Answers.Body: <p>This is a great question and a fun problem to mull over. Indeed the way it's written on <a href="https://en.wikipedia.org/wiki/Empress_(chess)#Value" rel="nofollow noreferrer">wikipedia</a> one gets the impression that amazon-vs-empress fortresses are trivial and easily attainable for the weaker side, where in fact it's very difficult to even come up with dummy examples where the empress stands any chance of holding (so not surprising not to see a example diagram there). In other word, most amazon vs empress (also known as <em>chancellor</em>) endgames are in fact winning for the side with the amazon (intuitively, the amazon has diagonal routes towards the <a href="https://en.wikipedia.org/wiki/Empress_(chess)#Movement" rel="nofollow noreferrer">weak points in the empress' safezone</a>), not even requiring the assistance of the king in order to dominate the empress+king duo. </p>
  
  <p><strong>Some intuition:</strong> You want to have the king of the side with the amazon to be well cornered by both the empress and the king, or at least for it to be in a different quadrant of the board compared to the amazon and near the opponent king (so all cases where the amazon and the king are somewhat central is highly likely to be winning already). The reason for this setup is that it allows you to defend <em>certain</em> checks with an <strong>empress-fork</strong> thus forcing the draw, while covering some key squares with the empress that otherwise would be exploited by the amazon to push the king towards the center and reach a winning situation again. One of the better scenarios for the amazon side would be: to <strong>pin</strong> the empress while having <strong>spare waiting</strong> moves with their king.</p>
  
  <p><em>Note that in the diagram editor here the rook and queen will denote the empress and amazon respectively. Additionally, certain moves that would be illegal in standard chess thus disallowed by the diagram editor here, as we're using rook and queen instead of fairy pieces, would be mentioned in the annotations.</em></p>
  
  <p>It's funny, if you just try to randomly set up any amazon vs empress ending, they're <em>almost always</em> losing for the empress-side. Now even though it is still possible to find some clear setups where the empress holds, it's important to note that fortresses of this kind constitute a minute fraction of the state-space of all legal amazon vs empress endgames. This observation is well corroborated by the statistics <a href="http://www.chessvariants.com/misc.dir/endgames.html" rel="nofollow noreferrer">provided here</a> for KA vs KC (C for Chancellor/empress): If you look at the third row in that table with the percentage breakdown, only <code>0.001052 %</code> of all legal KA vs KC configurations comprise fortresses, with most other draws being by <em>perpetual.</em> The latter types of examples are simpler to showcase specially if it's the empress' side to move, whereas fortresses with the amazon side to move are highly non-trivial. Below some examples for both types of draws are showcased (annotated): </p>
  
  <hr>
  
  <h2><strong>Fortress and perpetual examples:</strong></h2>
  
  <pre><code> [title "empress+king fortress example#1 - black to move"]
   [fen "k7/8/K7/6q1/R7/8/8/8 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Kb8 {the only amazon moves either force the draw immediately or lose on the spot because of the Rb6# threat. So only attempt is to move the king and try to escape but 2.Rb6+ Kc7 3.Rb5+ forces the draw with a fork. Thus, white is holding here.} 
  </code></pre>
  
  <p>Another example:</p>
  
  <pre><code> [title "empress+king fortress example #2 - black to move"]
   [fen "k7/6q1/K7/1R6/8/8/8/8 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qa1+ {Again all key squares c7-b7-a7 are well covered, and any check on the 6th row loses immediately, so black's only other attempt is Qa1+} 2.Ra5 Qf6+ (2...Qf1+ 3.Kb6+ Kb8 {and black even gets mated here: 4.Rc6+ Ka8 5.Rc8#}) 3.Kb5+ Kb8 {and 4.Rc6+ forking and forcing the draw}
  </code></pre>
  
  <p>Now to show you how sensitive these fortresses are, let's take example#2 again with the added difference that the amazon is now in a position to pin the empress on the <code>f1-a6</code> diagonal, and that suffices to break the fortress and win: </p>
  
