New frontiers in the Troitsky line,

or Kuzmichev's rules.

Below you may find the original PDF published by Vladmir Kuzmichev at (2-aug-2021) ChessStar.com

It 's about Two Knights against the King with one pawn. Especially the c-pawn.

Kuzmichev, Vladimir 1962 also presents some oriiginal studies by him with this theme.

First you may find the original Russian text in the PDF. Below that a (poor) translation.

The examples are replayable.

More about this theme on Arves: CQL Troitzky's ending and in EG142_Supplement.

If you have an iOS device (iPad or iPhone), browsing the PDF file will not work properly, Open the PDF here separately





New frontiers in the Troitsky line,

or Kuzmichev's rules


All practitioners, solvers and chess composers know the Troitsky line in chess.

Knowledge of this line helps to evaluate chess positions and transitions to the corresponding endgame during a chess game and when drawing up studies, analyzing positions.

Two knights against a pawn, almost two extra pieces, and of course the strongest side would like to realize such an advantage, but the weaker side has its own chances of a draw.

If there was no pawn, then there would be a draw immediately, and only its presence creates the intrigue of a possible loss.


Position 1

+ 0 + 8


Troitsky line for black pawns.


The meaning of the Troitsky line is that if the black pawn has moved to the next square forward, then the game for the owner of the pawn will most likely end in a draw. And if the black pawn has not yet crossed such a line and has been blocked by one of the opponent's knights on the Troitsky line earlier, then the game for the weaker side will be lost.

There are exceptions to all the rules, and they are very well noticed by chess composers in their studies and problems. Many problems can be cited where two knights checkmate the black king already in the situation when the pawn crossed the Troitsky line. Conversely, the weaker side achieves a draw if the pawn has not yet crossed the Troitsky line. But these are still rare interesting positions-exceptions.


In my study activities, I have repeatedly had to refer to the Troitsky line in both variations: one pawn against two knights is fighting for a draw and two knights against a pawn are fighting for victory. As a result, I was interested in positions with a black bishop pawn, which had already crossed the Troitsky line on the squares c4 and f4 and was blocked by a knight.


(for convenience, we will only talk about the black c4 pawn, since other mirror variations will be identical, including for the white pawn against two black knights)


It follows from Troitsky's line that if the black bishop pawn is on the square c4, then the game ends in a draw. Troitsky gives examples where White can win with such a pawn in a single, study way but, in my opinion, this is absolutely not the case. The author came to the conclusion that such a position with a pawn on c4 for the most part is always winning for

White with the exception of only two (!) significant points, which will be discussed below.

Accordingly, many studies by Troitsky and other authors that do not take into account the winning opportunities of the strongest side with such

pawns become erroneous, since winning in them is possible in many different ways.


Position 2

= 3 + 2

Draw, on any move.

The first exceptional case when Black draws with a blocked pawn on c4...

In this position (Position 2) if the black king hits the square b2, then Black always draws.

On Black's move, he only plays 1 ... Ka1! and the black king will infinity play across the fields a1, b2 until white stalemates him, also with the knight at 2.Se1.

It is precisely the use of stalemate opportunities in the corner a1 allows Black to save the game.

With White's move, an attempt to grab the square a1 knight 1.Sc2? - gives nothing. After black answer 1 ... Kb3 a positional draw occurs.

The knight from the field c3 cannot move, because the vacated field immediately the black pawn will rush with a draw.

The white king cannot move because of the loss of a knight. And the white knight from the field c2 the next move will be forced return back wherever he went, so as not to let the black king enter the field a1…


Position 3

+ 3 + 2


Position 4

+ 3 + 2


Position 5

+ 3 + 2


Position 6

+ 3 + 2


In the Positions 3, 4, 5, 6 always white wins, since the white king controls the important square b2 and does not let the black king into a draw a

stalemate is a sufficient condition to win. The white king and the second knight further drive the black king into a

mating net, which they create for the black king in any of the other three corners of the chessboard.

(a8, h8. h1).

In the Positions 3 and 4 standing on the knight at e4, can similarly be on other fields supporting the second knight a4, b5, d5, e2, d1.

And here is the finding of the knight on the supporting fields a2 and b1 in Positions 3 and 4 is results with draws zugzwangs due to the fact that such a knight will itself be limited in moves and will interfere with a winning maneuver for its king.

