Double Go / Limbo

In the variant of go known as Double Go where each player places two stones per move (and capturing occurs only after both stones are placed), a situation similar to ko fights can occur, known as 'limbo'.

For the purposes of this article, the placing of two stones will be considered to be one move, with each stone being half a move. This terminology is used for brevity; it is not necessarily standard.

In Double Go, the aim is to make groups with 3 eyes, as two eyed groups can be killed using a whole move.

Living groups may be divided into two sub-classes:

• Strongly alive groups - if the attacker uses half a move to attack a group, defence uses half a move
• Weakly alive groups - if the attacker uses half a move to attack a group, defence uses a whole move

 

Example - Strongly alive group

 

B uses half a move to attack, but W successfully defends with half a move

 

B uses a whole move to attack, but W still only needs to defend with half a move

 

Example - Weakly alive group

 

B uses half a move to attack, W needs a whole move to successfully defend

 

B uses half a move to attack, W fails to defend with half a move

 

Continuation - B uses a whole move to cut the W group into two parts, killing all the W stones.

 

B uses a whole move to attack, W needs a whole move to successfully defend

These examples might seem highly contrived, but they have been chosen to illustrate the weakly/strongly alive concept without needing any insights into life and death in Double Go.

Here is a smaller example, where some reading is necessary

W is weakly alive  

So why does any of this matter? Weakly alive groups are only alive in limbo!

An attacker can use half a move to attack a weakly alive group and half a move to make a limbo threat. Assuming the defender wants to use half a move to answer the limbo threat, he cannot defend the weakly alive group as he does not have two half moves remaining. Instead, the defender has to use his second half move to make a limbo threat of his own.

Eventually, the limbo threats of one player will be exhausted. If this is the original attacker, then the defender gets to save the group. If the defender runs out of limbo threats, the attacker gets to make a whole move and kill the original group.

Limbo threats are much like ko threats, the intention is to force a local reply to each threat. The difference is that there isn’t a ko shape and there are no issues about repetition. Kos may be used as limbo threats and vice versa.

This "limbo" concept seems to follow a set pattern: it takes 3 half moves to kill and 2 half moves to live, attacker having first move. ~srn347

Example  

The B group at the top has 3 eyes. The W group on the left is only alive in limbo.

Example – simple defence  

B meekly defends on the right and throws-in on the left

Example – simple defence  

W defends. W scores 4+1 and B scores 5. Jigo

Better defence?  

Herman: Correct me if I'm wrong, but isn't this a better defence for white? Now black's invasion has become pointless, and white gets 7 points, winning by 2. (Black must play at least one of the a marked points to prevent another point for white there)

Discussion

Example – B attacks, part 1  

Instead, B plays half a move into the weakly alive group and half a move as a limbo threat to attempt to cut-off the outside stones.

Example – part 2  

W cannot defend both attacks, so uses half a move to defend the cut and half a move to make an attack on the bottom B stones.

Herman: Shouldn't white's second half move be at a? That way, white can live and will not lose any stones... Discussion

Example – part 3  

B defends the cut and counter attacks against the right most W stones.

Example – part 4a, W defends the cut  

If W defends the cut, W runs out of limbo threats and all of the stones die

Example – part 4a cont., B kills everything  

Example – part 4b, W ignores the threat  

W saves the group, giving up the rightmost 3 stones

Example – part 4b, B captures  

B captures the 3 stones. W scores 4+1, B scores 9+4. B wins by 8.

Generalisation of limbo Limbo situations will occur in Go games with larger numbers of stones per turn. There will be many positions of type “X stones to attack, Y stones to defend”, where limbo can be created by distributing the attacking stones.

'Limbo' occurs in other double move connection games, such as TriGo, Double *Star (where the name 'limbo' is taken from) and Master-Y