  <pre><code> [title "Losing variation of example#2: Amazon pins can be deadly!"]
   [fen "k7/8/K7/1R6/8/8/7q/8 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qe2 {pinning and the Zugzwang's set: both remaining king moves are losing} 2.Ka5 (2.Kb6 Qc4+ 3.Ka6 Qa4+ 4.Ra5 Qb4#) Qc4+ 3.Ka6 Qa4+ 4.Ra5 Qb4# (1...Qa2+ 2.Ra5 {setting up a discovery on the king while covering c4} Qe6+ (2...Qe2+ 3.Kb6+ Kb8 {4.Rc6+ Kb8 5. Rc8# mate.}) 3.Kb5+ Kb8 {and the 4.Rc6+ fork forces the draw.})
  </code></pre>
  
  <p>Finally, here's one trivial perpetual example:</p>
  
  <pre><code> [title "Example#3 - perpertual, empress-side to move:"]
   [fen "7q/8/6k1/8/4K3/5R2/8/8 w - - 0 1"]
  
   1.Rf4+ Kg7 (1...Kg5 {leads to similar perpetuals}) 2.Rf5+ Kg6 {black can avoid the repetition but only at the cost of trading the remaining pieces.} (2...Kg8 {is followed by 3.Re7+ Kf8 4.Rc8+ winning the amazon}) 3.Rf4+
  </code></pre>
  
  <hr>
  
  <h2><strong>Other variations that look tempting:</strong></h2>
  
  <p>Here's one that resembles a fortress but it probably isn't. The ideal scenario for the empress side would be something like <code>1...Qf8</code> after which the empress would try to check the king endlessly as it cannot hide behind the amazon anymore. But white doesn't have to play <code>1...Qf8</code>, instead, staying on the long diagonal with a waiting move probably wins, e.g., <code>1...Qf6.</code></p>
  
  <pre><code> [title "Example#4: other attempts worth considering"]
   [fen "8/8/8/5q2/8/2k5/1R6/K7 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qf8 {Any amazon check results in a fork forcing a draw, so we have to either make king moves or waiting amazon moves.} (1...Qf6 2.Kb1 {any rook move also loses on the spot, e.g. 2.Rb1+ Kc2+ 3.Ka2 Qe6+ 4.Ka1 Qa6+} Qf1+ 3.Ka2 Qc1+ 4.Ka3 Qxb2#) 
  </code></pre>
  
  <p>Notice that any such pin with the king stuck in the corner is always losing, for instance:</p>
  
  <pre><code> [title "Example#5: Pin exploitation to dominate the empress"]
   [fen "8/8/8/3k4/3q4/8/1R6/K7 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qg1+ 2.Ka2 (2.Rb1 Qa7+ 3.Kb2 Qa4+ 4.Kc1 Qa2+ {winning the empress}) Qc1+ 3.Ka3 Qa1+ 4.Ra2 Qb1+ {winning the empress}
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2018-01-30T22:54:47.413Z
Answers.Text: In diagram 5, 1... Qe5 2. Kb1 Qc3+ 3. Ka1 Qc1+ 4. Rb1 Qc2# is mate
Answers.UserDisplayName: ericw31415
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2018-01-30T23:29:55.813Z
Answers.Text: @ericw31415 sure good find, even better. For the purposes of the discussions it sufficed to show at least one winning line in that diagram, so I didn't bother looking for a faster mate. Regarding your edited post: I've already shown fortress examples with a cornered king, but I will unfortunately not have the time to go through your specific examples, so my answer stands mainly for your original version of the question. Though I believe with the the discussions above you ought to be able to analyse any other similar setup on your own. If you're really stuck, feel free to drop a comment here.
Answers.UserDisplayName: Ellie
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2018-01-31T00:27:09.887Z
Answers.Text: Your answer is fine. I just edited in some examples to show that I have put in some research effort, and perhaps entice others to answer. If no one else has anything to say, I'll accept your answer.
Answers.UserDisplayName: ericw31415
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2018-01-31T00:46:05.563Z
Answers.Text: Very good that you've put in your own effort before posting. It has been a lot of fun thinking about this problem, thanks for posting ;)
Answers.UserDisplayName: Ellie
Answers.ContentLicense: CC BY-SA 3.0
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Answers.Text: NIce effort ! I think you could add a TL;DR saying that fortresses are very rare and the amazon almost always win.
Answers.UserDisplayName: Evargalo
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Answers.Text: @Evargalo thanks :), yeah I've said that but not captioned with tldr.
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Answers.UserDisplayName: Ellie
Body: <p>Many online sources (such as <a href="https://en.wikipedia.org/wiki/Amazon_(chess)" rel="nofollow noreferrer">Wikipedia</a>) state that the endgame of amazon (piece that moves like Queen and knight) and chancellor (piece that moves like rook and knight) is a draw. I did some experimenting but could not find such a position. Now Wikipedia says that a draw can be <strong>forced</strong> with a fortress that forces the stronger side to give perpetual check without giving such a position.</p>
  