When the knight is on the field a2 in Positions 3, if black moves, there will be a draw

1 ... Ka3 and white has no important move 2.Kc2? due to the emerging right stalemate, and on 2.Kb1 Kb3 3.Ka1 Kc2! -White has no winning move 4.Ka2 ?. While in Positions 3 knight on the field b1 when White moves, White does not there will be a necessary winning move 1.Kb1...

In Positions 4 finding a supporting knight on the field a2 immediately leads to a draw when White moves, since he has nowhere to move without losses. And when you find a supporting knight on the field b1 v Positions 4 a draw appears when black moves: 1… Kc1! 2.Ka2 Kc2! 3.Ka1 Kb3!- and white again has nowhere to go without loss.


Position 7

3 + 2

With black to move, draw

With white to move, win.


A special case of control by white fields b2 (Position 7) - when this square is controlled not by the white king, but by the second supporting knight (the white knight on a4 can stand similarly on the field d1).

In such a position, it is important to understand the following aspect: if the

black king moves freely across the squares c2-b3, then Black always makes a draw, since the black king creates a drawn fortress, endlessly playing on the squares a3-b3-c2-c1... This is the second exceptional case where the weaker side makes a draw.

1… Kb3! 2.Kb5 Ka3! 3.Kc5 Kb3! 4.Kd4 Kc2! 5.Ke3 Kc1!

6.Ke2 Kc2 7.Ke1 Kc1! - the white knights are hobbled and cannot move,

and the white king alone cannot oust the black king from the no-man's

fortress. (I want to draw your attention to the fact that this fortress was

first designated by me and no one had previously designated it. There

are a couple of works indicating a draw, when the king of the weaker side reached squares similar to c2 and c1 in such a position and a positional draw of the king on squares c2 and c1 against king of the strongest side, located on the e-file.

When White moves first in Position 7 white wins due to the fact that white the king blocks the draw path for the black king, then the white king takes control of the important square on its own b2 translating Position 7 to the winning Position 6 with 1.Kb4! Kc1 2, Ka3 Kc2 3.Ka2 Kc1 4.Sb6 Kc2 5.Sbd5 Kc1

6.Sb4 Kd2 7.Kb2 - white drove the black king out and wins.



Position 8

V. Kuzmichev (after A. TrotzkyOriginal

+ 3 + 2

It is easy to understand that in Position 8 the white king has no time to take control over the important field b2 before the black king.

And if the field b2 picks up the second knight when the white pawn is blocked on the square c4 by the second knight, then the black king manages to build a drawn fortress.

And yet White wins. How?


We look at the mainline:

1.Se7! c5 2.Sd5! c4 3.Sc3! Kf6 4.Sg3! (4.Sf2? Ke5! 5.Sfd1 Kd4! - and the black king builds a drawn fortress) 4… Ke5 5.Sge2! -white by built the first screen for the black king, from the fields d5, d4, e4, f4 making it go around and lose the pace

5… Kd6 6.Sd4! Kc5 7.Kc2! -and now Black has built more than just a shield,

but an impenetrable winning fortress

7… Kb6! -and a study by Troitsky appears in front of us (Position 9), which in

his book ("Collection of Chess studies" 1935) only such a construction with a

bishop pawn on the 4th rank with the second knight on the square c2

classified it as definitely winning for White.

Further, the white king is selected from under the opposition of the black king by an

accurate maneuver: 8.Kc8! Kc6 9.Kd8! Kd6 10.Ke8! Ke6 11.Kf8! Kf6 12.Kg8! Kg6 13.Sc

~ Kf6 14.Kh7! Ke5 15.Sc2! Kf6 16.Kh6 -and the black king releases the white king from under his opposition.


Position 9

+ 3 + 2

If White succeeds in creating such a position (Position 9), with a white knight on the field c2 and the black king outside the knights' strike line, then they always win.

White knights have created an impregnable fortress for the black king and thus do not let him into the lower left corner and to the important draw squares.

It should be noted that this is not the only position in which white can prevent the black king from reaching the field by playing and setting up screens. b2. Such positions are found in different places during the game. Several screens will be shown in Example 3...


Based on the foregoing, conclusions can be drawn and certain rules can

be formulated:


Rule 1

If the black king hits the b2 square, then it is definitely a draw.


