user618: Black here has 3 weak pawns; if any one of them falls - he will very likely lose the game as well. If he lets White king break through via c4-c5(d5) - White will also likely win. c5 pawn is a passed pawn, but is not going anywhere. Evaluations like "White is better" are not precise enough - is it possible via analysis to prove that White definitely wins? Or that Black definitely holds? Variations or plans for both sides are welcome.
White to move:
2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 64


user23298: Analysis with Houdini 6.02 signal a win for White(+4.53)at d= 41/103 after more than a quarter hour of analysis. The winning method is complicated though , and White should start with 1 Bg2. Thanks. A demo (analysis included for pure curiosity in my program )with New Engine gives + 5.58 at d= 56 , which confirms the endgame is slowly winning for White.

user17: This resembles another position referred to as the "Iron Duke." Based on my knowledge of the other position, I'd say that Black can draw.
The reason is that all the potential entry points (on the fourth rank) for the white king are either occupied or "covered" by Black.  There is a (blocked) White pawn on a4, b4 and d4 are covered by the Black pawn on c5, c4 is covered by the Black bishop, e4 and g4 are covered by Black pawns, and f4 and h4 have (blocked) White pawns. The Black bishop can continue to cover c4 while moving along the diagonal to protect whatever pawn White's bishop attacks (except for the one on a6, which the king can protect). 
If White tries for an exchange of Bishops on c4, the Black K "toggles" between c6 and d6, and the White king still can't break through.
Black can lose by moving his c pawn or wrongly moving his bishop. But as the position now stands, White can't force a win. Neither can Black for that matter. If he moves his (passed) c pawn, he has losing chances.

user1485: With new analysis from user AlwaysLearningNewStuff, we can conclude this is a win for white,
The first critical position appears after 1.Be2 Kb7.          
      [FEN "2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 1"]

       1.Be2 Kb7

Black defends everything at the moment, but the only moves that keep the position are Kb7-a7-b7. A simple triangle maneuver puts black in zugzwang.
     [FEN "2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 1"]

       1.Be2 Kb7 2.Kd2 Ka7 3.Kc2 Kb7 4.Kc3 

Now black is in zugzwang and loses. The following lines illustrate this:
[FEN "2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 1"]

1.Be2 Kb7 2.Kd2 Ka7 3.Kc2!+- Kb7 4.Kc3 ( 4...Ka7 5.Bc4 Be8 ( 5...Bg6 6.Be6 Kb7 7.Kc4 Kc6 8.Bd5+ Kd6 9.Bb7+- Bf7+ 10.Kd3 )( 5...Bxc4 6.Kxc4 Kb7 7.Kxc5+- ) 6.Be6 Bg6 7.Kc4 Kb7 8.Kxc5+- )( 4...Bg6 5.Bf3+! Kc7 6.Kc4 Kd6 7.Bb7+- Bf7+ 8.Kd3 )( 4...Be8 5.Kc4 Kc6 6. Bf3+ Kd6 7.Bb7+- Bf7+ 8.Kd3 ) c4! 5.Kd4! ( 5.Bxc4? Bxc4 6.Kxc4 Kc6= ) 5...Kc7 ( 5...Kc6? 6.Bxc4 Bxc4 7.Kxc4+- )( 5...Kc8? 6.Bxc4+- Bxc4 7.Kxc4 Kc7 8.Kc5 ) 6.Bf3! Kc8 7.Bd1!!+- Kc7 ( 7...Kd7 8.Bc2! Be6 9.Ke5+-)( 7...Kb7? 8.Bc2! Be6 9.Ke5 Bd7 ( 8... Bg8 9.Bxf5+- ) 10.Kd5!+- ) 8.Bc2! Be6 9.Ke5 Kd7 ( 9...Bd7 10.Kd5!+- ) 10.Bd1! Bf7 11.Kxf5 Kd6 12.Kf6!+-

The important idea is that after 4...c4! White uses 6.Bf3! to restrict Black king. After 6...Kc8 White repositions Bishop on c2 with the move 7.Bd1!! so the decisive zugzwang can be achieved. After 12.Kf6! the rest is a matter of technique.
The analysis is kept as brief as possible in order to preserve space, since the rest of the winning moves can be easily found with an engine. 

