Counting Liberties

British Go Journal No. 10. December 1969. Page 6.

Part 1

Masayoshi Fukuda, 6p

The following article is intended for relatively experienced players, who frequently have difficulty in determining how a fight will turn out because they dont know some of the formulae that more skilful players use.

Diagram 1a

Let us first consider Dia 1a. The two chains of 3 stones cannot both survive. In this simple example it is fairly easily seen that whoever attacks first will win. The least experienced player has learned that in simple situations like this he need only count the number of vacant points to which each group connects, and that the first play wins if the number is equal.

Diagram 1b

He should also know that, if his group has one more liberty than his opponents, then he can play elsewhere - there will be enough time if his opponent attacks first. Thus in Dia 1b, the white three stones are lost; Black can play elsewhere and need not take action here until White plays at one of the 4 liberties of the 7 stone black group. All Black needs do is count the liberties - he has 4, White has 3.

Although in simple situations the fighting power of a group is determined by the number of vacant points to which it connects, this is no longer valid in more complicated positions. From now on the tern "liberties" will be used to denote the fighting power of a group. An example will serve to clarify this point.

Diagram 1c

Consider Dia 1c. The cut-off black chain cannot live except by capturing the three white stones to the right - the capture of the three other stones does not lead to two eyes, since White will play back inside. Similarly, the three white stones to the right can only survive by killing the cut-off black chain. These whites have three liberties; how many has the black chain? If we play the sequence out, assuming that White starts, it will go as in Dias 1d & 1e.

Diagram 1d

Diagram 1e

Black 2 captures 3 white stones. White 7 captures 9 black stones. - Since Black came out just one move behind, we conclude that he too must have had three liberties. The reader can check this by playing the situation through giving Black the first move - in which case he wins.

Diagram 2a, b, c & d

In Dias 2a thru 2d we see four positions in which white chains enclose black stones, and are themselves enclosed by black groups. In Dia 2a, which is similar to the situation we have just been considering, three stones are enclosed; in B, four; in C, five; in D, six. In each case white cannot make two eyes against opposition after capturing the inside black stones. Such formations are painfully familiar to all beginners; the reader should be able to recognise them at once. How many liberties does the inside white group have in each case?

For the positions in Dia 1a thru 1c this inquiry is academic, since for simplicity these white stones have no fighting chance, but academic inquiries can be instructive.

Let us play through the sequence for Dia 2c (in Dias 2e thru 2j), with Black playing first and indicating the White opportunities to counter-attack, as distinguished from forced plays inside the group.

Diagram 2e 2 captures 5 stones.

Diagram 2f 4, 6, 8 elsewhere. 10 captures 4 stones.

Diagram 2g 12, 14 elsewhere. 16 captures 3 stones.

Diagram 2h 18 elsewhere. 20 captures 2 stones.

Diagram 2i 22 elsewhere. 23 captures 14 stones.

Diagram 2j Position at end of sequence.

We count seven missed moves before the death of the white stones; adding one we conclude that white had 8 liberties. If we go through a similar sequence for each position, we arrive at the table below. A few minutes spent in memorislng this table will save many anxious minutes of mental gymnastics during games.

It should be noted that in each case the trapped white stones have one inside and one outside vacant point to which they connect, and that each sequence starts with a play by the attacker filling the outside liberty followed by the defender capturing the attacker's stones on the inside.

Let us now consider some of the applications of the table, and then discuss the situations where, appearances to the contrary, it does not apply.

Diagram 3a, b, c & d

Dias 3a thru 3d show four positions where knowing the above table permits us to say after a very brief examination that first play wins, since in each case the fighting groups have an equal number of liberties. Dia 3a we have already analysed (in Dia 1c to 1f). Three captives, giving three liberties for Black; three obvious liberties for White.

In Dia 3b the 4:5 rule applies. White starts at 1, Black replies 2. These two plays are like the first two plays in preceding examples. From our table, then, we know that the Blacks have five liberties; the Whites also have five. Whoever plays first wins.

In Dia 3c, the situation is similar, illustrating the 5:8 rule - eight liberties for the Whites, and eight for the Blacks. In Dia 3d the 6:12 rule holds and the first player wins twelve liberties for the Whites, twelve for the Blacks. How many players could reach this conclusion about Dia 3d without knowing the 6:12 formula?

To be concluded.