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<title>MANGO (MAth Nerd GO)</title>
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<p>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 <b>MA</b>th <b>N</b>erd <b>Go</b></p>
<h2>Equipment</h2>
<p>To play Mango, you need the following:
</p>
<ul>
<li>A countable set, I.</li>
<li>A set, C, whose members are subsets of I, each of which contains exactly two members of I.
</li>
<li>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.</li>
<li>A real-valued function F, whose domain is I.</li>
<li>A function, T, whose range is {0,1}, that is defined on the real numbers.</li>
</ul>
<h2>Definitions</h2>
<p>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.</p>
<p> A finite subset, P, of C is called a PATH if the following
conditions are met:</p>
<ul>
<li>Each element of I has an index in P of 0, 1, or 2.</li>
<li>There are exactly two elements of I whose index in P is 1.</li>
</ul>
<p> 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.</p>
<p>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.</p>
<p> The ordered triple (E,B,W) is called the CONFIGURATION.</p>
<h2>Playing</h2>
<p>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.</p>
<p>The players must agree to an initial score for each player.</p>
<p>The players than decide who shall have the first turn.
Players alternate turns.</p>
<p>On a players turn, that player may do one of two things:</p>
<ul>
<li>The player may PASS.  It then becomes the other players turn.</li>
<li>The player may make a LEGAL MOVE.</li>
</ul>
<p>Note that a player *MUST* either pass or play a legal
move.  If there is no legal move, the player is forced
to pass.</p>
<p>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:</p>
<ol>
<li>A member, n, of E is selected.</li>
<li>n is removed from E and added to M.</li>
<li>All members of H that are not alive at the
end of the above step are removed from H and
placed in E.</li>
<li>All members of M that are not alive at the
end of the previous step are removed from M
and placed in E.</li>
</ol>
<p>A move is a LEGAL MOVE if the configuration, (E,B,W),
produced by the move is new.</p>
<p>The game ends when two consecutive turns are passes.</p>
<h2>Scoring</h2>
<p>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.</p>
<p>We use the symbol H to refer I-(E union M).</p>
<p>The player starts with the initial score agreed upon
at the start of the game.</p>
<p>For each m in M, the player receives F(m) points.</p>
<p>A player receives F(n) points for each member, n, of
E for which the following conditions both hold:</p>
<ol>
<li>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.</li>
<li>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.</li>
</ol>
<p>The player with the most points wins.</p>
<h2>Example</h2>
<p>To play ordinary 19x19 Go, with a 5.5 point Komi, the players
might agree to the following:</p>
<ul>
<li>I = { (x,y) | x and y are integers in [1,19] }</li>
<li>C = { {(x,y),(u,v)} | (x,y) and (u,v) are in I, (x-u)^2+(y-v)^2 = 1 }</li>
<li>B = W = {}</li>
<li>E = I</li>
<li>The initial scores are 0 for Black, 5.5 for White.</li>
<li>It is Blacks turn.</li>
<li>F((x,y)) = 1</li>
<li>T(z) = int((z+360)/361)</li>
</ul>
<p>To play a Go-like game on an infinite board, the playersmight agree to this:</p>
<ul>
<li>I = { (x,y) | x and y are integers }</li>
<li>C = { {(x,y),(u,v)} | (x,y) and (u,v) are in I, (x-u)^2+(y-v)^2 = 1 }</li>
<li>B = W = {}</li>
<li>E = I</li>
<li>Initial scores are Black:0, White:0.</li>
<li>It is Black to move.</li>
<li>F((x,y)) = exp(-x^2-y^2)</li>
<li>T(z) = 1 if z &gt; 1/1000, otherwise T(z) = 0</li>
</ul>
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