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This is one of those problems covered in endgame books on corresponding squares.
The win zone for White is if he gets his King to b5, g5, or h5.
The concept of corresponding squares is if one side is trying to win, and the other is trying to draw, if the side trying to win goes to a number, the defending side must be able to go to the corresponding number or he loses.
Since b5 is a winning square for White, The square c4 gets a 1. Since g5/h5 is also a win zone, and it takes 5 moves to get to h4, Black must be able to get to g6 in 5 moves, and so a6 is no good. The king must go to b6, so b6 gets a 1.
Now because d3 is adjacent to c4 (in king move terms) and a step closer to h4, it gets a 2, and Black must go a step closer to g6, so c7 gets a 2.
Now c3 is adjacent to both a 1 and a 2, so it gets a 3. The b7square is the only place from which Black can access 1 and 2, so it gets a 3.
Now d2 is adjacent to 2 and 3, and is 4 steps from h4, so it gets a 4 as does c8.
Now c2 is adjacent to the 2, 3, and 4, so it gets a 5 as does b8.
Now b3 is also adjacent to 1 and 3, so it also gets a 2, and b2 gets a 4.
The back rank for White gets another 2, 3, and 2. The afile for Black repeats the cfile, so a7 is a 2 and a8 is a 4.
So now the trick is the fact that no number on either side is adjacent to itself. Black currently sits on a 2. He can get to a 4 (a8), 5 (b8), 3 (b7), or 1 (b6), but he cannot get to a 2.
Therefore, White wins by going to a 2, which is 1.Kb1!
After 1.Kb1, if 1…Ka8, you go 2.Kb2. If Black tries to be coy and never leave the afile, the 1 becomes the problem. 2…Ka7 3.Kb3 and now 3…Ka8 fails to 4.Kc4 and Black cannot stop Kb5.
If Black goes to the bfile, with a 5 or 3, you go to your 5 or 3 and again, Black will get tempoed out either for the b5square, or the run to g5/h5.
So the biggest question is what if 1.Kb1 Kb6, since White cannot get to a 1 right away. From the 1, Black can get to 2 or 3, but not 4 or 5, so 2.Kc2 and now what for Black? If 2…Kc7, then 3.Kd3 wins. If 3…Kc8, then 4.Kc4. If 3…Kb6 or 3…Kb7, then 4.Ke2 and White gets to h5.
If 2…Kb7, then 3.Kc3 and now Black has is stuck. 3…Kc7 4.Kd3 already we said is losing. 3…Kb6 4.Kc4 and now White gets to b5 or 4…Ka6 5.Kd3 and he gets to h5.
It all has to do with the corresponding squares and 1.Kb1! Is the only way to do it!
1.Ka2? Kb7 (or Kb8) draws
1.Kb2? Ka8 draws
Very good!.
A good lesson in the distant opposition. I always visualized it as the opposition plus a knight move.
This position is covered in Fine’s Basic Chess Endings.
“Black must never remain more than one file to the left of White on his march to the K side. i.e. if the W king is on the e file, the black K must be at least on the d file. Consequently, with the B king on b6, and the W king on c4, black to play loses, since he must play to the b file to prevent Ws Kc4,b5”There’s an obscure oversight pointed out in Fine’s analysis by Norman Whitaker & Glenn E. Harleb’s in Selected Endings. What Fine overlooks is that after 1. Kb2 Kb7 2. Kc3 Kc7 in his preamble, he has exactly the same position as after 5 moves of the correct solution – but he now moves Kc4? and duly gets nowhere.
Norman T. Whitaker – a very strange mix of International Master, attorney, convicted felon/conman. https://texaschess.org/fromtheivyleaguetoalcatrazthelifeofinternationalmasternormantwhitaker/
 This reply was modified 12 months ago by TimmelJD.


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