A cycle is a move sequence that starts and ends at the same position. If at the end of the cycle it is the same player's turn as at the start, it is also a situational cycle (repeats the same situation). If cycles are allowed without restrictions on them, it is possible for a game to go on indefinitely.
Although it is easy to play cycles if both player cooperate in creating them, such cycles are not interesting, because it will be to the benefit of one of the players to break the cycle and play elsewhere. More interesting is to consider those cycles that can be forced by one player. Such cycles would allow a player that is behind to prevent a loss by forcing the game to go on indefinitely.
Such cycles are rare, and the only one that is seen regularly in actual play is the basic ko. repeating the position in a basic ko is prohibited under all rule sets.
Longer cycles, such as triple ko, eternal life or round-robin ko are not prohibited by all rule sets. This is probably because they are so rare that they were never considered. In rule sets that allow them, a game in which they occur is usually declared "no result", which is sometimes ruled as jigo (tie), and sometimes replayed.
We can classify cycles by the number of moves that are played before the position repeats. Shorter cycles are more common than longer cycles, and cycles with an even number of moves are far more common than those with an odd number of moves. If multiple cycles occur on the board simultaneously, they may affect each other, and combine to form a larger cycle (triple ko is an example of this).
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Single stone suicide
Send two, return one
Send three, return two
Round robin ko
The 1 move cycle is mostly a theoretical construct. In most rule sets it is impossible to play a one move cycle.
A single stone suicide constitutes a one move cycle. This can only occur in rule sets that allow suicide (but do not use positional superko) and is completely pointless even then, unless one wishes to pass without the other being allowed to end the game immediately.
The two move cycle is the most common cycle, as this is the basic ko.
The basic ko, prohibited under all rule sets, repeats the position after two moves. Cycles that are longer than this are usually called long cycles
The sending two, returning one sequence is an example of a three move cycle. Under territory scoring rules, this sequence constitutes a loss for white, as white has give two prisoners to black's one. Under area scoring, it does not constitute such a loss, but still loses sente
Under rules that use area scoring, but that do not prevent long cycles, this pattern would allow the white player to force the game to go on indefinitely, unless the black player is willing to take a local loss (in the above case, by ignoring and allowing white to capture, turning the seki into a live white group and losing black 6 points). Under territory scoring rules, this is not the case, as each repetition of the cycle gains black points. Black can then play through the cycle a few times until he has gained enough points to compensate for the local loss.
The sequence known as eternal life constitutes a four move cycle.
If either positional or situational superko is used, is prohibited in the above sequence. Under basic ko rules, this cycle could repeat indefinitely.
This sequence, known as sending three, returning two, shows a five move cycle. As with all cycles that have an odd number of moves, this sequence loses sente. Unlike the sending two, returning one scenario however, is mandatory for black as the only way to avoid losing his group.
Under positional superko rules, is prohibited.
This basic triple ko example shows a six move cycle (actually a combination of three 2-move cycles) Under superko rules, is prohibited.
The round robin ko constitutes an eight move cycle.
Molasses ko (an eight move sequence, but not a cycle in itself) also results in a ten move cycle when combined with a basic ko.