618: <p>Black here has 3 weak pawns; if any one of them falls - he will very likely lose the game as well. If he lets White king break through via c4-c5(d5) - White will also likely win. c5 pawn is a passed pawn, but is not going anywhere. Evaluations like "White is better" are not precise enough - is it possible via analysis to prove that White definitely wins? Or that Black definitely holds? Variations or plans for both sides are welcome.</p>

<p>White to move:</p>

<pre><code>2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 64
</code></pre>
 23298: <p>Analysis with Houdini 6.02 signal a win for White(+4.53)at d= 41/103 after more than a quarter hour of analysis. The winning method is complicated though , and White should start with 1 Bg2. Thanks. A demo (analysis included for pure curiosity in my program )with New Engine gives + 5.58 at d= 56 , which confirms the endgame is slowly winning for White.</p>
 17: <p>This resembles another position referred to as the "Iron Duke." Based on my knowledge of the <em>other</em> position, I'd say that Black can draw.</p>

<p>The reason is that all the potential entry points (on the fourth rank) for the white king are either occupied or "covered" by Black.  There is a (blocked) White pawn on a4, b4 and d4 are covered by the Black pawn on c5, c4 is covered by the Black bishop, e4 and g4 are covered by Black pawns, and f4 and h4 have (blocked) White pawns. The Black bishop can continue to cover c4 while moving along the diagonal to protect whatever pawn White's bishop attacks (except for the one on a6, which the king can protect). </p>

<p>If White tries for an exchange of Bishops on c4, the Black K "toggles" between c6 and d6, and the White king still can't break through.</p>

<p>Black <em>can</em> lose by moving his c pawn or wrongly moving his bishop. But as the position now stands, White can't force a win. Neither can Black for that matter. If he moves his (passed) c pawn, he has losing chances.</p>
 1485: <p>With new analysis from user AlwaysLearningNewStuff, we can conclude this is a win for white,</p>

<p>The first critical position appears after <code>1.Be2 Kb7</code>.          </p>

<pre><code>      [FEN "2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 1"]

       1.Be2 Kb7
</code></pre>

<p>Black defends everything at the moment, but the only moves that keep the position are <code>Kb7-a7-b7</code>. A simple triangle maneuver puts black in zugzwang.</p>

<pre><code>     [FEN "2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 1"]

       1.Be2 Kb7 2.Kd2 Ka7 3.Kc2 Kb7 4.Kc3 
</code></pre>

<p>Now black is in <em>zugzwang</em> and loses. The following lines illustrate this:</p>

<pre><code>[FEN "2k5/5b2/p7/P1p2p1p/P4P1P/2K2B2/8/8 w - - 0 1"]

1.Be2 Kb7 2.Kd2 Ka7 3.Kc2!+- Kb7 4.Kc3 ( 4...Ka7 5.Bc4 Be8 ( 5...Bg6 6.Be6 Kb7 7.Kc4 Kc6 8.Bd5+ Kd6 9.Bb7+- Bf7+ 10.Kd3 )( 5...Bxc4 6.Kxc4 Kb7 7.Kxc5+- ) 6.Be6 Bg6 7.Kc4 Kb7 8.Kxc5+- )( 4...Bg6 5.Bf3+! Kc7 6.Kc4 Kd6 7.Bb7+- Bf7+ 8.Kd3 )( 4...Be8 5.Kc4 Kc6 6. Bf3+ Kd6 7.Bb7+- Bf7+ 8.Kd3 ) c4! 5.Kd4! ( 5.Bxc4? Bxc4 6.Kxc4 Kc6= ) 5...Kc7 ( 5...Kc6? 6.Bxc4 Bxc4 7.Kxc4+- )( 5...Kc8? 6.Bxc4+- Bxc4 7.Kxc4 Kc7 8.Kc5 ) 6.Bf3! Kc8 7.Bd1!!+- Kc7 ( 7...Kd7 8.Bc2! Be6 9.Ke5+-)( 7...Kb7? 8.Bc2! Be6 9.Ke5 Bd7 ( 8... Bg8 9.Bxf5+- ) 10.Kd5!+- ) 8.Bc2! Be6 9.Ke5 Kd7 ( 9...Bd7 10.Kd5!+- ) 10.Bd1! Bf7 11.Kxf5 Kd6 12.Kf6!+-
</code></pre>

<p>The important idea is that after <code>4...c4!</code> White uses <code>6.Bf3!</code> to restrict Black king. After <code>6...Kc8</code> White repositions Bishop on <code>c2</code> with the move <code>7.Bd1!!</code> so the decisive <em>zugzwang</em> can be achieved. After <code>12.Kf6!</code> the rest is a matter of technique.</p>

<p>The analysis is kept as brief as possible in order to preserve space, since the rest of the winning moves can be easily found with an engine. </p>
