Hex

Hex begins with the board vacant of counters

Alternate Names

Con-Tac-Tix, Polygon, Nash, Join, John (as a pun of the name of one of the discoverers and also a slang name for bathrooms which are often filled with hexagonal tiles).

No. of Players

Two

Equipment

A Hex board and several each of black and white (here, blue and red) counters are required for play. The standard Hex board is a rhombus composed of eleven hexagons to a side. A rhombus of any size, however, can be used and children or amateurs may enjoy playing hex on boards with sides as small as three hexagons. The mathematician Nash believed that a Hex rhombus composed of fourteen hexagons to a side was best.

History

Hex was invented in 1942 by the Danish mathematician and poet Piet Hein while attending the Neils Bohr Institute for Theoretical Physics in Copenhagen, Denmark. Hein published the game in Denmark under the name of Polygon. It was independently discovered and studied extensively by the famous mathematician John Nash in 1948, then a graduate student at Princeton University. Hex is widely considered to be one of the most ingenuitive positional board games invented with a simplicity of rules balanced by a complexity of play unseen in any other game, short of Go. Hex was probably the first board game of its kind, and since its invention numerous games using the connection mechanism of Go have spawned including Y, Poly-Y, Star, and Havannah.

Objective

The objective is to be the first player to build an unbroken line of counters of one’s own color connecting the two opposite sides of the board assigned to that same color. A draw is impossible in this game.

Play

Alternate turns entail the placement of one friendly counter at any vacant cell on the board. Once played, the counters cannot move or be captured.

Strategy

Variations

A Hex board can be made from any size of a rhombus board with equal numbers of hexagon cells along each side as small as 3x3 hexagons per side.

By simply placing two rows of counters on the board one can create a smaller version of the game.

Some players prefer to play counters on intersections rather than inside of the cells. This is achieved by playing on a hex board composed of triangles as shown here.

Playing counters at the intersections on this board is equivalent to playing inside the cells on the board shown above. Other than the visual appearance, this does not change the dynamic of the game at all.

Other boards have also been utilized for similar connection games:

Octagons, published by R. Wayne Schmittberger in 1992, utilizes the following board and works great as a paper and pencil game. The rules are very similar to Hex. Using the Pie Rule and alternating turns, each player may use their respective color to fill in one half of an octagon or two squares. The first player to build an unbroken line of cells colored with their color that connects their two sides of the board is the winner.

Octagons Board

Pex, invented by David J. Bush and Marjorie Rice, is nearly identical to Hex, except that it's played on a rhombus-shaped board composed of irregular pentagons, rather than regular hexagons. Exactly half of the board's pentagonal cells are adjacent to seven other cells, while the other half each are adjacent to only five neighbors.

Pex Board

The Shannon Switching Game, invented by Claude Shannon, the "father of Information Theory", is more akin to a recreational mathematical exercise or a paper and pencil game than a positional board game but is, nonetheless, closely related to Hex and the family of Connection Games. In this game, two points are connected by a network of line segments. The two players are called cut and short. With alternating turns, "cut" may remove or erase one line segment while "short" colors one line segment that "cut" may not erase. When all of the lines have either been erased by "cut" or colored by "short" the game is over. "Short" wins if the two points are still connected whereas "cut' wins if they are not.

A very simple version of the Shannon Switching game begins

1. Cut removes one line segment

2. Short marks one line segment

3. Cut removes another

4. Short marks another

5. Cut removes the last line segment. This ends the game with cut as the winner.

A Shannon Switching Game of Greater Complexity

Gale, invented by David Gale and published by Hasbro in 1960, is also known as Bridg-It, The Game of Gale, or Birdcage. The board is a set of two grids of differently colored dots overlaid at an offset. With alternating turns, players draw links of their own color between two orthogonally adjacent dots of their own color, attempting to join their two sides, as in Hex.

Gale begins with a vacant board.

The game ends with blue as the winner.

