How Many Live Groups can there be on a Go Board?

It is sometimes asked what is the maximum possible number of groups in a game?

This is normally asked for computer programming purposes to get a practical rather than a theoretical limit. However the theoretical question is not easy to solve.

The question, the maximum possible number of groups on the board, has two interpretations (at least).

1: Maximum number independently alive

2: Maximum number alive in seki (or similar)

Both of these are difficult to define unambiguously, without 134 densely printed pages of Japanese-style rule-bookishness. However assume that we all know what we mean.

The group shown here can be considered         O . O . .
as either one or two groups (of 1 and 6        . O O . .
stones, but assume that it is clearly          O O O . .
one, in the spirit of the regular rules.       . . . . .

Whereas something like this (on a 4x4 board),
would have to be counted as four groups,       . # O O
two black and two white. The separate halves   # . O #
of the black groups are connected by a         O O . #
uni-colored eye; this does not 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.

Possible answers seem to be 28 and 60, respectively.

Here are solutions.

1:

. O # . # O . O # . # O . O # . # O .
O O # # # O O O # # # O O O # # # O O
. O # . # O . O # . # O . O # . # O .
O O # # # O O O # # # O O O # # # O O
# # O O O # # # O O O # # # O O O # #
. # O . O # . # O . O # . # O . O # .
# # O O O # # # O O O # # # O O O # #
. # O . O # . # O . O # . # O . O # .
# # O O O # # # O O O # # # O O O # #
O O # # # O O O # # # O O O # # # O O
. O # . # O . O # . # O . O # . # O .
O O # # # O O O # # # O O O # # # O O
. O # . # O . O # . # O . O # . # O .
O O # # # O O O # # # O O O # # # O O
# # O O O # # # O O O # # # O O O # #
. # O . O # . # O . O # . # O . O # .
# # O O O # # # O O O # # # O O O # #
. # O . O # . # O . O # . # O . O # .
. # O . O # . # O . O # . # O . O # .

28 groups fully alive.

2:

# . O # . O # . O # . O # . O # . O O
# . O # . O # . O # . O # . O # . O O
# # O # # O # # O # # O # # O # # O O
O O # O O # O O # O O # O O # O O # #
O . # O . # O . # O . # O . # O . # #
O . # O . # O . # O . # O . # O . # #
O O # O O # O O # O O # O O # O O # #
# # O # # O # # O # # O # # O # # O O
# . O # . O # . O # . O # . O # . O O
# . O # . O # . O # . O # . O # . O O
# # O # # O # # O # # O # # O # # O O
O O # O O # O O # O O # O O # O O # #
O . # O . # O . # O . # O . # O . # #
O . # O . # O . # O . # O . # O . # #
O O # O O # O O # O O # O O # O O # #
# # O # # O # # O # # O # # O # # O O
# . O # . O # . O # . O # . O # . O O
# . O # . O # . O # . O # . O # . O O
# # O # # O # # O # # O # # O # # O O

60 groups alive in seki.

An alternative solution based on regular pattern of stones, that instead of 15 intersections for the groups in the centre only uses 12, still only gets one extra fully alive group for a total of 29:

# # # # O O . O # # # # # # # # o o .
# # . # O . O O # # O O # # . # o . o
# . # # O O # # O O . O # . # # o o o
# # O O # # . # O . O O # # O O # # #
O O . O # . # # O O # # O O . O # # .
O . O O # # O O # # . # O . O O # . #
O O # # O O . O # . # # O O # # o # #
# # . # O . O O # # O O # # . # o o o
# . # # O O # # O O . O # . # # o # #
# # O O # # . # O . O O # # O O o # .
O O . O # . # # O O # # O O . O # . #
O . O O # # O O # # . # O . O O # # #
O O # # O O . O # . # # O O # # o o o
# # . # O . O O # # O O # # . # o . o
# . # # O O # # O O . O # . # # o o .
# # O O # # . # O . O O # # O O # o o
O O . O # . # # O O # # O O . O # # #
O . O O # # o o # # . # O . O O # # .
O O o o o o o o # . # # O O # # # . #

The 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?

For the seki groups option an improved formation is:

. # . O . # . O . # . O . # . O . # .
# # O O # # O O # # O O # # O O # # #
O O # # O O # # O O # # O O # # O O O
. O . # . O . # . O . # . O . # . O .
O O # # O O # # O O # # O O # # O O O
# # O O # # O O # # O O # # O O # # #
. # . O . # . O . # . O . # . O . # .
# # O O # # O O # # O O # # O O # # #
O O # # O O # # O O # # O O # # O O O
. O . # . O . # . O . # . O . # . O .
O O # # O O # # O O # # O O # # O O O
# # O O # # O O # # O O # # O O # # #
. # . O . # . O . # . O . # . O . # .
# # O O # # O O # # O O # # O O # # #
O O # # O O # # O O # # O O # # O O O
. O . # . O . # . O . # . O . # . O .
O O # # O O # # O O # # O O # # O O O
# # O O # # O O # # O O # # O O # # #
. # . O . # . O . # . O . # . O . # .

which has 63 live groups alive in seki.

Using the small strings of stones definition it is believed some claims have been made to have more than 130 strings alive in seki. However we only have a solution with 108 strings:

. O # # O # . # O # O O O O O # # . #
O . # # O # # . O # O . O . # O # # .
# # . O O # # O . # # O O # . O O # #
# # O . # O O # # . O # # O O . # O O
O O O # . O O # # O . # # O O # . O O
# # # O O . # O O # # . O # # O O . #
. # # O O # . O O # # O . # # O O # .
# . O # # O O . # O O # # . O # # O O
O O . # # O O # . O O # # O . # # O O
# # # . O # # O O . # O O # # . O # .
O O # O . # # O O # . O O # # O O . #
O . O # # . O # # O O . # O O # . O O
O O O # # O . # # O O # . O O # # # #
O . # O O # # . O # # O O . # O O # .
O # . O O # # O . # # O O # . O O # #
# O O . # O O # # . O # # O O . # O O
# # O # . O O # # O O . # O O # . O O
. # # O O . # O O # . O # # # O O . #
# . # O O # . O O . # O # . # O O # .