy. Since there are only three lines connected to an intersection rather than four, it is easier to force your opponent into a capture situation. So the player who initiates a forcing action has a greater advantage in Hex Go. However, keep in mind that your opponent may sacrifice stones in one area to gain a greater advantage in another. Also, since unconnected stones are easier to capture, safe territories are of greater importance.

Ending a Game As the game progresses, it becomes more difficult to acquire new territory. The game ends when both players agree that further play will not enable them to acquire new territory.

Scoring After the decision has been made to end the game, the board is checked for the following:

* Stones which are surrounded by an opponent's stones and are not connected to safe units of stones are removed by the opponent and counted as captured stones.

* Vacant intersections not surrounded by a continuous boundary of the same color stones are considered neutral. These intersections are filled in with unplayed stones of either color, and they do not count as points for either player.

* The player whose continuous safe boundary of stones is closest to the vacant intersections receives the points for those intersections.

* The final score for each player is the number of surrounded intersections minus the number of stones lost in capture.

Handicap play When two players have unequal skill levels, play can be adjusted to provide even competition by allowing the weaker player to place one or more handicap stones before the first move. The number of handicap stones is established by mutual agreement based on previous matches.

While handicap stones may be played at any position, they are traditionally placed at the locations shown to the right in the following order: D4, R16, D16, R4, D10, R10, H4, M16, H16, M4, H10, M10 If there is only a slight difference in skill levels, the weaker player may compensate by playing first in every game or in two out of every three games.