Rule 2

If b2 is controlled by the second knight (from the fields a4, d1), then Black makes a draw only if his king moves freely along the adjacent squares b3-c2,

as a result of which a draw fort is created for him.


Rule 3

If the white king controls the b2-square and can occupy it, then White always wins.


Rule 4

If the stronger side during the game creates positions in which the king of the weaker side does not enter the squares similar to b2, then the stronger side always wins.


You can limit yourself only to rules 1 and 2, in which the conditions for reaching a draw by the weakest side are indicated in the sense that if these conditions are not met, then in all other cases the strongest side with two knights always wins. But for a quick assessment of the position, it is more convenient to use all four rules in the complex.

The fourth rule is a special case of the third rule, since if White is able to prevent the black king from reaching the field b2, then this means that

the white king is always able to capture the key square b2, translating the fourth rule into the third.

It is quite easy to take these rules into account in the course of a chess game, composing or analyzing studies.

It is unlikely that these squares with bishop pawns (c4, f4 - for black pawns and c5, f5 - for white pawns) can be directly included in the Troitsky line.

Most likely, it will be more convenient to consider such fields as additions to Troitsky's line - "Kuzmichev's field" with the indicated rules.


Position 10

+ 2 + 8

Pawns in black - Troitsky's line; Pawns in blue - Kuzmichev's fields

It should also be kept in mind that positions where the white king blocks the black pawn a little earlier - on the field c5...

The logic of winning is like this - on the field c3 the knight is set, and the white king leaves the blocking square taking control of the blocking square b2…



Here are some of my studies on the new rules and their variations.

Example 1

V. Kuzmichev

Chess school, 2018

= 2 + 3

An example on the topic of a very unfortunate finding of knights in the fields a7, b8

1.Kb6? Sac6? 2.c5! - and Black is forced to let the white king go to no

man's fields,

But: 1… Sbc6! 2.c5 Kb8! - and black wins, for example: 2… Ka6 3.Sc8 Kb5 4.Kb7


1.c5! Sac6 2.Kb6! = =

1… Sbc6 2.Ka6 !! Kc7 - correct stalemate

2… Kb8 3.Kb6! Ka8 4.Kc7! - breaking Black with a draw

3… Kc8 4.Ka6! Kd7 5.Kb7 (b6) Sc8 6.Ka8! Sd6 (cxd6) - another correct stalemate.

6… S8e7 7.Kb7 Sd5 8.Ka8! Sf4 9.Kb7 Se610.Ka8! Sxc5 / Sd8 are correct stakes.

(Similarly: 8… Se3 9.Kb7 Sc4 10.Ka8! S4a5 - another correct echo-stalemate)

6… Kd8 7.Kb7 S8e7 8.Ka8! Ke8 9.Kb7 Kd7 10.Ka8! Sd5 11.Kb7 Sc7 - blocked the a square for

the white a8 but 12.Kb6! and Black can do nothing - a draw.


Example 2

V. Kuzmichev

Chess school, 2018 (version)

= 2 + 3

1.c4! Sd8 2.Kf5!

(2.Kf4? Sc6! 3.Ke4 Sc3 +! 4.Kd3 Sa4? 5.c5! Sb2 6.Kc3! Sd1 7.Kc4!

Ke7 8.Kb5! Kd7 9.Kb6! - and the white king reached the drawn square b7

9… Sb2 10.Kb7! Sc4 11.Ka8! S4a5 is a correct stalemate: 4..Sa2! - and black wins)

2... Ke7 - the black king and knight are ready to block the white pawn on field c6 (2.c5? Sc6!)

White finds an interesting solution:

3.Ke4 !! - let's go to catch the black knight on b1

3… Sd2 + 4.Kd5! Kd7 5.c5! Sc6 - the white pawn is blocked in time, and the white king will not be admitted to the field b7 but - unexpectedly perfect stalemate in the center of the board!

Other options:

3… Sc6 4.Kd3! Sb4 5.Kd4! Kd7 6.Kc5 !! - a paradoxical move.

From the point of view of the Troitsky line, it is necessary to move the pawn forward faster so that it crosses this line as soon as possible, and with this paradoxical move White not only blocks his own pawn, but the next move the pawn will also not be able to move, since it still needs to be freed - two tempos seem to be lost, but:

6… Sc6 7.Kb6! - and a draw, since the white king made his way to the important squareb7 and now he is not afraid of blocking a pawn with a knight on the field With6...

