Territory: by Ing ChangKi
The definition of territory determines the scoring of a Go game. Various
counting methods used nowadays include the Japanese method (JM) of comparing
vacant points, the stone-counting methods (SCM) of the Ming Dynasty (1368-1644
A.D.), and the most recent fill-in-to-count (FITC) method. Due to the differences
in the definition of territory, the results obtained by these methods are
slightly different. The so-called Japanese method of comparing vacant points
is currently the most popular method being used. It was introduced to Europe
and America by the Japanese. Players who were not familiar with the history
of Go misunderstood that this method was invented by the Japanese and named
it the 'Japanese way of counting' . However, according to the research of
Professor Yang Lien-sheng of Harvard University, the JM was actually invented
by the Chinese, and was widely practiced from the Han Dynasty (206 B.C.-220
A.D.) through the early period of the Ming Dynasty. (As to what counting
method was used before the Han Dynasty, there is no written record.) approximately
during the Nan-Pei Dynasty (the Southern and the Northern Kingdoms, 420-589
A.D.), this counting method was first introduced into Japan. For more than
1500 years these scoring rules remained unchanged. On the other hand, the
Chinese had slowly switched to the stone-counting method after the Ming
Dynasty. They had totally abandoned the method of comparing vacant points
for about 600 years. The reason the Chinese had modified their counting
method will be discussed later. The definition of the JM is based on the
concept of 'stone efficiency'. Therefore, it only considers vacant points
surrounded by a live group of stones as territory. The stones and the vacant
points within a seki are not counted. Such reasoning stemmed from the fact
that the 'purpose' of the game is to secure territory (or vacant points).
the moves (or stones) are merely 'intruments' used to accomplish this 'purpose'.
Threefore, one should only count the vacant points surrounded by the stones,
but not the stones themselves. (By replacing the captured stones into the
vacant points of the same color during counting, the number of black and
white stones are considered to be approximately even. Thererfore, there
is no need to count the stones.) As mentioned before, the vacant points
associated with a seki are also not counted as territory. the majority of
go players don't understand the reason behind this. According to the Japanese,
there are the following reasons:
1) seki is rare (who cares if the counting is not exact?);
2) even with the presence of seki, the number of vacant points associated
(for both sides) are approximately the same;
3) seki is complicated and sometimes it is not clear who owns which vacant
points. Since everything is an approximation to start with, why count the
vacant points? For the sake of simplicity, one can apply this method, but
it is far from precise. If one analyses this kind of counting scientifically,
there are two main defects:
a) Theorectical Defect. Mathematically, it is not correct to
assume that the number of stones (including captured stones) plus the vacant
points associated with a seki are the same (for both sides). thus, they
should cancel out. If one argues that the stones are not counted, because
they are not considered territory, then why doesn't one count the vacant
points of a seki?
b) Practical Defect. Since stones are not counted, the actual territory
becomes less if one places a stone in his own vacant space. During games,
numerous disputes can arise because of this (e.g., a player refuses to eliminate
a ko in his own territory). the only solution is to make various unreasonable
adjudications for various situations (e.g., bent four in the corner is dead
and not related to the game, etc.). In the two games played by Go Seigen
against Iwamoto Kaoru and Takagawa Kaku, conflicting adjudications were
given.
This kind of defect is due to a lack of understanding of the definition
of territory. The stone-counting method of the Ming Dynasty is based on
the concept of the 'survivability of the stones'. except for the two basic
eyes and shared spaces (of a seki), all stones and vacant points are counted
as territory. the purpose of the game (according to this method) is to make
as many stones survive on the board as possible (without killing your own
groups). therefore, the stones of a living group are automatically counted
as territory. Besides the two basic eyes (for each living group) and the
shared spaces (of a seki), one can fill all the rest of the vacant oints
with stones, so they are also counted as territory. the stone-counting method
of the Ming Dynasty and the method of comparing vacant points are based
on two completely oposite points of view.
Let's summarize the advantages and disadvatages of the SCM:
1) Advantages. By counting stones instead of vacant points, one does not
lose any points by playing within one's own territory. Without such disputes,
there is no need for adjudications. This was the main reason why the Chinese
switched from the JM to the SCM.
2) disadvantages. Not counting eyes and shared spaces, SCM required a 'tax
on groups': the players with more groups has to give up one stone (for the
two eyes not counted) for every excess group. this method mixes the life
and death of stones with the counting of territory. During the progress
of the game, not placing stone(s) into one's basic eyes or shared spaces
of a seki is understandable, but for the sake of counting there should be
no problem. The eyes and the shared spaces are part of the living groups
and should be scored as territory. The fill-in-to-count method is based
on the concept of 'the division of the board's interesections'.
There are 361 intersections on the board. the purpose of the game is to
capture as many intersections as posssible. Any point out of the 361 should
belong to either White or Black. In other words, the sum of the terrritories
must equal 361 points, and the difference between the territories must equal
the final score. this is simple and straightforward. With 180 each of Black
and White stones, there is one vacant point left on the board after all
the stones are filled in. The only vacant point left should lie in the winner's
territory. the FITC method is simpler and more accurate than the JM. Unlike
the SCM, the FITC method preserves the board position. the SCM generally
destroys the position totally, and is not easy to double check results in
case of disputes.
The Japanese rules theorist Kaise Takaaki stressed that one should retain
the JM because of its simplicity and beauty (neatness of the final board
position). Indeed, simplicity, beauty and preciseness are the crucial requirements
for any counting system. The defiinitions of the three counting methods
discussed in this article can be represented by the following equations,
with T1, T2 and T3 being the scores of the JM, SCM and FITC respectively.
T1 = P-(S + d) T2 = P - Q T3 = P where P is the sum of the mumbers of stones
and vacant points, S is the number of stones, d is the number of vacant
points associated with a seki, and Q is the number of eyes and shared spaces.
Note that from the above equations T3 - T1 = S + d and T3 - T2 = Q. The
difference in score between FITC and JM is the number of stones plus the
vacant points of a seki. the difference in score between FITC and SCM is
the number of eyes and shared spaces.
Finally, I should like to summarize the three counting methods discussed
in this article togehter with an illustration of the following diagrams:
1) the JM Method of Comparing Vacant Points. This method counts the vacant
points within the living groups s territory. the vacant points of a seki
are not considered as territory. the number of stones on both sides are
assumed to be equal and thus not counted.
2) the SCM of the Ming Dynasty. This method counts both the stones and vacant
points as territory , except the two basic eyes required for each living
group and the shared spaces.
3) FITC Method. This methodd considers all the stones and vacant points
as territory. the sum of territory for both sides is equal to 361 points.