  <p>Examples of positions I've tried are (the queen represents amazon and the rook represents chancellor):</p>
  
  <pre><code>[Title "Centralized king"]
  [FEN "8/8/8/3kr3/8/8/8/KQ6 w - - 0 1"]
  
  1. Qb4+ Ke6 2. Qf4 {Forces the chancellor into a pin} Kd6 3. Kb2 {Waiting move, and now the king must leave the chancellor} 1-0
  </code></pre>
  
  <p>or</p>
  
  <pre><code>[Title "Cornered king"]
  [FEN "kr6/8/8/8/8/8/8/KQ6 w - - 0 1"]
  
  1. Qe4+ Ka7 (1... Rb7 2. Qe8+ Ka7 3. Qc6+ Ka8 4. Qc8+ Rb8 5. Qc7#) 2. Qc5+ Rb6 (2... Ka8 3. Qc7#) 3. Ka2 {Any king move is followed by Qxb6#} 1-0
  </code></pre>
  
  <p>or</p>
  
  <pre><code>[Title "King on rim"]
  [FEN "4k3/4r3/8/8/8/8/8/KQ6 w - - 0 1"]
  
  1. Qb8+ Kf7 2. Qd6+ Kf8 3. Kb2 Kg8 4. Qxe7+ Kh8 5. Qf6#
  </code></pre>
  
  <p>All three of these examples lead to trivial wins for White. Could someone post a fortress in this endgame, if such a fortress exists at all?</p>
  
  <hr>
  
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2018-02-06T13:12:13.653Z
Comments.Text: @user929304 Whoops, I've been pretty inactive lately
Comments.UserDisplayName: ericw31415
Tags: <endgame><chess-variants><fortress><unorthodox-pieces>
Title: Amazon vs Chancellor endgame
ThreadText: # Amazon vs Chancellor endgame
  
  (Asked by: username_0 on 2018-01-29)
  
  Many online sources (such as [Wikipedia](https://en.wikipedia.org/wiki/Amazon_(chess))) state that the endgame of amazon (piece that moves like Queen and knight) and chancellor (piece that moves like rook and knight) is a draw. I did some experimenting but could not find such a position. Now Wikipedia says that a draw can be **forced** with a fortress that forces the stronger side to give perpetual check without giving such a position.
  
  Examples of positions I've tried are (the queen represents amazon and the rook represents chancellor):
  
  ```
  [Title "Centralized king"]
  [FEN "8/8/8/3kr3/8/8/8/KQ6 w - - 0 1"]
  
  1. Qb4+ Ke6 2. Qf4 {Forces the chancellor into a pin} Kd6 3. Kb2 {Waiting move, and now the king must leave the chancellor} 1-0
  
  ```
  
  or
  
  ```
  [Title "Cornered king"]
  [FEN "kr6/8/8/8/8/8/8/KQ6 w - - 0 1"]
  
  1. Qe4+ Ka7 (1... Rb7 2. Qe8+ Ka7 3. Qc6+ Ka8 4. Qc8+ Rb8 5. Qc7#) 2. Qc5+ Rb6 (2... Ka8 3. Qc7#) 3. Ka2 {Any king move is followed by Qxb6#} 1-0
  
  ```
  
  or
  
  ```
  [Title "King on rim"]
  [FEN "4k3/4r3/8/8/8/8/8/KQ6 w - - 0 1"]
  
  1. Qb8+ Kf7 2. Qd6+ Kf8 3. Kb2 Kg8 4. Qxe7+ Kh8 5. Qf6#
  
  ```
  
  All three of these examples lead to trivial wins for White. Could someone post a fortress in this endgame, if such a fortress exists at all?
  
