Addictions

The choirs we are in have a series of appearences the coming time so I'm rehearseling a lot.
I developed a new addiction: ear training and learn how to recognize relative pitch.
I use a very nice freeware program, functional eartrainer, which has an interesting philosophy.
Further I started with rithm exercises, which I found here.

Usually I manage to stay away from mainstream hypes like Sudoku and the like, but this is fun.
The addiction will probably not last very long because the little program works so well that I will achieve my goals in a few weeks.
Margriet couldn't resist the Sudoku hype.

So not much chess lately.

Abbreviations we use

40/120, 20/60, SD/30 - means 40 moves in 2 hours followed
by 20 moves in 1 hour followed by Sudden Death in 30 minutes.
G/30 - means 30 minutes for the whole game
SD/30 - means 30 minutes for the whole game
G/5(5) - means 5 minutes game with 5 seconds per move increment.

ACT - Amsterdam Chess Tournament
B - Bishop
BCC - Boylston Chess Club
BOOC - Bishop Of Opposite Color
BOSC - Bishop Of the Same Color
CC - Correspondence Chess
CCT - Corus Chess Tournament
CET - Chess Endgame Training 2.0 from Convekta
C-K - Caro - Kann
CPT - Chess Position Trainer
CTA - CT-Art 3.0 from Convekta
CT(f)B - Chess Tactics for Beginners from Convekta
CTS - Chess Tactics Server
CVT - Chess Vision Trainer from Fussy Lizzard
DG - Historian of the Knights
DLM - De La Maza
ECO - Encyclopedia of Chess Openings
Elo - Rating based on prof. Arpad Elo's system
FCE - Fundamental Chess Endings from Muller and Lamprecht
FICS - Free Internet Chess Server
FIDE - Federation International Des Echecs
FM - FIDE Master
GM - GrandMaster
GPA - Grand Prix Attack
ICC - Internet Chess Club
ICT - Intensive Course Tactics from George Renko CD 1 and CD 2
IG - Icelandic Gambit
IGM - International GrandMaster
IM - International Master
IQP -Isolated Queen's Pawn
K - King
KG - King's Gambit
KIA - King's Indian Attack
KID - King's Indian Defense
LPDO - Lose pieces drop off
M - Master
MDLM - Michael De La Maza
N - Knight
Na3 - Sodium Attack
NCO - Nunn's Chess Openings
NIC - New In Chess
NM - National Master
OCD - Obsessive Compulsive Disorder
OCL - Online Chess League
OTB - Over The Board; rated game with real opponent
p - pawn
PCT - Personal Chess Trainer
Q - Queen
QC - Quick Chess
QGA - Queen's Gambit Accepted
QGD - Queen's Gambit Declined
QP - Queen's pawn
R - Rook
RD - Rating deviation used by CTS and FICS
RR - Round Robin pairings
Sac - Sacrifice
SCID - Shane's Chess Information Database
SOCES - Secrets Of Chess Endgame Strategy by Lars Bo Hansen
SOPE - Secrets Of Pawn Endings from Muller and Lamprecht
SS - Swiss System pairings
STC - Slow Time Controls at FICS, ICC, USCL
TCT - Tasc Chess Tutor
TD - Tournament Director
TWIC - The Week In Chess
USCF - United States Chess Federation
USCL - United States Chess Live
WCT - Winning Chess Tactics from Yasser Seirawan
WFM - Woman FIDE Master
WGM - Woman GrandMaster
WIM - Woman International Master
YO - year old

I will make a permanent link of this post in my sidebar.
Do you know more abbreviations we frequently use?

Lucena and Philidor

Today I studied Lucena's and Philidor's positions.
It is all very basic and simple.
The weird thing though is that I read a 3-page article of Jeremy Silman about Philidor's position and there were quite a few errors in his analysis. Since I use the Nalimov tablebace this appears to be common among endgame authors.

Indeed with a day or two studying the very basics of rook endgames a big step forward can be made. Once you know what to study, that is.
In some way rook endings feel somewhat more familiar to me than pawn endings. The area is very broad though. I'm glad I finally started with it.

I like to end with an encouraging citation of John Nunn - Secrets of practical chess.
"The ability to play rook and pawn endings well is a great distinction between a master player and a club player. In simultaneous displays, it is noticable how the club players routinely lose completely equal rook endings, and how easely a master manages to escape with a draw from lost rook endings."

