FEN

"Geometry is the archetype of the beauty of the world." Johannes Kepler

FEN is a two player, area majority/influence game played on polygon tessellations or graphs and is easily played using paper and pencil. The game has no luck elements but feels like a push-your-luck game. FEN is similar in appearance to Dots and Boxes or ConHex but play is deeper.

FEN is based on the observation that all closed polygons have an odd number of components: one face (interior) and an equal number of edges (line segments) and nodes (points). See the Cell Components figures on the right. Edges and nodes can be shared by adjacent polygons but faces are unique to each polygon. Instead of using just edges to surround a polygon, control of half of the edges and/or nodes of the polygon are required to capture a polygon face. On a turn, players must choose to claim one node, or a fixed number of edges dependent on the graph structure, or conditionally, one or more faces!

Rules for playing on a triangular board are presented here with other board shapes discussed later.

Triangle Cell ComponentsTriangles have 7 components: 1 face, 3 edges and 3 nodes.

Square Cell ComponentsSquares have 9 components: 1 face, 4 edges and 4 nodes.

Two-player Graph

Setup

Create or select a size-7 triangle graph or game board.

Recommended two player game board. Players will be able to place 1 node tile or up to 3 edge tiles. (49 triangle faces)

Many graphs are possible, for example, a limping hex board size 2 by 3 for a shorter two players.(37 triangle faces)

Another two player game board. Players will be able to place 1 node tile or 2 edge tiles. (49 square faces)

Game Components

To play FEN using a game board

A triangular board of size 7
Adequate numbers of face, edge, and node tiles in two colors, or

FEN can also be played with paper, pencil, and colored pens. Click here to download FEN Triangle and Square Printout.png, and print the file.

Concepts

Face - The interior of a polygon.
Edge - One of the line segments enclosing the interior of the polygon.
Node - A point (represented by a circle) where two or more edges meet.
Majority control of any polygon is achieved by claiming half of its nodes and edges and then claiming the face of the polygon.

Game Play

Players choose colors and decide who goes first. Turns alternate. No passing.

On a turn, a player may perform one of the following actions. Tiles may only be placed on empty cells.

Claim one node cell by placing a tile on it.
Claim three edges by placing tiles on them, such that no two of these edges belong to the same triangle. See figure Legal and Illegal Placements below.
Capture one or more triangles by placing face tiles provided the player has previously claimed any combination of at least three nodes and/or edges on each of the triangles.

The pie rule can be applied. On the second player's first turn, the player can choose to swap colors in lieu of placing any influence tiles.

Game End

The game ends when one player claims ownship of a majority of the triangles i.e., 25 triangles. This player wins the game.

Legal and Illegal PlacementsPlacement restrictions only apply to pairs of edges when they are placed. Edges placed on earlier turns are ignored.

Edges placed during a turn can share nodes but not faces.

Board Variations

Finite tesselations

Two-player FEN can be played on virtually any finite tessellation which is created using an odd number of polygons: regular, semi-regular, and custom tilings. It is expected that some tessellations will work better than others. Triangles were chosen for the two player game because the least number ot node and edge components are needed to aquire a majority. The following caculations can be used to determine how many nodes and edges per turn a player may place.

After creating a new board containing a odd number of polygons, the number of nodes (n) and edges (e) playable per turn must be determined based on the structiure of the board.

Examine all the nodes and determine the number of faces they are associated with. Find the count that occurs the most (the mode, N). For example, for a regular hex board, N = 3.
Examine all the edges and determine the number of faces they are associated with. Find the count that occurs the most (the mode, E, usually 2). For example, for a regular hex board, E = 2.
Find the smallest number divisible by both N and E. (the least common multiple, LCM). For a hex board, the smallest number is 6 which is divisible by N=3 and E=2.
The number of nodes per turn (n) will be LCM / N, for a regular hex board, n = 6 / 3 = 2.
The number of edges per turn (e) will be LCM / E, for a regular hex board, e = 6 / 2 = 3.

With n and e in hand, the game can begin. The rules are as presented above with one addendum. When placing multiple nodes (like edges), no two may be applied to the same polygon.

Please note that for now these game rules may not be duplicated and distributed via the web. All rights are reserved. Those that wish to program or sell this game in any form should contact the author at philleduc.pled@gmail.com for permission or a license to do so.

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