# Half-point ko

Difficulty: Intermediate   Keywords: Ko

Chinese: 半劫 (bànjié); 单劫 (dānjié)
Japanese: 半劫 (hankō); 半コウ (hankō)
Korean: 반패

A half-point ko, also known as a ⅓-point ko since that is its true miai value, is the common term for a ko in which nothing is at stake beyond the fate of the stone inside the ko.

A half-point ko

This shows a idealised half-point ko.

The name half-point ko is used because it is the translation for both the Japanese term 半劫 (hankō) and the Chinese term 半劫 (bànjié), both of which literally mean "half ko." But if we do not know the winner of the ko fight, who gets the point at stake is a 50-50 proposition, and thus the term half-point ko makes sense. Charles Matthews has proposed minimal ko as an alternative name.

It is rare to have a play smaller than this but larger than a Japanese dame, but Miai Values List / 0.00 to 0.99 shows a ko with a miai value of ⅙. A ko whose miai value is really ½ point exists: see real half-point ko.

## Calculation of the miai value

### Calculation from points per move

One way to calculate the miai value of this ko is as follows. The difference between winning and losing the ko is 1 point. There are 3 moves between the winning positions, 1 move for each player to win the ko and 1 move to take the ko. So each move is worth 1/3 point, on average.

### Calculation from a sum of three kos

Three kos

Another way of calculating the miai value of a minimal ko is by considering the sum game of three kos.

Black plays first ( at )

Black plays first. takes a ko. connects since it's no use to fight the ko. Since and are miai, there is no use to try to win both kos, so each player connects their own ko. As Black capture one stone, the value of this position is 1 point.

White plays first ( at )

White plays first. connects a ko, and captures. Then and each connects their own ko. Again, the value of this position is 1 point.

The value of this position is 1 point. Dividing the value over the three kos, the average territory value (count) of each ko is 1/3 point.

## Playing order of third-point kos

At the end of the game, there are often a number of ⅓-point kos. These should not be played until there is nothing else left but dame.

The priorities for playing them are unaffected by which ko/superko rules are in force; these priorities are:

1. Connect any ko in which you lead.
• Since any in which your opponent leads is miai with this, there can be no point taking that.
3. Make a threat (which the opponent should answer) to contest the very last ko.
• This is thus the only one for which a ko fight should be played.

### Result of third-point kos

The result of correct play in the 1/3-point kos, depends on how many more of them initially favour one player, whom we call the “leader”; we correspondingly call there opponent the “trailer”. (If they are level, it makes no difference who we treat as the leader.)

The above playing order leads to the the following results:

• Miai pairs of opposite kos cancel (in an even number of moves).
• Any excess for one player falls in triplets in the ratio 2:1 to the player leading and trailing respectively.
• The net result is thus that the leader wins 1/3 of the number of kos in this phase.
• Each triplet takes 4 moves to resolve (connect, take, connect, connect).
• After the triplets, there may still be 1 or 2 kos remaining; the result depends on how many there are, whether the leader has sente and possibly the result of a ko fight for the very last ko.
• The leader will have sente at that point if they did before starting to play the kos, since the previous phases take an even number of moves.
• If there is a ko fight, it is started by the opponent of the leader, the leader wins it only when they have strictly more ko threats than their opponent.
• The following table shows the net number of these the leader wins, where “±ko” means a ko fight and should be counted as +1 if they win or -1 if they lose:
Result of residual →-point kos after triplets
Kos remaining → 0 1 2
Sente   0 1 1 + ±ko
Gote   0 ±ko 0

### Value of third-point kos

We can calculate the value (in various senses) of the 1/3-point kos, depending on how many more initially of them favour the leader; the value may also depend on sente and ko threats, as explained above.

We state the value as positive for the leader.

We shall call the excess e of kos favouring the leader:

 e = 3 q + r
where q (the quotient) is the number played in triplets and r (the remainder) is 0, 1,  or 2.

We also define

 s  as the net number of the remainder (r) won by the leader, as shown in the table in the previous sub-section.

We therefore calculate n, the net number of 1/3-point kos won by the leader, as

 n = q + s  (the leader wins 2q of the triplets against q won by the trailer, and s of the remainder)

Some of the kos may change hands; this only happens in those favouring the leader and the number (h) where this happens is:

 h = q + s - r (the trailer wins 1/3 of the triplets and those of the remainder r that the leader does not, i.e. -(r-s))

The final net area score on the debatable points to the leader is twice the net number of kos they win, i.e.:

 a = 2 n , i.e. 2(q+s)

The change in the area score during play in the kos is twice the number of them won by the trailer (if we previously scored each open ko as 2 points the the player leading it); this is:

 a_c = 2 h , i.e.  2(q+s-r)

The final net territory score in the kos is the net number of captives taken in them. This depends on the net number of captives originally taken by the players and on the number that change hands. It is:

 t = c - h

The change in the territory score during play in the kos is the number of those that change hands; this is:

 t_c = -h

PJT: I hope it is now correct, clearer and more useful than when Bill made the comment below! N.B. I may yet try putting the various values in a table to see if that makes them clearer. I also wonder if it would be better to make h negative.

Bill: Could you present this in a way that will be clear to non-mathematical go players? Thanks. :)

Patrick: Fair enough; it seems to have come out rather more complicated than it is worth, I fear. I shall have a go.

Bill: Thanks, Patrick. :)

## Omission of half-point ko fights in professional game records

Often go games end with a fight over a half-point ko. These fights are not shown in professional game records, as a rule. Instead the winner of the ko is indicated (e.g. “White wins & fills ½-point ko”), or can be determined from the final score.