The problems start all over again.

Life is picking up, so I decided to reorganize my study somewhat.
I have covered the first 7 chapters of SOPE pretty well.
That means that I assimilated the very basics of pawn endings.
There are 3 more chapters to go to be more complete.
I decided to do them at a later stage.
Now I started full throttle with rook endings.
And all the time consuming problems as I met in pawn endings start all over again.
Just because the lack of educational skills by the authors of endgame books.

The most important part I have to master initially is K+R+p vs K+R.
That is what I'm studying.
I use the following books:

• Theoretical and practical endings - GM Max Euwe
• Fundamental chess endings - Muller and Lamprecht
• Secrets of rook endings - John Nunn

In rook endings there is always a chance that I can sacrifice my rook in order to queen my pawn. That makes only sense if I master the remaining endgame Q vs R.
That is won in 99% of the cases.
25 years ago Max Euwe (as all grandmasters in that time) thought it was a very simple ending which lasted for about 20 moves, so there wasn't much need to make always the best move.
I tried his system against the computer, but I couldn't win a single time.

The reason for this is that the computer has altered this endgame from simple to complex.
The defensive side has much more possibilities than once was thought.
The endgame Q vs R new style is covered by John Nunn in his book Pawnless Endings.
I don't have that book, but Muller and Lamprecht used Nunn's analysis in their book Fundamental chess endings (FCE).
If I use that system to play the endgame against my computer, I am confronted with the fact that not all lines are covered.

So this is my problem:
I would be happy to learn the endgame "old style", the simple way.
Because I assume most opponents don't know how to defense against it in the most resilient way.
There plea two arguments against this:
First I can't train this simple method against the computer so how can I master it?
Second I am started studying endgames in the first place to squeeze points out of the endgame.
Hoping to win drawn endgames and hoping to draw lost endings.
By learning how to defend Q vs R endings I have a good chance to draw such lost ending.

The other option is to learn the Q vs R ending "new style".
The problem with that is that the method as described in FCE is not complete. So I have to complete the analysis myself. Which is probably a lot of work.
The books give me no clue how often it is possible to get this ending on the board. So I don't know how useful it is to learn it.

Once again I have to spill my time to find out everything on my own.
Oh, well.

New Knight

Please all give a Warm Welcome to our newest Knight, Pendrax the Slothful.
He is going to take his time doing the circles.

Dragonslayer seems to have gone 404. Marriage can have a heavy impact.
Enfin we will speak him within half a year as Dragon the Stop-light or so I assume:)

Mastering endgame positions.

Yesterday I trained K+B+N vs K.
I downloaded a few chess engines to emulate different resistance.
It took me about 3 hours to master the ending.

2 - 3 hours is a representative time to master any complex endgame position.
There are about 150 basic pawn endings, so it will take 150 x 3 = 450 hours to master them.
My initial estimation of 6 months comes rather close when studying 3 hours a day (150 days ~ 5 months)

Everybody agrees that rook endings are more difficult than pawn endings.
Probably that means that there are more than 150 basic rook positions and not that the average time to master a rook position exceeds 3 hours.
So far I counted 190 basic rook positions, but I haven't a good insight in the whole area yet.

This makes it very clear why endgame study isn't popular at clublevel.
It just takes a serious amount of time.
Lately I compared two kinds of positions (40 in total). Both with 2 rooks and a lot of pawns.
20 positions were rook endings, 20 positions were mate in #x)
At first sight the positions looked equally complicated.
It was striking that the rook endings costed me an enormous amount of effort while I solved most mates within seconds. Clearly a matter of pattern recognition.

The beginning of a good foundation.

Seven weeks now I'm working on pawn endings and I'm getting a feeling for the basics now.
I spilled a lot of time due to authors with little educational skills, but I worked my way around it.
It is just a matter of familiarizing myself with about 150 basic positions.
It was a lot of work to gather these positions and to sift the non-educational from the useful positions.

I made a cautious start with rook endings. It is already clear that I have to go thru the same process of sifting there (bleah).
It is difficult to play off endings against the computer.
The first time I entered a position in Arena it reacted even before the very first move with "I resign!"
Since the user interface of Arena is almost as bad as Fritz', it took me an hour to find out how to disable resigning.
Most chess engines don't work very well in endgame positions (as Montse found out lately) so I use the Nalimov tablebases. Problem with that is that it often says something like "oh, a resilient defence is lost with mate in 42, but if I walk straight to the middle of the board with my king it is mate in 43", thus choosing the less resilient defense to play against.

