Single-stone suicide  

A single-stone suicide is the only possible type of one-move cycle apart from a pass. It is only legal in rule sets that allow suicide and do not use positional superko; even then the only possible point of it is to pass without allowing one’s opponent to end the game immediately. It costs one point under territory but not area scoring and loses sente.

2-move cycles

Two successive passes form a 2-move situational cycle, if “move” in the definition is taken to mean turn rather than board play. If we allow multiple moves by one player, another possibility is suicide of two stones.

The only other possible type of two-move cycle is the basic ko, which is also the only cycle to occur commonly in actual play and is prohibited under all rule sets. Cycles longer than this are usually called long cycles

Basic ko

This is the only possible type of two-board-move cycle, since the second stone must capture the first in order to recreate the original position.

3-move cycles

Send two, return one

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 she has give two prisoners to Black’s one. Under area scoring, it does not constitute such a loss, but still loses sente.

If positional superko is used, is prohibited. This is intentional, but not ideal: in exceptional circumstances, positional (as well as situational) superko can lead to counter-intuitive results. See Rules Beast 1 for a case where both superko rules surprisingly force a connection in a seki to avoid a capture; RP’s Cycle Completion proposes a remedy.

Under rules that use area scoring without banning long cycles, this pattern would allow White to force an endless game unless Black can afford 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 one point. Black can play through the cycle a few times until he has gained enough to compensate for the local loss.

4-move cycles

Eternal Life

The sequence above, known as eternal life, constitutes a four-move cycle. If either positional or situational superko is used, is prohibited. Under basic ko rules, this cycle could repeat indefinitely.

Sending three, returning one is also a 4-move cycle.

5-move cycles

Send three, return two

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.

6-move cycles

Triple ko

This basic triple ko example shows a six-move cycle (actually a combination of three 2-move cycles). Under superko rules, is prohibited.

Eternal ko is also a six-move cycle.

8-move cycles

Round robin ko

This “round robin ko” constitutes an eight-move cycle.

The quadruple ko is also an eight-move cycle.

Note that molasses ko is not an eight-move cycle because it needs to be combined with at least a basic ko elsewhere to bypass the ko rule.

10-move cycles

Triple ko stones cycle

The triple ko stones cycle (not the same as triple ko) constitutes a ten-move cycle. Mentioned at http://home.snafu.de/jasiek/ko.pdf.

Shimada’s ko also constitutes a ten-move cycle. Mentioned at http://www.britgo.org/rules/molasses.html and http://harryfearnley.com/go/shimada/intro.html.

Round robin ko (an eight-move cycle) and Molasses ko (an eight-move sequence, but not a cycle in itself) can both be made into ten-move cycles by combining them with basic kos.

12-move cycles

 

This sequence leads to an enlarged round robin ko, a twelve-move cycle.

14-move cycles

continued  

This quadruple ko stones cycle (not the same as quadruple ko) is a fourteen-move cycle. It was discovered by Matti Siivola. Mentioned at http://home.snafu.de/jasiek/ko.pdf.

N-move cycle

Pinwheel ko

Pinwheel ko is an infinitely scalable cyclic pattern.

Cycles and combinatorial game theory

In CGT we analyse local positions separately as abstract games and then analyse how they may be combined into a sum of games. When considering cycles some issues arise:

• The standard definition of an abstract game forbids all forms of repetition.
  • Games allowing repetition are known as “loopy” games. For go, we need to consider sums of loopy games in which repetitions are forbidden in the sum (though not in the summands); the precise form of this prohibition depends on the rule set we are analysing.
• When considering a sum of games we allow successive moves by the same player in one summand (with intervening moves elsewhere by another).
  • The definition of “cycle” above tacitly assumes that play within one cycle alternates between the two players, as it does in almost all the examples below.
  • When repeated plays by one player are allowed, suicide of n stones in an n-point space is an n-move cycle.

See also

• Ko — A general Definition by RobertJasiek
• Cycle Balance Law? — In a situational cycle the difference in the number of passes is the negative difference of removed stones.
• Directed Acyclic Graph — A mathematical construct sometimes used to detect cycles and transpositions.
• Superko — The various concepts and proposals used in rules to prevent indefinite play by forbidding cycles.
• Stoical Go — A Go variant where ko and superko rules are replaced with a simpler rule rendering all known forced cycles impossible.
• Incursion Go — An improvement on Stoical Go.
• Capture Go — A Go variant where the goal is to be the first to capture a certain number of stones. This automatically includes its own superko resolution.
• Pinwheel Ko — An infinitely scalable cyclic pattern.
• Named positions and sequences that yield cycles:
  • 1-cycle: Single-stone suicide
  • 2-cycle: Basic ko
  • 3-cycle: Sending two, returning one
  • 4-cycle: Eternal life
  • 5-cycle: Sending three, returning two,
  • 6-cycle: Triple ko, Eternal ko
  • 8-cycle: Round robin ko, Quadruple ko, Molasses ko (with repeated moves by both sides)
  • 10-cycle: Triple ko stones cycle, Shimada’s ko
  • 14-cycle: Quadruple ko stones cycle