3… Sc3 + 4.Kd4! Sa4 5.c5! Sc6 + 6.Kc4! Sb2 7.Kb5! Kd7 8.Kb6! - and the

white king made his way to the no-man's square again b7 (8… Sc4 + 9.Kb7! S4a5 + 10.Ka8! Kd8 is a correct stalemate)

4… Se2 5.Kc5 !! Kd7 6.Kb6! Sd4 7.Ka7! with a little forking:


7… S4c6 8.Ka8! Se6 9.c5! (9 ... Sxc5 - echo correct stalemate)

9… Sed4 10.Kb7 Kd8

(Another transition to a correct stalemate occurs after 10 … Sb3 11.Ka8! S4a5 - an echo correct stalemate)

11.Ka8! - draw

(11.Ka6? - provoking a new correct stalemate 11 ... Kc7, but 11 ... Kc8! 12.Kb6 Kb8! - and Black drives out the white king with a win,

for example: 13.Ka6 Sb3 14.Kb6 Sba5! 15.Ka6 Kc7! 16.Kb5 Kb7)


7… Kc8 8.c5 !!

(8.Ka8? S8e6 !! 9.c5 Sb5! 10.c6 Sc7 # is the correct checkmate)

8… S4c6 + 9.Ka8! Kd7 - echo correct stalemate.



Example 3

V. Kuzmichev


+ 3 + 2

In this sketch, the solution and the thematic false trail to the motive of

building a no-man's fortress are artistically connected.

1.Shf3? Kf7 2.Kc6! Kg6 3.Se5 +!

(3.Kd5? Kh5! 4.Ke4 Kg4! 5.Kf5 Kh3! 6.S5h4 Kg3!

7.Kd3 Kf2! 8.Kd2 Kf1! - Black created a draw fortress 6.Kf5 Kh3! = =)

3… Kg5

(3… Kh5 4.Sdf3! - White set a screen for Black)

4.Sdf3! Kf5 5.Kd5!- and White, having placed an echo-screen, wins by not letting

black king on the fields g3-f2 But:

1… Kd7 !! 2.Sf5 Ke6! 3.S5h4 - white knight picked up the key field g2

3… Kd5! 4.Kc7 Ke4 5.Kd6 Ke3! 6.Ke5 Kf2!- the black king can't break

into the no-man's corner h1, but he surprisingly builds a no-man's

fortress between two white knights

7.Kf5 Kg3! 8.Kg5 Kh3! 8.Kf5 Kg3!(8.Kxf4 - perfect stalemate)

9.Ke4 Kf2! 10.Kd3 Kf1! 11.Kd2 Kf2! - a draw, since the white knights cannot

leave their squares, and the white king alone cannot oust the black king from this no-man's fortress.

The right decision:

1.Sdf3! Kd7 2.Kb8 !! Kd6

(Black cannot lose the tempo 2 ... Kd8 as White's knights in this case

manage to build a winning fortress, for example

3.Sg2 Kd7 4.Sgf1! Ke6 5.Sd3! Kf5 6.Sf2!)

3.Kc8 !! Kd5

(3… Kc6 4.Kd8! Kd6 5.Ke8! Ke6 6.Kf8! Kf6 7.Kg8! - the black king got on the

screen in the squares g7-g6-g5-f5-e5 and will be forced to lose the necessary tempo to bypass it, skipping white King to capture important squares, white wins)

4.Kd7! Ke4 5.Ke6! Ke3 6.Kf5! Kf2 6.Kg4! - and the white king has time to prevent black to build a no-man's fortress for the king

6… Kf1 7.Kh3! Kf2 8.Kh2 Kf1- White wins, since now on a knight

can help the white king, for example:

9.Sg6 Kf2 10.Se5 Kf1 11.Sg4 Ke2 12.Kg2 - white took a winning position...



Example 4

V. Kuzmichev


= 2 + 3

The match fields in this ratio are very amazing!


(1.Kf6? Sd7 +! 2.Ke7 Sc5! - Black blocked a pawn on

the Troitsky line with a win

1.c5? Sbc6! - Black blocked a pawn on the Kuzmichev square

with a win)

1… Kc3 2.Ke6 !!

