Answers.Body: <p>It takes four captures to promote a pawn in monochromatic. Only 15/16 of each sides' pieces are capturable. Thus, in theory, each side may only promote three pawns at most, or a total maximum of six combined.</p>
  <p>And it is indeed possible, albeit tricky. Utilizing en passant to change a pawn's lane color is a crucial aspect.</p>
  <pre><code>[FEN &quot;&quot;]
  
  1. a4 a5 2. b4 b5 3. bxa5 bxa4 4. g4 g5 5. h4 h5 6. Rh3 Rh6 7. Rb3 Rb6 8. axb6 axb3 9. bxc7 bxc2 10. cxb8=N Qb6 11. d4 Qc5 12. dxc5 d5 13. cxd6 Bf5 14. Bf4 Bh7 15. Bh2 gxh4 16. gxh5 Ra6 17. Ra3 cxb1=N 18. dxe7 Rg6 19. Rg3 hxg3 20. hxg6 gxh2 21. gxh7 f5 22. e4 fxe4 23. f4 exf3 24. Be2 fxe2 25. exf8=B hxg1=N 26. hxg8=N exd1=Q
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-21T13:54:25.990Z
Answers.Text: I hoped this would appeal to you Rewan :)
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-22T08:26:42.490Z
Answers.Text: This is long known to "monochromators", though :-)
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-25T14:54:26.063Z
Answers.Text: @HaukeReddmann I wouldn't be surprised but do you have a reference?
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-25T19:10:52.303Z
Answers.Text: @Laska: Sorry, no. That I "know" something just means I'm a know-it-all, and often "needs citation". :-) You have to ask an actual expert.
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-26T22:27:00.780Z
Answers.Text: Oh @HaukeReddmann you "guess-it-all". I agree it’s an obvious idea, but I have searched in PDB, and there appears to be nothing that addresses this task in the 342 mono problems with solutions nor the 104 without solutions. I am speaking with Thierry Le Gleuher about sth else, and may I ask him
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-21T11:50:01.330Z
Answers.LastActivityDate: 2023-12-22T02:15:05.327Z
Answers.LastEditDate: 2023-12-22T02:15:05.327Z
Answers.UserDisplayName: Rewan Demontay
Answers.Body: <p>I want to complement @RewanDemontay's accepted answer.</p>
  <p>It's clear that the number of promotions can't be more than 6, because each promotion involves 4 captures, plus the promoted unit itself which is not eligible for captures. (32-2)/(4+1) = 30/5 = 6.</p>
  <p>One can divide the pieces into 4 sets of 8. How many promoted pawns can share the same square colour as their king? There are several patterns possible from simple counting, but only two can work out:</p>
  <p><strong>Case 1</strong></p>
  <div class="s-table-container">
  <table class="s-table">
  <thead>
  <tr>
  <th style="text-align: center;"></th>
  <th style="text-align: center;">White dark sq</th>
  <th style="text-align: center;">White light sq</th>
  <th style="text-align: center;">Black light sq</th>
  <th style="text-align: center;">Black dark sq</th>
  </tr>
  </thead>
  <tbody>
  <tr>
  <td style="text-align: center;">King</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Promoted pawns</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">2</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured units</td>
  <td style="text-align: center;">6</td>
  <td style="text-align: center;">4</td>
  <td style="text-align: center;">6</td>
  <td style="text-align: center;">4</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured e.p.</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">2</td>
  </tr>
  <tr>
  <td style="text-align: center;">Total</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  </tr>
  </tbody>
  </table>
  </div>
  <p><strong>Case 2</strong></p>
  <p>From counting we get:</p>
  <div class="s-table-container">
  <table class="s-table">
  <thead>
  <tr>
  <th style="text-align: center;"></th>
  <th style="text-align: center;">White dark sq</th>
  <th style="text-align: center;">White light sq</th>
  <th style="text-align: center;">Black light sq</th>
  <th style="text-align: center;">Black dark sq</th>
  </tr>
  </thead>
  <tbody>
  <tr>
  <td style="text-align: center;">King</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Promoted pawns</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">1</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured units</td>
