Answers.Body: <p>The g6-break comes to mind, as in</p>
  <pre><code>[fen &quot;8/5ppp/8/5PPP/8/8/8/8 w - - 1 0&quot;]
  
  1.g6 hxg6 2.f6 gxf6 3.h6
  </code></pre>
  <p>We can adapt this. Here, White wins starting with the break 1.g6</p>
  <pre><code>[fen &quot;8/pppppppp/8/4PPPP/8/8/8/k1K5 w - - 0 1&quot;]
  
  1.g6 hxg6 2.f6 exf6 3.exf6 gxf6 4.h6
  </code></pre>
  <p>Same idea in the center, now five pawns are needed, but we get a minority attack on either side.</p>
  <pre><code>[fen &quot;8/pppppppp/8/2PPPPP1/8/8/8/k1K5 w - - 0 1&quot;]
  
  1.e6
  </code></pre>
  <p>This breaks works on fifth row as well, given Black can neither check (wK on c3) nor escort his own pawn in due time (wK on c1).</p>
  <pre><code>[fen &quot;8/8/pppppppp/8/2PPPPP1/1K6/8/k7 w - - 0 1&quot;]
  
  1.e5 dxe5 2. f5 exf5 3. gxf5 gxf5 4. d5 cxd5 5. cxd5 f4 6. Kc3
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-10-05T19:44:17.837Z
Answers.Text: Amusingly, on an infinite board White can win with a constant 2/3 of Blacks pawns. (I doubt this value can be improved.)
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-10-06T13:10:00.457Z
Answers.Text: @HaukeReddmann not sure what this means: as the second diagram shows, five connected pawns on the 5th rank beat arbitrarily many pawns on the 7th (assuming that Queening first by a few tempi is a winning advantage).
Answers.UserDisplayName: Noam D. Elkies
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-10-06T19:03:31.207Z
Answers.Text: @NoamD.Elkies: Ah, I see. Any pawn two rows away from the last white obviously has no relevance in the fight. My 2/3 assumes these pawns don't count.
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-10-05T11:18:16.407Z
Answers.LastActivityDate: 2022-10-06T09:00:13.197Z
Answers.LastEditDate: 2022-10-06T09:00:13.197Z
Answers.UserDisplayName: Peter Fischer
Body: <p>Consider the following well-known motif:</p>
  <pre><code>[FEN &quot;8/1k6/5pp1/7p/7P/6P1/8/7K b - - 0 1&quot;]
  </code></pre>
  <p>Black wins even with Ka8, but being true Patzer:
  1...g5?? 2.g4! Kmoch's crosslever, 1-0.</p>
  <p>Now in the critical position after 1...g5, pawns are hanging. Can you conceive a position with the following conditions:</p>
  <ul>
  <li>White, a pawn down, to move wins by a breakthrough</li>
  <li>Everything beyond a pawn-free file isn't counted in &quot;pawn down&quot;</li>
  <li>No free White pawns</li>
  <li>If a White pawn has passed a Black one, the latter isn't counted for &quot;pawn down&quot; either</li>
  <li>Black doubled pawns are OK</li>
  <li>No captures possible in the position</li>
  <li>Kings far away, play no role</li>
  </ul>
  <p>(Note: After thinking ten seconds, I know the answer, but I still leave it as a puzzle for you. Feel free to think about <em>how many</em> pawns White can be down for a sufficient breakthrough.)</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-10-05T02:15:41.350Z
Comments.Text: Your diagram seems to be upside down.
Comments.UserDisplayName: bof
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-10-05T07:25:50.850Z
Comments.Text: @bof: Yup, that seems to be the default when BTM. I'd wish the replayer had a little black or white dot or arrow to indicate the right to move and the orientation of the board.
Comments.UserDisplayName: Hauke Reddmann
Tags: <puzzles><pawn-endgame>
Title: Minority Breakthrough
ThreadText: # Minority Breakthrough
  
  (Asked by: username_0 on 2022-10-04)
  
  Consider the following well-known motif:
  
  ```
  [FEN "8/1k6/5pp1/7p/7P/6P1/8/7K b - - 0 1"]
  
  ```
  
  Black wins even with Ka8, but being true Patzer:
  1...g5?? 2.g4! Kmoch's crosslever, 1-0.
  
  Now in the critical position after 1...g5, pawns are hanging. Can you conceive a position with the following conditions:
  
  * White, a pawn down, to move wins by a breakthrough
  * Everything beyond a pawn-free file isn't counted in "pawn down"
  * No free White pawns
  * If a White pawn has passed a Black one, the latter isn't counted for "pawn down" either
  * Black doubled pawns are OK
  * No captures possible in the position
  * Kings far away, play no role
  
  (Note: After thinking ten seconds, I know the answer, but I still leave it as a puzzle for you. Feel free to think about *how many* pawns White can be down for a sufficient breakthrough.)
  
  **Tags:** puzzles, pawn-endgame
  
  > **Comments:**
  > **username_1**: Your diagram seems to be upside down.  
  > **username_0**: @username_1: Yup, that seems to be the default when BTM. I'd wish the replayer had a little black or white dot or arrow to indicate the right to move and the orientation of the board.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2022-10-05. Score: 4 | Accepted answer
  
  The g6-break comes to mind, as in
  
  ```
  [fen "8/5ppp/8/5PPP/8/8/8/8 w - - 1 0"]
  
  1.g6 hxg6 2.f6 gxf6 3.h6
  
  ```
  
  We can adapt this. Here, White wins starting with the break 1.g6
  
  ```
  [fen "8/pppppppp/8/4PPPP/8/8/8/k1K5 w - - 0 1"]
  
  1.g6 hxg6 2.f6 exf6 3.exf6 gxf6 4.h6
  
  ```
  
  Same idea in the center, now five pawns are needed, but we get a minority attack on either side.
  
  ```
  [fen "8/pppppppp/8/2PPPPP1/8/8/8/k1K5 w - - 0 1"]
  
  1.e6
  
  ```
  
  This breaks works on fifth row as well, given Black can neither check (wK on c3) nor escort his own pawn in due time (wK on c1).
  
  ```
  [fen "8/8/pppppppp/8/2PPPPP1/1K6/8/k7 w - - 0 1"]
  
  1.e5 dxe5 2. f5 exf5 3. gxf5 gxf5 4. d5 cxd5 5. cxd5 f4 6. Kc3
  
  ```
  
  > **Comments:**
  > **username_0**: Amusingly, on an infinite board White can win with a constant 2/3 of Blacks pawns. (I doubt this value can be improved.)  
  > **username_3**: @username_0 not sure what this means: as the second diagram shows, five connected pawns on the 5th rank beat arbitrarily many pawns on the 7th (assuming that Queening first by a few tempi is a winning advantage).  
  > **username_0**: @username_3: Ah, I see. Any pawn two rows away from the last white obviously has no relevance in the fight. My 2/3 assumes these pawns don't count.  
  
  ---
  
  