On the 2.Ke5? Black is already winning

beautifully 2… Kb3 !! 3.Kd5 Ka3 !! 4.Ke6 Ka4 !!

5.Kd6 Kb4! 6.c5 Sbc6! 7.Kc7 Kb5! 8.Kc8 Ka6! 9.Kc7 Ka7!- and black wins

6.Kc7 Sa6 +! 7.Kb6 Sc5!

2… Kb3

2… Kb4 3.Kd6! Sbc6 4.Kd7 !! Kc5 5.Kc7! -another interesting a

positional opportunity for a draw to the weaker side

4.Kc7? Kc5! 5.Kc8 Kb6! -and Black is already winning

3.Ke7 !! Ka4 4.Kd8! Kb4 5.Kc8! Sbc6 6.Kd7 !! Kc5 7.Kc7! = =

1 ... Sb7 2.c5! Sc6 3.Ke6! Kc3 4.Kd7! Sbd8 (Sba5) 5.Kc7! Kb4

6.Kb6! Kc4 7.Ka6! Kd5 (7 ... Kxc5 is a perfect stalemate)

8.Kb6! Ke6 9.Kc7! Ke7 10.Kc8! - no man's fortress (10… Se6

11.Kb7! Kd7 12.Ka8! Sed8 is a correct stalemate).



Example 5

V. Kuzmichev


+ 3 + 2

The white knights on their own do not manage to block the black pawn

with a win, and then the white king comes to their aid.

1.Kg5! f5 2.Kf4! Kd7 3.Sa5! Kd6 4.Sb6! Kc5

5.Sd7 +! Kb5 6.Sb3!

(6.Sb7? Kc6! 7.Sbc5 Kd6! = =

7.Sdc5 Kb6! = =)

6... Kc4 7.Sd2 +!

(7.Sc1? Kc3! 8.Se2 Kd2! 9.Sg1 Kf1!

10.Sf3 + Kf2! 11.Sde5 Kg2! 12.Sd3 Kh1!

13.Sde1 - correct stalemate)

7… Kd3 8.Sf3! Ke2 9.Kg3! f4 + 10.Kg2!

- and White wins by Kuzmichev's rule.



Example 6



= 2 + 3

A curious draw tactic is used in this study to achieve a draw.

1.Kd2! Ka3 2.Kc3 !! Ka4 3.Kc4 !!

(3.Kd4? Kb5! 4.c4 + Ka6! 5.c5 Sc6 +! 6.Kd5 Kb7! - Black wins according to Kuzmichev's rule)

3… Se3 + 4.Kc5! Sf5 5.c4! Ka5 - and unexpected perfect stalemate

on the middle of the board!

3… Ka5 4.Kc5! Ka6 5.Kd6! - the white king simply drives away the black knight from field c6

5… Sc8 + 6.Kc7! Sa7 7.c4! - but now you can pawn ahead 7… Sf4 8.c5! -

and Black will not be able to block the white pawn, a draw.

The white king first occupied a stable foothold, preventing Black from blocking

the white pawn, and only then the pawn began to move.

I would like to draw your attention to the fact that several examples show a wiser mechanism for achieving a draw for the weaker side, namely, not

to rush forward with a pawn ahead in order to quickly cross the Troitsky line - so you can lose. This intersection must first be well prepared by the

king so that when the pawn crosses the Troitsky line, the king will have an advantage, firstly, to capture the key drawable squares, and secondly, to

disperse the knights and prevent them from blocking the pawn.



Example 7

V. Kuzmichev


+ 3 + 2

An example of how knights can, in an study manner, fight against a

bridgehead captured by the king:

1.Sd4 +! Kc3! 2.Se2 +! Kd2 3.Sf4!

(3.Sg3? Kc2! 4.Sf5 c5! 5.Se3 + Kc3! 6.Sd5 + Kd4! 7.Sc7 Ke3!

8.Sd1 Kd2! 9. ~ c4 = =

6.Sfd1 Kd2! = =

4.Sfe4 c5! 5.Se2 Kb1 !! 6.S4c3 + Ka1!

7.Sc1 c4 = =

6.S2c3 + Ka1! 7.Sd2 c4 = =

And white just surprisingly fails to put a seemingly inevitable checkmate

on the black king)

3… Kc3 4.Se4 +! Kd4 5.Sg3 !!