  <td style="text-align: center;">3 or 4</td>
  <td style="text-align: center;">6 or 7</td>
  <td style="text-align: center;">3 or 4</td>
  <td style="text-align: center;">6 or 7</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured e.p.</td>
  <td style="text-align: center;">2 or 1</td>
  <td style="text-align: center;">1 or 0</td>
  <td style="text-align: center;">2 or 1</td>
  <td style="text-align: center;">1 or 0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Total</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  </tr>
  </tbody>
  </table>
  </div>
  <p>Now looking a little deeper, the White promotion on light squares must capture PBNR or eBNR (where e means en passant pawn). The latter is not possible, since e, N &amp; R must all be captured on 6th or 8th rank. Similarly for the Black dark square promotion. So the only possibility is:</p>
  <div class="s-table-container">
  <table class="s-table">
  <thead>
  <tr>
  <th style="text-align: center;"></th>
  <th style="text-align: center;">White dark sq</th>
  <th style="text-align: center;">White light sq</th>
  <th style="text-align: center;">Black light sq</th>
  <th style="text-align: center;">Black dark sq</th>
  </tr>
  </thead>
  <tbody>
  <tr>
  <td style="text-align: center;">King</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Promoted pawns</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">1</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured units</td>
  <td style="text-align: center;">4</td>
  <td style="text-align: center;">7</td>
  <td style="text-align: center;">4</td>
  <td style="text-align: center;">7</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured e.p.</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Total</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  </tr>
  </tbody>
  </table>
  </div>
  <p><strong>Case 3</strong> (not possible)</p>
  <div class="s-table-container">
  <table class="s-table">
  <thead>
  <tr>
  <th style="text-align: center;"></th>
  <th style="text-align: center;">White dark sq</th>
  <th style="text-align: center;">White light sq</th>
  <th style="text-align: center;">Black light sq</th>
  <th style="text-align: center;">Black dark sq</th>
  </tr>
  </thead>
  <tbody>
  <tr>
  <td style="text-align: center;">King</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Promoted pawns</td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">1</td>
  <td style="text-align: center;">2</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured units</td>
  <td style="text-align: center;">5</td>
  <td style="text-align: center;">4</td>
  <td style="text-align: center;">4</td>
  <td style="text-align: center;">6</td>
  </tr>
  <tr>
  <td style="text-align: center;">Captured e.p.</td>
  <td style="text-align: center;">0</td>
  <td style="text-align: center;"><strong>3?!?</strong></td>
  <td style="text-align: center;">2</td>
  <td style="text-align: center;">0</td>
  </tr>
  <tr>
  <td style="text-align: center;">Total</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  <td style="text-align: center;">8</td>
  </tr>
  </tbody>
  </table>
  </div>
  <p>This &quot;asymmetric pattern&quot; (and its mirror image) are not possible, because it requires too many en passant. Each promoting pawn can capture e.p. at most once.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-12-25T15:24:40.800Z
Answers.LastActivityDate: 2023-12-26T09:02:53.997Z
Answers.LastEditDate: 2023-12-26T09:02:53.997Z
Answers.UserDisplayName: Laska
Body: <p>A whimsical question just occurred to me. I think I have an answer, but I thought I would share it here as a puzzle.</p>
  <p>What is the maximum number of promoted units that can appear simultaneously on the board in a monochromatic game?</p>
  <p>(with reference to <a href="https://chess.stackexchange.com/questions/43400/what-are-monochromatic-retrograde-analysis-problems">What are &quot;monochromatic&quot; retrograde analysis problems?</a>)</p>
  <p>Please find a game to illustrate this.</p>
  