  ---
  
  **Tags:** endgame, chess-variants, fortress, unorthodox-pieces
  
  > **Comments:**
  > **username_0**: @user929304 Whoops, I've been pretty inactive lately  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2018-01-30. Score: 5 | Accepted answer
  
  This is a great question and a fun problem to mull over. Indeed the way it's written on [wikipedia](https://en.wikipedia.org/wiki/Empress_(chess)#Value) one gets the impression that amazon-vs-empress fortresses are trivial and easily attainable for the weaker side, where in fact it's very difficult to even come up with dummy examples where the empress stands any chance of holding (so not surprising not to see a example diagram there). In other word, most amazon vs empress (also known as *chancellor*) endgames are in fact winning for the side with the amazon (intuitively, the amazon has diagonal routes towards the [weak points in the empress' safezone](https://en.wikipedia.org/wiki/Empress_(chess)#Movement)), not even requiring the assistance of the king in order to dominate the empress+king duo.
  
  **Some intuition:** You want to have the king of the side with the amazon to be well cornered by both the empress and the king, or at least for it to be in a different quadrant of the board compared to the amazon and near the opponent king (so all cases where the amazon and the king are somewhat central is highly likely to be winning already). The reason for this setup is that it allows you to defend *certain* checks with an **empress-fork** thus forcing the draw, while covering some key squares with the empress that otherwise would be exploited by the amazon to push the king towards the center and reach a winning situation again. One of the better scenarios for the amazon side would be: to **pin** the empress while having **spare waiting** moves with their king.
  
  *Note that in the diagram editor here the rook and queen will denote the empress and amazon respectively. Additionally, certain moves that would be illegal in standard chess thus disallowed by the diagram editor here, as we're using rook and queen instead of fairy pieces, would be mentioned in the annotations.*
  
  It's funny, if you just try to randomly set up any amazon vs empress ending, they're *almost always* losing for the empress-side. Now even though it is still possible to find some clear setups where the empress holds, it's important to note that fortresses of this kind constitute a minute fraction of the state-space of all legal amazon vs empress endgames. This observation is well corroborated by the statistics [provided here](http://www.chessvariants.com/misc.dir/endgames.html) for KA vs KC (C for Chancellor/empress): If you look at the third row in that table with the percentage breakdown, only `0.001052 %` of all legal KA vs KC configurations comprise fortresses, with most other draws being by *perpetual.* The latter types of examples are simpler to showcase specially if it's the empress' side to move, whereas fortresses with the amazon side to move are highly non-trivial. Below some examples for both types of draws are showcased (annotated):
  
  ---
  
  ## **Fortress and perpetual examples:**
  
  ```
   [title "empress+king fortress example#1 - black to move"]
   [fen "k7/8/K7/6q1/R7/8/8/8 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Kb8 {the only amazon moves either force the draw immediately or lose on the spot because of the Rb6# threat. So only attempt is to move the king and try to escape but 2.Rb6+ Kc7 3.Rb5+ forces the draw with a fork. Thus, white is holding here.} 
  
  ```
  
  Another example:
  
  ```
   [title "empress+king fortress example #2 - black to move"]
   [fen "k7/6q1/K7/1R6/8/8/8/8 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qa1+ {Again all key squares c7-b7-a7 are well covered, and any check on the 6th row loses immediately, so black's only other attempt is Qa1+} 2.Ra5 Qf6+ (2...Qf1+ 3.Kb6+ Kb8 {and black even gets mated here: 4.Rc6+ Ka8 5.Rc8#}) 3.Kb5+ Kb8 {and 4.Rc6+ forking and forcing the draw}
  
  ```
  
  Now to show you how sensitive these fortresses are, let's take example#2 again with the added difference that the amazon is now in a position to pin the empress on the `f1-a6` diagonal, and that suffices to break the fortress and win:
  