(5.Sg5? Ke3! 6.Sg2 Kf2! = =)

5… Ke3 6.Sgh5 !! c5 7.Kb2! c4 8.Sd5 +! Kd2 9.Sc3! -

and White wins according to Kuzmichev Likewise 7… Kd4 8.Se2! ~ 9.Sc3! with a win Echo variation:

3… Ke3 4.S4h3! c5 5.Kb2! c4 6.Sd1 + Kd2

7.Sc3! - and White again blocked the pawn according to Kuzmichev with a win.

Or: 5… Ke2 6.Se4! c4 7.Sc3 +! - with a win.

Or: 3… c5 4.Sd5! c4 5.Se4 +! Kc1 6.Sdc3 / Sec3 - White wins according to Kuzmichev.



Example 8

V. Kuzmichev

Chess school, 2018

= 2 + 3

Black will easily block the pawn on Kuzmichev's square with a win.

And White's thought revolves around trying to create a preliminary foothold with the white king, and for this, an attack on knights can be used.

1.Ke4! Sf7

(1… Se7 2.Ke5! Shg6 3.Kf6! = =)

2.Kf5! Sh4 +

And it seems that after:

3.Ke6 Sd8 + 4.Kd7 Sf7 5.Kc7 Se5 6.c4 - whites has achieved it's goal, dispersed the knights, seized the bridgehead, ensuring a drawn pawn advance, but:

3… Sg5 +! 4.Kd5 Kb7! 5.c4 Sg6 !! 6.Kd6 Sf7 +! 7.Ke6 Sd8 +!

8.Kd7 Sc6! -Black still captured the Kuzmichev square with a win. Ot they manage to do it even easier after:

6.c5 Se7 +! 7.Kd6 Sc6!

And where, then, is the promised draw again?

And the draw hits on a completely different end of the world, and you will have to ride to that end of the world on a completely different knight!

3.Kg4! Sg2 4.Kf3! Se1 5.Ke2! Sc2 6.Kd3! - the white king having made a circle returned to his place where it has started, but it did not return alone, but brought his knight with him.

6… Sa3 7.c4 !! - quite unexpectedly, a black knight finds itself in a spacious cage

7… Ka5 8.Kc3!

(8.c5? Se5 +! 9.Ke4 Sc6! 10.Kd5 Kb5! 11.Kd6 Sc4! 12.Kc7 S4a5! - and Black has time to grab the b7 square with his knight at the last moment, winning according to Kuzmichev

13.Kc8 Ka6! 14.Kc7 Ka7!)

8… Sd8 9.Kc3! Sb1 10.Kc2! Sa3 11.Kb3! - draw with eternal pursuit knight

Or: 6… Sa1 7.c4! Sb3 8.c5! Se5 + 9.Kc3! Sa5 10.c6! - and the pawn passed the dangerous field, draw.



Example 9

V. Kuzmichev


+ 3 + 2

And finally, a funny position.

The black king is already in the corner, and the black pawn cannot be held, it would seem, all the signs of a draw. But, theoretically, the no-man's escape corner suddenly becomes a dangerous place.

1.Sd5! Ka2

(1… c4 2.S3b4! c3 3.Se3! c2 4.Sexc2 # is a perfect checkmate)


(Trying to block a pawn will result in a draw

2.Sc3 +? Kb3! 3.Kd2 c4 +! 4.Sc ~ Kb2! - and a draw by Kuzmichev's rule

2.S3f4? c4! 3.Sc3 + Ka1! - and a draw according to Kuzmichev's rule)

2… c4 3.Sc1 +! Ka1 (3 ... Ka3 4.Sc3! - win according to Kuzmichev's rule)

4.Sb4! - if White were to move again now, it would be a draw, but it

is Black's move 4… c3 5.Sb3 # - another regular checkmate with another knight.


It is important to remember that the lower left corner for the king of the weaker side becomes salutary only when the pawn was blocked or blocked at the moment by the knight on the square c4, from that moment on, the corner becomes a saving one, and until that moment, when the king is in the corner, there is a chance to run into a mating attack. With all other pawns on other files, this is exactly what happens - in all corners of the chessboard, the weaker side receives a mating attack and only a bishop pawn that has reached a square of the same c4 and blocked by a knight, creates a draw saving corner beside him for his king.