If I want to relax from the heavy stuff I do some fun endings against the computer like K+N+B vs K, Q vs N, Q vs R and the like.

An undeserved win

When I'm getting better at endgames it becomes clear that I have to get used to undeserved wins and undeserved draws.
Now I have accepted long games as a good alternative for a crushing kingside attack I play much more relaxed. If an attack reports itself I go for it, if not, I quietly trade off all pieces.
Yesterday I played an important game. Only the queens and a lot of pawns were left on the board.
At a certain moment my opponent could force the exchange of queens. But he misevaluated the remaining pawn ending as lost for him. I knew it was actually lost for me.
So he hadn't the courage to trade off the queens, which gave me the opportunity to queen a pawn of my own and he resigned.

He had played the opening and the middlegame well. It is clear that his performance in the endgame nose dived. I assume that that holds true for most people at club level.
In the land of the blind the one-eyed is king. The so called endgame experts at clublevel are most of the time the one-eyed. So getting some real endgame skills will definitely pay off.

Brushing up the confidence.

To brush up my confidence here a few other examples of simple pawn endings misplayed by the masters. Can you do it better?

diagram 1 Antsigin - N. Zhuravliov USSR 1952





















White to move.
White resigned here in drawn position!

diagram 2 Berger - Mason Breslau 1889





















White to move
White played here 1. Ke4?? and lost the drawn position

diagram 3 I. Rogers - E. Levi Canberra 1996





















Black to move.
Black played here 1. ... b5?? and lost the drawn game.

Finding my way

Finding my way in the endgame jungle takes me an enormous amount of time.
To give an example: previous year it took me 6 weeks to find out where to start.
I would expect there were lots of books out there that could help me to cut down on this time consuming process. A book that saves me 6 weeks research by just telling me where to start and why.
But I haven't found any appropriate books yet.
SOPE is a good reference work, but it does little to cut down on the efforts.

Especially the habit to treat the exceptions without giving the general principles is wide spread among endgame authors. In such cases I have to work my way back.
First I have to realize that the given example is an exception.
Then I have to analyze it. After that I have to generalize it. Then I can formulate the general idea behind it, often only to find something trivial that could be explained to me in a few well chosen sentences.
This process costs me days, every time again.
Take for instance the pathetic case in the following diagram:

diagram 1





















White to move
If it is for showing the beauty of endgame compositions it is great of course.
But to use it in a book that is supposed to learn me the basics of pawn endgames it is simply bizar. It took me hours to solve, and another few hours to find out that I'm not going to get this on the board within a lifetime.
I just don't get the stupidity of the authors to waste my time with such nonsensical cases.
You can compare it with struggling to learn the basics of arithmatic and that they throw in a few integral equations "to show you the beauty of mathematics."
BTW to save your time the answer is 1. Kg7!!@#$%

Because I have no feeling for endgames, I have to investigate everything.
Only after this research I can say if it was worth the effort.
In most of the cases it is not.
But I'm obliged to continue the research, because no author tells me the shortcuts.

Other authors only tell me possible shortcuts, like the Lucena position for instance, without relating it to the whole. So I learn tids and bits without a relating framework. Hence I cannot tell if it is important what they are trying to tell me.
With a big chance to learn something I will probably not use within a lifetime.
Like K+B+N vs K for instance.

To find the right direction I have read a lot of books the past year. To very little avail.

John Watson - Secrets of modern chess strategy.
I don't know what is secret about a written book. He makes a painstaking effort to proof that in modern chess general rules often are broken in specific positions by mere calculation. It's a good book, it doesn't improve my chess though.

John Nunn - Understanding chess move by move. I don't see a way to apply the things I learn here in my own games. It's a good book, it doesn't improve my chess though.

Yasser Seirawan - Winning chess strategies. Most positional subjects he treats are already common knowledge. He has a few interesting chapters on pawn play, which I already incorporated in my play. Improved my chess a little indeed.

Max Euwe - Theoretical and practical endgames. Makes all the errors of endgame books I allready mentioned. I use it as reference. Sort of.

John Nunn - Secrets of rook endings. Excellent reference work. But a lot has to be done before it helps my endgame play. And I mean a lot.