  ```
   [title "Losing variation of example#2: Amazon pins can be deadly!"]
   [fen "k7/8/K7/1R6/8/8/7q/8 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qe2 {pinning and the Zugzwang's set: both remaining king moves are losing} 2.Ka5 (2.Kb6 Qc4+ 3.Ka6 Qa4+ 4.Ra5 Qb4#) Qc4+ 3.Ka6 Qa4+ 4.Ra5 Qb4# (1...Qa2+ 2.Ra5 {setting up a discovery on the king while covering c4} Qe6+ (2...Qe2+ 3.Kb6+ Kb8 {4.Rc6+ Kb8 5. Rc8# mate.}) 3.Kb5+ Kb8 {and the 4.Rc6+ fork forces the draw.})
  
  ```
  
  Finally, here's one trivial perpetual example:
  
  ```
   [title "Example#3 - perpertual, empress-side to move:"]
   [fen "7q/8/6k1/8/4K3/5R2/8/8 w - - 0 1"]
  
   1.Rf4+ Kg7 (1...Kg5 {leads to similar perpetuals}) 2.Rf5+ Kg6 {black can avoid the repetition but only at the cost of trading the remaining pieces.} (2...Kg8 {is followed by 3.Re7+ Kf8 4.Rc8+ winning the amazon}) 3.Rf4+
  
  ```
  
  ---
  
  ## **Other variations that look tempting:**
  
  Here's one that resembles a fortress but it probably isn't. The ideal scenario for the empress side would be something like `1...Qf8` after which the empress would try to check the king endlessly as it cannot hide behind the amazon anymore. But white doesn't have to play `1...Qf8`, instead, staying on the long diagonal with a waiting move probably wins, e.g., `1...Qf6.`
  
  ```
   [title "Example#4: other attempts worth considering"]
   [fen "8/8/8/5q2/8/2k5/1R6/K7 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qf8 {Any amazon check results in a fork forcing a draw, so we have to either make king moves or waiting amazon moves.} (1...Qf6 2.Kb1 {any rook move also loses on the spot, e.g. 2.Rb1+ Kc2+ 3.Ka2 Qe6+ 4.Ka1 Qa6+} Qf1+ 3.Ka2 Qc1+ 4.Ka3 Qxb2#) 
  
  ```
  
  Notice that any such pin with the king stuck in the corner is always losing, for instance:
  
  ```
   [title "Example#5: Pin exploitation to dominate the empress"]
   [fen "8/8/8/3k4/3q4/8/1R6/K7 b - - 0 1"]
   [StartFlipped "0"]
  
   1...Qg1+ 2.Ka2 (2.Rb1 Qa7+ 3.Kb2 Qa4+ 4.Kc1 Qa2+ {winning the empress}) Qc1+ 3.Ka3 Qa1+ 4.Ra2 Qb1+ {winning the empress}
  
  ```
  
  > **Comments:**
  > **username_0**: In diagram 5, 1... Qe5 2. Kb1 Qc3+ 3. Ka1 Qc1+ 4. Rb1 Qc2# is mate  
  > **username_1**: @username_0 sure good find, even better. For the purposes of the discussions it sufficed to show at least one winning line in that diagram, so I didn't bother looking for a faster mate. Regarding your edited post: I've already shown fortress examples with a cornered king, but I will unfortunately not have the time to go through your specific examples, so my answer stands mainly for your original version of the question. Though I believe with the the discussions above you ought to be able to analyse any other similar setup on your own. If you're really stuck, feel free to drop a comment here.  
  > **username_0**: Your answer is fine. I just edited in some examples to show that I have put in some research effort, and perhaps entice others to answer. If no one else has anything to say, I'll accept your answer.  
  > **username_1**: Very good that you've put in your own effort before posting. It has been a lot of fun thinking about this problem, thanks for posting ;)  
  > **username_2**: NIce effort ! I think you could add a TL;DR saying that fortresses are very rare and the amazon almost always win.  
  > **username_1**: @username_2 thanks :), yeah I've said that but not captioned with tldr.  
  
  ---
  
  