Originally posted by David113My guess is a king move.🙂
[fen]6k1/1p6/1P1p4/3p4/3Pp2p/4P2p/1K5P/8[/fen]
White's turn. What is the only drawing move?
Looking at it I think the white king has to track the black king and always be able to end up stopping the black king from penetrating into white's territory, so I would start with 1 K-a2 which puts the white king on the opposed distant diagonal, then follow the king in whichever direction the black king goes.
Okay, there are only king moves available. On first look, the key square is c6 here. Looking at whites "area" from a1 to a5 and b1 to b5 it seems that black has the same "area" rotated by 90° from d7 to h7 and from d8 to h8. Therefore White cant move into opposition right now. But I guess all this doesnt matter at all because:
White's king will move from a4 to b4 to a4 forever. As soon as Black moves Kc6 White answes Ka5 and Black has to go back. So the first move doesnt matter. What did I miss?
edit: the 90° rotation wasnt correct there. gotta be mirrored like sonhouse said. however, i still think my above answer shows you can move whatever you want 🙂
Ka1. White needs to make sure he can reach the 'corresponding' squares of the black king (extension of 'remote opposition'😉 in order to be able to counter every black king move to either side: Kg4 must be met with Ke2, and Kc6 must be met by Ka5. I am not an expert, but I think this is an example of 'heterodox' opposition: away from the pawns, either king must try to chose opposite colour to achieve his objective (draw for white, win for black). And b2 and f7 are the starting points to either side.
A problem by Ebers (1930), though in the position I have the White king is at c3, and the Black at h7. And after 1. Kb2 Kg8 we arrive at the posted position. Mephisto's answer is right, White need to preserve access to corresponding squares in order not to let the Black king in its half
Originally posted by Mephisto2Can you show me where Ka1 draws whereas ka2 loses? I don't see a differance. Like you said, if black gets to g4, from Ka2 the king also gets to e2 just as easily and likewise can still get to b5 after black gets to d7. I can see triangulation ways of getting to b5 or e2 from almost any position on the white side. For instance, 1 kc1, still stops the black king from entering even though it can't get to b5, getting to a5 still stops the black king entering even after getting to c6.
Ka1. White needs to make sure he can reach the 'corresponding' squares of the black king (extension of 'remote opposition'😉 in order to be able to counter every black king move to either side: Kg4 must be met with Ke2, and Kc6 must be met by Ka5. I am not an expert, but I think this is an example of 'heterodox' opposition: away from the pawns, either king ...[text shortened]... ive (draw for white, win for black). And b2 and f7 are the starting points to either side.
Originally posted by sonhouseLet's try a few lines.
Can you show me where Ka1 draws whereas ka2 loses? I don't see a differance. Like you said, if black gets to g4, from Ka2 the king also gets to e2 just as easily and likewise can still get to b5 after black gets to d7. I can see triangulation ways of getting to b5 or e2 from almost any position on the white side. For instance, 1 kc1, still stops the black k ...[text shortened]... can't get to b5, getting to a5 still stops the black king entering even after getting to c6.g
1) after 1.Ka1, black can try several things
1a) ... Kf7 2.Kb2! Ke7 (Kf6 then Kc2; Ke6 then Kc3; Kg6 then Kc1; Kf8 then Ka2 etc...) 3.Kb3 Kd7 4.Kb4 and white is in time to reach a5 4. ... Ke6 5.Kc3 and white is in time to follow and reach e2 in time
1b) ... Kg7 2.Kb1 Kf6 3.Kc2 and if Kg5 then Kd1 or if Kf5 then Kd2 or if Kg6 then Kc1 and alternatively if Ke7 then Kb3 etc...
Note that white mimicks the black king move on the corresponding square of opposite colour.
Now let's try 1.Ka2. Black will immediately play 1. ... Kf8, heading towards the kingside and taking the opposite colour opposition. Now if
2a) Kb2 Kf7 3.Kc3 (check out other options are analog) Ke6! and white is in zugzwang. He has to make a choice towards either kingside or queenside, and leave the other path open. Black wins.
or if
2b) Ka3 then Ke8 (or Kg7) 3.Kb3 Ke7 4.Kc3 Ke6! and white is in zugzwang again.
Note that black here choses the opposite colour square each move.
This is not comprehensive, but it should give an idea.
Originally posted by Mephisto2That is one interesting problem. I still have to digest it all. I went through most of your notes and still am not totally convinced a1 is the only way to a perfect draw but I am studying it you can be sure!
Let's try a few lines.
1) after 1.Ka1, black can try several things
1a) ... Kf7 2.Kb2! Ke7 (Kf6 then Kc2; Ke6 then Kc3; Kg6 then Kc1; Kf8 then Ka2 etc...) 3.Kb3 Kd7 4.Kb4 and white is in time to reach a5 4. ... Ke6 5.Kc3 and white is in time to follow and reach e2 in time
1b) ... Kg7 2.Kb1 Kf6 3.Kc2 and if Kg5 then Kd1 or if Kf5 then Kd2 or if Kg6 ...[text shortened]... he opposite colour square each move.
This is not comprehensive, but it should give an idea.
This seems to be the most complicated series of king manouvers that can happen on the board with only pawns and kings. Do you know of any more of equal but differant kinds of manouvers leading to draw?
Originally posted by sonhouseIt looks more complicated than it is. The starting point is that there are two threats (invasion points for the black king), and there are pivotal squares between the two: a2 and f7 are two of them. If white cannot move Ka2 as a response to Kf7, then he is in zugzwang and has to make a choice, leaving the other way open. From a2 and f7, you can easily derive the other 'corresponding' squares. The idea is not different from the basic 'opposition', extended to 'remote opposition' and 'corresponding squares' up to 'corresponding domain', the latter being the set of connected corresponding squares. The fact that in this case the corresponding squares are of opposite colour, makes it in fact very easy to play this without much thinking, once you determined the correspondence. You could blitz it out.
That is one interesting problem. I still have to digest it all. I went through most of your notes and still am not totally convinced a1 is the only way to a perfect draw but I am studying it you can be sure!
This seems to be the most complicated series of king manouvers that can happen on the board with only pawns and kings. Do you know of any more of equal but differant kinds of manouvers leading to draw?
There are several different maneuvers that are interesting to study: diagonal moves (Reti maneuver), blocking squares, all types of opposition and their interaction, .... Most 'pawns and king' studies involve one or more of them. Many have been posted here before by others.