Alexander Kotov - Think like a grandmaster. Good that someone tries to write from a different angle. It's a good book, it doesn't improve my chess though.

Baburin - winning pawn structures. I haven't formed an opinion yet.

Capablanca - Chess fundamentals. I read only a few pages, but I'm already enthousiast. Finally someone who separates the important from the irrelevant!
I had read a lot ABOUT him, and based on that I actually had drawn the conclusion that he was of no use to me.
Wrong! He has a very comprehensible way of writing. And I look forward to read more from him. I wouldn't be surprised if it would boost my chess.

SOPE	Titel of chapter	#repeated
Chapter 1	Kp vs K	4
Chapter 2	Kp vs Kp	4
Chapter 3	Race of the passed pawns	3
Chapter 4	Small number of pawns	1
Chapter 5	Unique features rook's pawn	1
Chapter 6	Fortresses, stalemates, underpromotion	1

I'm busy with chapter 4 from SOPE now.

Getting grip on K+p vs K+p

I worked for a month and a half on K+p vs K+p (chapter 2 of SOPE).
Chapter 2 is not about the race of the passed pawns (=chapter 3) or the rook pawns (=chapter 5) but only about two pawns on the same file or on adjacent files.
Finally I am beginning to feel that I get some grip on these positions.
I hadn't expected that mastering these seemingly simple positions would cost me so much time.

But underestimating simple looking positions is common all the way up to grandmasters.
Take for instance the next diagram from a game between I. Rogers and A. Shirov (Groningen 1990):

diagram 1.





















White to move and win.
There was no time trouble involved.
Shirov (black) had headed for this position because he thought it was an obvious draw.
Can you find the win for Ian Rogers?

Now I'm so far that in these positions the first move I look at is often the right move.
I'm still not fail proof yet but that will come in the near future.

So time is ripe to get along with K+pp vs K+p.
At least I have the tools now to judge all the transitions to K+p vs K+p.

Man, this is hard work!

Interference of the kings

Montse asked me a few days ago if there wasn't a simple method to map out the interference by two kings. Since then I played thru about a hundred postions.
From that I distilled two simple empyric rules.
I can't give a 100% guarantee that everything is covered with these two rules but at least it comes pretty close.
I will put them to the test with real positions in the near future.

The interference starts at the moment the kings are close to each other, at one square distance.
I haven't taken a look yet what happens when the kings are approaching each other.
There are 12 positions with the kings close two each other (12 hours of the clock).
I have analysed two positions: 3 o'clock and 4 o'clock.
All other positions can be derived by mirroring.

The two kings have each their own target where they head for.
On the paths to their targets the kings can shoulder away each other.

diagram 1





















White to move.
red = the target of the black king
blue = a blue square can be conquered by the white king in time. That is to say, at the moment that black conquers the red square.
yellow = the white king comes one move to late due to being shouldered away. Would be blue if there wasn't a black king
none = out of reach of the white king, no matter if there is a black king in the way or not.
position = 3 o'clock

The distance from the black king to its target (red square) = 3
Diameter of the blue area = 3 + 1 (premove) = 4
Distance of the yellow file from the white king = 4

Empyric rule #1 (position 3 o'clock):
The yellow area with targets that white cannot reach in time has the shape of a file at distance x+1 from the white king.

Where x = distance black king to target.
No matter where the target of the black king is.
That is to say, as long as it is on the files a, b, c or d

Another example:

diagram 2





















White to move.
x = black king - red square = 4
yellow file = 4 + 1 = 5 moves away from the white king
ofcourse the red square is surrounded by yellow squares, since the white king can't reach them due to the black king.
So that is all very simple.

Now let's have a look at the 4 o'clock position.

diagram 3





















red = target black king
blue = targets white, white is in time
yellow = white comes 1 move to late due to shouldering
none = out of reach anyway

rule #2 (4 o'clock):
The yellow area can be calculated exactly the same way as in rule #1, with one difference:
The southern part (1 rank below the red target) can be reached in time, hence that part of the file is blue.

And again, the red target can be anywhere on the a, b, c or d file!
That's all folks!

Dance of the Kings

To get a better picture of the interference between the kings in the endgame, it is useful to have a few expedients for better communication.
The first is the windrose.

diagram 1.





















A king can walk in only 8 directions. From now on I call these directions as their name in the windrose.

