From: ts@cup.portal.com (Tim W Smith)
Newsgroups: rec.games.go
Subject: Re: alternative goban topologies? (was: Go vs other board games
Date: Sun, 30 Aug 92 20:53:15 PDT
Organization: The Portal System (TM)
  <37246@sdcc12.ucsd.edu>

> >   Also, I don't think any person in the world can invent a better board game
> >   than go using such simple playing equipment and rules.  A better game
> >   would have to be in the third group with may be different types of pieces.
> 
> Or maybe a different type of board... has anybody tried playing go on a
> board with hexagonal or triangular grid, where stones could have at most
> three (or six, respectively) liberties?  And how about a board with no

Here's my version of go.  It uses simpler equipment and rules than the
standard game.  Furthermore, it includes many of the generalizations
commonly suggested.  It is called Mango.  I don't know why, yet (I'm
still working on what that is an acronym for).

---------- Cut Here For Official Rules Of Mango ----------

Equipment:

	To play Mango, you need the following:

		1. A countable set, I.

		2. A set, C, whose members are subsets of I,
		each of which contains exactly two members of I.

		3. 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.

		4. A real-valued function F, whose domain is I.

		5. 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:

		1. The player may PASS.  It then becomes the
		other players turn.

		2. 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 players
	might 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

--Tim Smith