Maximum Number Of Live Groups

Two Groups  

It's already been mentioned that things like might be considered as either one or two groups, but I'll assume that it's clearly one, in the spirit of the regular rules.

Four Groups  

• derPlumps: This should be "6 groups" according to diagram 1

Whereas something like this (on a 4x4 board), would have to be counted as four groups, two black and two white. The separate halves of the black groups are `connected' by a uni-colored eye; this doesn't hold for the two white bits.

Anyway, none of this appears to be relevant for the answers to questions 1 and 2 above; on a 19x19 board at least.

I get answers of 28, and 60, respectively.

As always, I would be delighted to see anyone write in with improvements.

Here are my solutions.

29 groups alive  

But I think with a bit more thought I can fine tune this.

Next question is how can you PROVE that a certain number is the maximum - apart from trying every combination of alive stones on a board and counting.

TapaniRaiko: One can count the points used by each group. They should add up to at most 361. In the corner a group takes up at least 8 points. In the middle, it takes up 12. In the border, it takes up 10 points consuming 4 points of the border or 11 points consuming 3 of the border. It seems that this leads to a maximum of 33 groups.

30 groups with room to spare  

I could not find a way to rearrange the marked area to any shape that could contain a live white group. (circled stones would be black then)

Anyway, 30 groups fit quite nicely :-)

 -- Bass

31 groups alive  

Only took one more day of iterating to find this.

-- Bass, 12.9.2002

108 strings, but not alive !  

There are two white groups in this diagram with three liberties. (I circeled them to make them stand out.) From this configuration, white can capture most of the board.

Basic shape pattern  

I'm now convinced that the best patterns will have a basic shape like that shown here, which must be the max possible density of live groups in an infinite board. (These groups are only alive in seki, of course.)

This density is noticably better than any other solutions seen on the net.

It's a bit tricky fitting in stuff round the edges of a finite board, though. In view of the messy nature of my solution, and the alleged Dutch publication of a 130-odd group formation, I'm far from sure this is best possible. It must be close though.

So here's my solution:

128 groups (counting orthogonal connections only)  

• 128 groups (counting orthogonal connections only)
• 114 groups (counting 2 bits joined by an eye as the same)

Interestingly, (in Chinese counting), the result of the game is an exact tie ! ( 151 points each, with 59 neutral).

Strip  

Here we have shown only a single strip. These strips can be put together to cover an infinite board. This second configuration allows the boundary connectivity to change phase.

132 groups (in the sense of connected components)  

This arrangement has 132 groups (in the sense of connected components).

HandOfPaper: Shouldn't the phrase you want be " monochromatic connected components "? Otherwise, you would have only 8 of them...

If we allow connections though empty intersections, then the connected components combine to form 65 areas. The three circles are empty intersections that behave as false eyes (as opposed to seki points).

Corner rearrangement  

The false eyes at F6 and Q16 are the centers of "windmills". These windmills are there to solve phase problems with the edges of the board. Each one forms a single area with 4 connected components. The false eye at T1 is the center of another windmill. This rearrangement of that corner makes this clear.

This rearrangement does not change the area or component counts.

181 strings at 19x19 - but seki?  

If one player moves, is there any safe way to the opponent to win the game? Or, on the contrary, is there any safe way the first moving player wins the game? Perhaps it is something you could call a "global seki"? - Although i´ve been trying to analyse it for quite a while, I still don't know anything about that. So, if you're sure that it's not stable at all, which means moving first has a positive value here, just remove it. :) --rubilia

(Sebastian:) Well, that's what it's a game for - let's play it: Odd Dots Go Ongoing Game

blubb: For more discussion on this idea see Global Seki, and about this particular position as well as the amazingly complex resulting game, see DotsGo.

Random reader?: If you turn the board by 45 degrees, the resulting game is equivalent to a diagonal-go on a diamond board. Diagonal go means white stones have liberties and capture in one slant-direction, while the black stones capture along the crossing slant. This is a fair-game, and it is similar to go, although more complicated. This is not technically "life and death", it is a go variant.

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Maximum number of (not alive) groups?

lefuet: if we drop the restriction of live groups, how many can be fit on a goban? heidi_row brought up an upper bound: 0.8 times 361. A bid less because of the edge. This can be motivated as follows: One stone could have at most 3 groups as neighbor and an empty intersection. So here we have 5 intersections (the stone and his 4 neighbor intersections) of wich 4 contain different groups. So on every 5 intersections there can be a maximum of 4 different groups. So 0.8 * 361.

276 not live groups (288 - 12 on the edge)  

This is such a pattern. Because of the edge some stones have to be removed. On every edge 3 stones have to be removed making this 288 - 12 = 276 groups. This is at least close to the optimum.

277 strings  

Gunnar Farnebäck: The maximum number is 277 on 19x19. For more information, see http://computer-go.org/pipermail/computer-go/2006-December/007398.html