A General Definition of Ko

  Difficulty: Advanced   Keywords: Ko, Theory

This page summarises the general definition of ko invented by Robert Jasiek in his paper [ext] Ko and relies on several of his terms.

A ko consists of connected intersections that are related to each other by one or several cycles having specific characteristics.

Table of contents

Types of Ko Intersections

The intersections of a ko are its ko intersections. A ko is defined by its ko intersections. There are three types of ko intersections:

  • basic-ko-intersection
  • local-ko-intersection
  • global-ko-intersection

A ko intersection can be of one, two or three of these types.


basic ko  

A basic-ko-intersection is one of two adjacent intersections on which a cycle of two successive board-plays could occur if rules did not prohibit it.

The diagram shows an example, with the ko intersections marked.

Common Conditions for Local-ko-intersection and Global-ko-intersection

Local-ko-intersection and global-ko-intersection are defined by presuming the Default Restriction Rules. An intersection that shall be a ko intersection is related to one or several cycles (i.e. a set of cycles) by these criteria:

  • each of the cycles starts from the same position (local-ko-intersection: some particular position; global-ko-intersection: the current position)
  • each of the cycles has at least one play creating the current position (i.e. the analysed position)
  • each of the cycles has a play that is on the intersection

A formal version of the definitions can be found in Robert Jasiek's paper [ext] Ko.


  • A player can answer-force one of the cycles by moving first in it if the opponent does prevent area improvement of the player exactly on the cycle's intersections.


  • The komi is given.
  • The player having the turn is given.
  • The player can answer-force one of the cycles if the opponent does prevent the player's win.

Examples of Kos

Kos Having Basic-ko-intersections

The basic-ko-intersections are marked.  

All basic-kos currently on the board as such consist of basic-ko-intersections.

Kos Having Local-ko-intersections

The local-ko-intersections are marked.  

There are two kos and each consists of four connected intersections. The combination of the two kos as a combined shape is also known under the name round robin ko

The local-ko-intersections are marked.  

The squared intersections are local-ko-intersections only. The circled intersections are both basic-ko-intersections and local-ko-intersections. At any time, a triple ko shape has only two basic-kos but altogether consists of three kos.

Kos Having Global-ko-intersections

Komi = 14.5. Black to move. The global-ko-intersections are marked.  

The squared intersections are global-ko-intersections only. The circled intersections are both basic-ko-intersections and global-ko-intersections. So far only one game has been reported in that purely global-ko-intersections occurred.

Historical Context

Relatively little about non-trivial kos is known before the 20th century. Most of what is known was related to rules disputes or other strange historical incidents (like Nobunaga's alleged triple ko). Apparently only a few long cycle ko shapes had occurred or were otherwise known at that time: triple ko, eternal life, moonshine life.

Things changed a bit during the first half of the 20th century when rules researchers mainly in the USA or in Japan studied a few more shapes. E.g., the Japanese 1949 Rules listed some incl. round-robin ko.

The next phase of discoveries was from the late 60ies to the 80ies. Then prominent researchers in rules and ko were Ing Chang-ki and Ikeda Toshio. Especially Ing took matters very seriously, devised several revisions of Ing Ko Rules, collected shapes sent to him from throughout the world and discussed them in Ing rules booklets and related pamphlets. E.g., triple round-robin ko and triple ko stones (which, according to rumours had been discovered by Matti Siivola) could be seen. Although Ing tried very hard and spent much of his life in an understanding of rules and ko, his so called definitions were heavily flawed and incomplete. However, they were also a major breakthrough for the next generation of researchers.

A note in between: Kos given due to cycles with four or more plays are rare to extremely rare. Therefore most of the related shapes did not occur in games and not even in famous tsumego problem collections but rather were discovered by rules experts because they educated and so enabled themselves to discover more and more long cycle kos.

About during the early or mid 1990ies, the current age of the internet and the researchers using it has emerged. This accelerated research significantly. The topic ko is very complicated though. Therefore still almost all important research has been made by a few very active researchers, especially Bill Spight and Robert Jasiek. Research has followed two major paths: the combinatorial game theory and thermography path (people like Spight) and the rules and definition of terms path (mostly by Jasiek). While thermography explains how to evaluate kos in terms of values, definition of terms explains how to identify kos in contrast to what is not a ko. In future, research will also have the task of combining both paths.

Page creation: Robert Jasiek.

A General Definition of Ko last edited by on December 16, 2011 - 21:20
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