Twixt is a connection game invented by Alexander Randolph. It was marketed by 3M and popular in the 1960's and 70's. It is played on a grid of 24x24 holes with the four corners missing. Two opposite sides of the board are colored with one player's color while the other two have the opponent's color, as in Hex. Each player commences with fifty pegs and fifty links of a distinctive color in hand. Using the Pie Rule and alternating turns, the two players place a peg of their respective color in any vacant hole on the board with the exception of the two rows (or columns) that border the opponent's colored side. If possible, the player may also add or rearrange their links on the board. A player's links may join any two of their pegs at opposite corners of a six-hole rectangle (3x2 or 2x3). Links, however, may not cross previously placed links. Multiple links may be added in a single move and friendly links previously placed may be rearranged on any turn, so long as they are legal by the aforementioned rules. The first player to connect his or her two sides of the board is the winner. Unlike, most connection games, draws are possible in this game.

Chameleon was discovered by Cameron Browne and published in 2005 in his Connection Games: Variations on a Theme. It was independently discovered by Randy Cox and Bill Taylor. The game is played with any board and counters used for Hex but gives players the option of placing a counter of either color on the board on their turn. A player wins the game by connecting their two sides with counters of either color, i.e. the player attempting to connect the two blue sides will win if they are connected by red counters. If a counter is placed that simultaneously creates a bridge between both player's edges, the winner is the player who placed the final counter.

Nex, invented by Joao Pedro in 2004, is played with any board and counters used for Hex. It does, however, require, an additional set of counters of different color than the other two which are considered neutral counters. It is played identically to Hex with one exception: on their turn, a player has one of two options:

  • He may place a counter of his own color and a neutral stone at any two vacant positions on the board or...

  • He may replace two previously placed neutral counters with counters of his own color, and exchange a different counter of his own color on the board to a neutral counter.

Hecks is a connection game that can be played online at the Wolfram Demonstrations Project, http://demonstrations.wolfram.com/HecksBoards/. It is similar to Hex, but here the hexagonal tiles are replaced by randomly generated irregular polygons. With alternate turns, White and Black fill a tile with the objective of creating a connecting bridge between opposite sides of the board, which need not be declared in advance.

Snapshots of the Randomly Generated Hecks Game

Join, described by Craige Schensted and Charles Titus in their book Mudcrack Y and Poly-Y, could be said to be the simple recognition that you can put a border around any board composed of adjacent cells, divide the border into four segments, and color the segments alternately with two opposing colors to create a connection game. Alternate turns to fill the cells with two different colors, and when the board is filled in completely, the two border pieces of one of the two colors will be joined, and the two border pieces of the other color will not. Note that a division between any two border segments always falls on a definite area region of the board and never on the line separating two adjacent cells.

Join Boards

Poly Join is an extension of Join. Here, the border is divided into eight, twelve, or some larger multiple of four segments. When a Poly Join board is filled with the two colors, the border segments will be joined into groups. The total number of groups will be more than half the original number of border segments. Since the number of border segments is a multiple of four, the number of groups will be odd, and thus one player will connect his border pieces into fewer groups than the other player. The player who joins his border pieces into the smallest number of groups wins. Diagram Needed?

Poly Join Boards

see pp 150, 180-181 of Mudcrack Y and Poly-Y (9 boards total)

Hex was probably the first connection style board game, and several great games evolved from it. These include Y, Poly-Y, Star, *Star and many others.

Sources

  1. Schensted, Craige and Charles Titus. Mudcrack Y & Poly-Y. NEO Press, 1975. ISBN 0-911014-23-3

  2. Browne, Cameron. Connection Games: Variations on a Theme. A K Peters/CRC Press, 2005. ISBN 978-1568812243

  3. Wolfram Demonstrations Project. http://demonstrations.wolfram.com/HecksBoards/

  4. Schmittberger, R. Wayne. New Rules for Classic Games. John Wiley & Sons, Inc. New York, 1992. ISBN 0-471-53621-0