Here is an attempt to describe the rules of Go precisely. This is actually for a generalization of Go that I call MANGO, which stands for MAth Nerd Go

## Equipment

To play Mango, you need the following:

• A countable set, I.
• A set, C, whose members are subsets of I, each of which contains exactly two members of I.
• Three subsets of I, called E, B, and W, such that their union is I, and the intersections of any pair of them is empty.
• A real-valued function F, whose domain is I.
• A function, T, whose range is {0,1}, that is defined on the real numbers.

## Definitions

If P is a finite subset of C, and n is a member of I, then the INDEX of n in P is the number of elements of P that contain n.

A finite subset, P, of C is called a PATH if the following conditions are met:

• Each element of I has an index in P of 0, 1, or 2.
• There are exactly two elements of I whose index in P is 1.

An element of I whose index in a path is 1 is called an ENDPOINT of the path. An element of I whose index in a path is 2 is called an INTERIOR point of the path.

Let S be one of the sets B or W. Let s be a member of S. Let L be the set of all members, e, of E, such that there is a path whose endpoints are s and e, and whose interior points are all in S or E. Let z be the sum over L of F. Then s is ALIVE if T(z) = 1.

The ordered triple (E,B,W) is called the CONFIGURATION.

## Playing

The players must first obtain a Mango set. This consists of agreeing to the sets I, C, E, B, and W, and the functions F and T.

The players must agree to an initial score for each player.

The players than decide who shall have the first turn. Players alternate turns.

On a players turn, that player may do one of two things:

• The player may PASS. It then becomes the other players turn.
• The player may make a LEGAL MOVE.

Note that a player *MUST* either pass or play a legal move. If there is no legal move, the player is forced to pass.

A MOVE consists of performing several actions. In the following, if it is Black's turn we will use the symbol M to refer to the set B and the symbol H to refer to the set set W. If it is White's turn, M will be W and H will be B. Here are the actions that are taken by a player on that players turn:

1. A member, n, of E is selected.
2. n is removed from E and added to M.
3. All members of H that are not alive at the end of the above step are removed from H and placed in E.
4. All members of M that are not alive at the end of the previous step are removed from M and placed in E.

A move is a LEGAL MOVE if the configuration, (E,B,W), produced by the move is new.

The game ends when two consecutive turns are passes.

## Scoring

Each player uses the following procedure to compute his score. We will use the symbol M to refer to B if the player is Black, and to refer to W if the player is White.

We use the symbol H to refer I-(E union M).

The player starts with the initial score agreed upon at the start of the game.

For each m in M, the player receives F(m) points.

A player receives F(n) points for each member, n, of E for which the following conditions both hold:

1. There exists a path with n as one endpoint and the other endpoint in M, and which contains no members of H as interior points.
2. All paths that contain n as one endpoint and a member of H as the other endpoint contain a member of M as an interior point.

The player with the most points wins.

## Example

To play ordinary 19x19 Go, with a 5.5 point Komi, the players might agree to the following:

• I = { (x,y) | x and y are integers in [1,19] }
• C = { {(x,y),(u,v)} | (x,y) and (u,v) are in I, (x-u)^2+(y-v)^2 = 1 }
• B = W = {}
• E = I
• The initial scores are 0 for Black, 5.5 for White.
• It is Blacks turn.
• F((x,y)) = 1
• T(z) = int((z+360)/361)

To play a Go-like game on an infinite board, the playersmight agree to this:

• I = { (x,y) | x and y are integers }
• C = { {(x,y),(u,v)} | (x,y) and (u,v) are in I, (x-u)^2+(y-v)^2 = 1 }
• B = W = {}
• E = I
• Initial scores are Black:0, White:0.
• It is Black to move.
• F((x,y)) = exp(-x^2-y^2)
• T(z) = 1 if z > 1/1000, otherwise T(z) = 0