Tags: <chess-variants><pawn-promotion><retrograde-analysis><monochromatic>
Title: Max number of promoted units on the board in a monochromatic game
ThreadText: # Max number of promoted units on the board in a monochromatic game
  
  (Asked by: username_0 on 2023-12-21)
  
  A whimsical question just occurred to me. I think I have an answer, but I thought I would share it here as a puzzle.
  
  What is the maximum number of promoted units that can appear simultaneously on the board in a monochromatic game?
  
  (with reference to [What are "monochromatic" retrograde analysis problems?](https://chess.stackexchange.com/questions/43400/what-are-monochromatic-retrograde-analysis-problems))
  
  Please find a game to illustrate this.
  
  **Tags:** chess-variants, pawn-promotion, retrograde-analysis, monochromatic
  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_1 on 2023-12-21. Score: 3 | Accepted answer
  
  It takes four captures to promote a pawn in monochromatic. Only 15/16 of each sides' pieces are capturable. Thus, in theory, each side may only promote three pawns at most, or a total maximum of six combined.
  
  And it is indeed possible, albeit tricky. Utilizing en passant to change a pawn's lane color is a crucial aspect.
  
  ```
  [FEN ""]
  
  1. a4 a5 2. b4 b5 3. bxa5 bxa4 4. g4 g5 5. h4 h5 6. Rh3 Rh6 7. Rb3 Rb6 8. axb6 axb3 9. bxc7 bxc2 10. cxb8=N Qb6 11. d4 Qc5 12. dxc5 d5 13. cxd6 Bf5 14. Bf4 Bh7 15. Bh2 gxh4 16. gxh5 Ra6 17. Ra3 cxb1=N 18. dxe7 Rg6 19. Rg3 hxg3 20. hxg6 gxh2 21. gxh7 f5 22. e4 fxe4 23. f4 exf3 24. Be2 fxe2 25. exf8=B hxg1=N 26. hxg8=N exd1=Q
  
  ```
  
  > **Comments:**
  > **username_0**: I hoped this would appeal to you username_1 :)  
  > **username_2**: This is long known to "monochromators", though :-)  
  > **username_0**: @username_2 I wouldn't be surprised but do you have a reference?  
  > **username_2**: @username_0: Sorry, no. That I "know" something just means I'm a know-it-all, and often "needs citation". :-) You have to ask an actual expert.  
  > **username_0**: Oh @username_2 you "guess-it-all". I agree it’s an obvious idea, but I have searched in PDB, and there appears to be nothing that addresses this task in the 342 mono problems with solutions nor the 104 without solutions. I am speaking with Thierry Le Gleuher about sth else, and may I ask him  
  
  ---
  
  ### Answer by username_0 on 2023-12-25. Score: 1
  
  I want to complement @username_1's accepted answer.
  
  It's clear that the number of promotions can't be more than 6, because each promotion involves 4 captures, plus the promoted unit itself which is not eligible for captures. (32-2)/(4+1) = 30/5 = 6.
  
  One can divide the pieces into 4 sets of 8. How many promoted pawns can share the same square colour as their king? There are several patterns possible from simple counting, but only two can work out:
  
  **Case 1**
  
  |  | White dark sq | White light sq | Black light sq | Black dark sq |
  | --- | --- | --- | --- | --- |
  | King | 1 | 0 | 1 | 0 |
  | Promoted pawns | 1 | 2 | 1 | 2 |
  | Captured units | 6 | 4 | 6 | 4 |
  | Captured e.p. | 0 | 2 | 0 | 2 |
  | Total | 8 | 8 | 8 | 8 |
  
  **Case 2**
  
  From counting we get:
  
  |  | White dark sq | White light sq | Black light sq | Black dark sq |
  | --- | --- | --- | --- | --- |
  | King | 1 | 0 | 1 | 0 |
  | Promoted pawns | 2 | 1 | 2 | 1 |
  | Captured units | 3 or 4 | 6 or 7 | 3 or 4 | 6 or 7 |
  | Captured e.p. | 2 or 1 | 1 or 0 | 2 or 1 | 1 or 0 |
  | Total | 8 | 8 | 8 | 8 |
  
  Now looking a little deeper, the White promotion on light squares must capture PBNR or eBNR (where e means en passant pawn). The latter is not possible, since e, N & R must all be captured on 6th or 8th rank. Similarly for the Black dark square promotion. So the only possibility is:
  
  |  | White dark sq | White light sq | Black light sq | Black dark sq |
  | --- | --- | --- | --- | --- |
  | King | 1 | 0 | 1 | 0 |
  | Promoted pawns | 2 | 1 | 2 | 1 |
  | Captured units | 4 | 7 | 4 | 7 |
  | Captured e.p. | 1 | 0 | 1 | 0 |
  | Total | 8 | 8 | 8 | 8 |
  
  **Case 3** (not possible)
  
  |  | White dark sq | White light sq | Black light sq | Black dark sq |
  | --- | --- | --- | --- | --- |
  | King | 1 | 0 | 1 | 0 |
  | Promoted pawns | 2 | 1 | 1 | 2 |
  | Captured units | 5 | 4 | 4 | 6 |
  | Captured e.p. | 0 | **3?!?** | 2 | 0 |
  | Total | 8 | 8 | 8 | 8 |
  
  This "asymmetric pattern" (and its mirror image) are not possible, because it requires too many en passant. Each promoting pawn can capture e.p. at most once.
  
  
  ---
  
  