Most activity in endings take place with the kings circleing around each other at one square distance. It is useful to have a convenient naming system for all 12 different rotation positions.

diagram 2





















To that end we think of it as a clock with the black king (the opponent) in the middle.
The white king is at the end of the short hand.
Horizontal opposition is 3 o'clock or 9 o'clock.
Vertical oppositon is 6 or 12 o'clock and so on.

The movements of the kings can be described with vectors (length = # moves, direction = according the windrose) and rotation.

With these tools we can easy describe new rules.

diagram 3.


begin position
(12 o'clock)









end position
(12 o'clock)


Black to move.
Begin position = Kg8, Kg6 (= position 12 o'clock)
When the black king walks south west and the white king follows at 1 square distance, there can be no rotation. When the black king arrives at b1, the white king will still be in the 12 o'clock position.

Rule #1: when two kings walk in the same direction at close distance, they can't encircle each other.

And derived from rule #1:
Rule #2: When you want to encircle the enemy king, you can only do so at the expense of a tempo.

diagram 4.


Begin position
12 o'clock









End position
1 o'clock



White to move.
Blue = move 1
Red = move 2
Green = move 3 - 5

White uses a tempo to rotate from 12 o'clock to 11 o'clock.
Black starts to walk in SE-direction.
White follows at 1 o'clock, which will be the endposition too.

Let's have a look at a real position.

diagram 5.





















White to move and draw.
It is clear white can't keep his pawn on the board.
How many extra tempo's has white?
The first spare tempo is the start tempo.
When both kings walk in the same direction, there can be no rotation.
The last 2 moves of black are in western direction, which provides white with another spare 2 tempo's. Don't bother to much about this, a little further this will be shown in a few diagrams.

Thanks to Takchess we now all know that you have to imagine the final position first.
See diagram 6.

diagram 6.


Begin position
12 o'clock







Final positon.
Kings 90 degrees rotated
3 o'clock



When the white king reaches the c2 right after the black king has eaten the white pawn, white can hold the draw.
In the start position the white king is at 12 o'clock and in the final position at 3 o'clock.
So white has to rotate with the black king in the center 3 times (3 x 30 degrees = 90 degrees).
At the expense of 3 spare tempi.
Hence the start move of white must be 1. Kh8!!

I'll give you all the important moves in the next diagrams.

diagram 7





















1. Kh8!!
1st rotation from 12 o'clock to 1 o'clock at the expense of the start tempo.
See diagram 7


diagram 8.





















2. ... Kf5
3. Kg7 Ke4
4. Kf6 Kd3
5. Ke5 Kc2
6. Kd4
Translation over 4 moves in SW-direction.
No rotation. See diagram 8


diagram 9.





















6. ... Kb2
7. Kd3
The translation of the black king to the west (Kc2-Kb2-Kxa2) will cost him 2 additional tempo's
The white king rotates around the black king from 1 o'clock to 2 o'clock at the cost of the first additional tempo. See diagram 9.

diagram 10.





















7. ... Kxa2
8. Kc2 =
The white king rotates further from 2 o'clock to 3 o'clock.
Draw!
See diagram 10.

Can you see for yourself now why 1. Kf8? loses the game?

Composition generator "Grigoriev"

I tested my empyrical formula's in a lot of situations with the Nalimov tablebases.
They didn't let me down.
Let me show you a random situation with two blocked pawns on the same file.

diagram 1





















Where would you put the white king if you had the choice?
First you have to draw the draw zone.

diagram 2





















The basic measure is the distance between the black king and the white pawn = 4 moves
The draw zone is based on the fact that the white king conquers the key square when the black king snacks the white pawn.

Center of the draw zone = c4 = key square black pawn
Diameter of the draw zone = 4 + 1 = 5 moves (white can afford to lose one tempo)

Then you draw the win zone.

diagram 3





















Center of the win zone is the black pawn.
Diameter win zone = 4 moves.

Where you don't win or draw, you lose.

diagram 4





















Red = loss zone
Then you have to find out where one king shoulders the other away when they walk on their ideal paths to their respective targets.

diagram 5





















Yellow = interference zone
When the kings collide the impeded king loses a tempo.
If the white king hinders the black king, the win zone increases by one move.
When the black king hinders the white king, the loss zone increases by one move.

diagram 6.





















The only thing you have to do now is to drop your white king on an interesting place and you have 3 brand new Grigoriev-like studies.
With hardly any calculation!
One solution guaranteed!
White to move.

