2-player 3-color go

[le_4TC]

There are many multi-player variants which take advantage of the fact that the go rules generalize easily to multiple colors. But when playing with more than 2 players, there are lots of new complications (how to define objectives? how much diplomacy to allow?). Here’s a variant which combines the fun of multiple colors with the simplicity of 2 players:

Before starting, place some number of neutral stones on the board. Then play normally with black and white. No more neutral stones are placed throughout the game, but the stones already on the board behave just like they would in a 3-player game (for instance, they are removed from the board once they run out of liberties).

Unlike the last variant I proposed (Playing on the union of two boards), this is super convenient to play on VGS. One of the players takes the seat of the blue player to place some predetermined stones (while black and white pass). Once all the blue stones are placed, blue resigns the game. Then black and white can continue playing normally, without having to worry about passing for blue.

Here’s an example game I played against myself: https://go.kahv.io/#2488 (not the highest quality moves, just a proof of concept).

There are many fun starting positions to try out. I think a nice one to start with would be 5 blue stones on the star points of a 13x13 board. This will make corner joseki quite interesting!

If you’d like to try it out with me, just take the black seat in this game: https://go.kahv.io/#2505 (or feel free to propose your own starting position, with handicap if you prefer).

(Edit: That seat is now taken, just let me know if you’d like to play as well, and we’ll start a new board!)

[le_4TC]

This sort of endgame shape (at D10 and G7) seems to be a common occurence:

Note that it would be a mistake for black to play C10. Instead black should wait for white to play C10, and then play D10.

(although black C10 could possibly be useful as a point-losing ko threat sometimes. If white did not have a stone at B8, C10 would be big threat, and could potentially help winning a big ko, making the point-loss worth it)

[le_4TC]

Played two more games just now, one regular one pixel go. Some interesting endgame positions below. Click on boards to view full games.

The black stones at B2 and C2 are dead, despite the fact that the blue stone at C3 will eventually be black if white plays on! There are two equivalent ways to end the game (after the exchange N5 J7 of course):

Option 1: Agree that the black stones are dead, C3 is alive, and D3 is dame.
Option 2: Alternatively, white may want to capture the blue stone, to claim D3 as a point. Then white must start with B1 and C1, and then C2 and D3 to capture. This allows black to get C3.

Option 2 gives one point extra for white and for black. Thus it doesn’t matter in a two player game, but if blue were actually a player, black and white would usually prefer getting a point extra each, while taking a point from blue. Anyways the choice is in white’s hands, black can not force option 2.

This was the final position in the pixel go game. Two interesting things:

1. The blue stones on top are alive, because either side capturing them loses points. For instance, black can capture to gain 1 point at C12, but then white would make 4 points by filling the empty space left. So this is different from the situation in the other game where white could choose whether to leave it or not, here both sides have to leave the blue stones alive. This sort of situation is quite special for pixel go (in regular go, black can capture to gain 1 point, and then black and white will split the 4 points evenly, so it’s a gain for black).

2. Check out that cool seki at the bottom! Actually it would be a seki even if only C3 were blue, and even without the pixel variant. So this could easily happen in a non-pixel game too. For instance:

I really like being able to look at a position like this and saying with certainty that it is a seki. If it were a 3-player game, things would be much more messy. Maybe white wants to reduce blue’s score, and doesn’t care about black’s score? Then white should capture the blue stones. Maybe white plays directly after blue, and blue wants to reduce black’s score? Then blue could suicide to let white kill black.

So 2-player games are nice and simple in comparison. It’s a seki. There are two point-losing ko threats for white. That is all there is to say about the position.

[le_4TC]

There are so many new interesting situations possible in this variant! Check out this tsumego:

What is the status of this black group?

Solution

Black’s only move to live outright is this B2 bomber (by playing C2, not B2, unfortunately )

This is a seki-like shape. It’s funny that even though it’s “eye vs no eye”, the seki is stable since no-one is playing blue. Black has no territory here, and will just score 7 points for the alive black stones.

Black also has the option to fight this ko:


At first glance it looks like black takes the ko first, but after black captures white can take back immediately since A2 has then changed from blue to white! So this is a ko which white takes first, and it’s alot heaver for black. Black only stands to gain 5 extra points by playing this way, but if black loses the ko not only does he lose 7 points, white also gains 12 points. So basically black can gain 5 points or lose 19 points, a total swing of 24 points.

Note that counting points is more subtle here than in typical area scoring rules, not all gained points are gained from the other player!

If white plays first in the inital position, the only way to kill outright is to start with C2, and then after black plays elsewhere, B2.

Now there are two options to kill, and it’s impossible to say which is better in isolation. C2 is better to capture black quickly, but leaves one big ko threat. B2 does not leave any ko threat, but in the future black can capture at A1 to gain one extra liberty (which could matter if the outside white group is not independently alive).