This is of course a relative simple situation since the pawns are sitting ducks.
At the moment I'm trying to do the same with one or more ranks between the pawns.
I don't understand all aspects yet, but I allready found a few interesting things.
Heck, at this pace I'll never reach Lucena's position!
Btw, I ordered Secrets of rook endings from John Nunn.

Endgame decomposer

Before I move on to the more complicated pawn endings with 3 pawns or more, I want to have a clear picture of all endgames with 2 pawns.
Since a position with 3 pawns can become a position with 2 pawns by saccing or conquering a pawn.

Why am I making such a big issue of this?
Yesterday I came across a study of TCT with Kpp vs Kp.
The author of TCT is of master level.
He made fun of the majority of people who would move the h-pawn in that position, which would result in a draw instead of a win.
He told that moving the g-pawn instead would be an easy win.
So I checked it with my Nalimov tablebase and that move was a draw too!
The only winning move was a counter intuitive king move he hadn't mentioned.

So if a master, who has spend hours at the preparation of a study position for his students, still makes such errors, these positions must not be underestimated!
Margriet and I couldn't find the logic behind the illogical-looking king move.
But it was clear that knowledge of the underlying 2-pawn positions was essential.
And thus I went back to the basics.

My previous post showed an easy looking yet complicated position with two pawns.
The tree of analysis of that position consists of 5 branches and 40+ leaves.
That is an enormous amount of calculation that has to be done if you encounter such positions in an OTB game.
I guess that a lot of people will dismiss this position by counting the material and say that this is a dead draw. And indeed, if both sides overlook the subtleties of the position, it maybe will proof to be a draw!

If such an easy position leads allready to so much calculation, how am I ever going to master this? That's why I decided to look after a method to prune the tree of analysis drastically.
I think I have found at least a part of this method.
In my last post I showed you the following diagram.

diagram 1





















White to move and draw.
green = white wins
blue = draw
red = black wins

White has of course to move to the blue zone with 1.Kg3!!
How can these zones be determined in any given position of Kp vs Kp?
I did this with the help of my Nalimov endgame tablebase.
But since that isn't allowed in an OTB game, we have to find another method.
This is my try.

Let's have a look at the different underlying zones.
To begin with the green winning zone.

diagram 2





















White to move.

If white steps on a green square he will be in time to conquer the black pawn.
This is a position where I have neglected the interference between the white and the black king.
The geometry of the zone is quite determined by the distance between the white king and black pawn (5), compared with the distance between the black king and the white pawn (5)
Notice that the squares a3, a4, a5, b3 and c3 omit from the winning zone.
That's because I haven't taken the interference between the kings into account.
If I do take that into account, the white king will shoulder away the black king and these squares have to be added to the green zone.

Let's have a look at the underlying draw zone.
In this position I neglect the interference between the kings and I neglect the fact that white can win by conquering the black pawn.
The basis of the position is the keysquare b4.
If white reaches b4 at the moment the black king snacks the white pawn, white can hold the draw.

diagram 3





















White to move.
blue = draw zone
yellow = key square

The geometry of the position depends solely on the distance between the white king and the key square (6) and the distance between the black king and the white pawn (5)
The former (6) can be greater than the latter (5) because white starts to move, but the black king is allowed to reach b6 before the white king reaches b4, without disturbing the draw.

Now we have to look at a more complicated point, the interference of the kings.

diagram 4





















red = shortest path of the black king to the white pawn (5 moves)
yellow = the "shoulder of the black king"

There are a lot of ways to get to the white pawn. All of them with a length of 5 moves. All of them go thru the red zone. The path that hinders the white king the most will be along the blue arrow. I call this the ideal path of the black king.
The white king has 141 methods to get to b5 in 6 moves.
The ideal path of the white king is to stay away from the ideal path of the black king as far as possible.

diagram 5





















green = key square
yellow = the zone where the kings could interfere.

To come any further I formulate the following hypothesis:

If the ideal path of the black king doesn't interfere with the ideal path of the white king, the black king has to leave his ideal path (= make it longer) to cause problems for the white king.
The white king can answer this by leaving his ideal path too (= make it longer), without changing the result.

I'm not going to try to prove this hypothesis. In the oncoming studies of Kp vs Kp I will see in an empyrical way if it is true and useful or not.

I hope you could stay awake :)