Answers.Body: <p>This is the best arrangement that I've been able to find by hand. It covers 248 squares at least once; missing the 8 squares which are the 2 knight moves outwards from each of the center 4 squares. If I've counted right, I believe it covers 148 of those 248 squares at least twice each. I don't expect triple-covered squares to be a tiebreaker for this question and it's difficult to do by eye, so I admit to not personally counting which and how many squares are covered at least 3 times each in this arrangement.</p>
  
  <p><a href="https://i.sstatic.net/9sBOX.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9sBOX.png" alt="The best arrangement that I&#39;ve managed to find by hand"></a></p>
  
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Answers.CreationDate: 2020-05-30T01:33:01.580Z
Answers.LastActivityDate: 2020-05-30T01:33:01.580Z
Answers.UserDisplayName: RadiantDarkBlaze
Body: <p>The 4 compound leapers I'd like to look at are:</p>
  
  <ul>
  <li><p>(1,2), (0,3), (2,2); knight+<a href="https://en.wikipedia.org/wiki/Threeleaper" rel="nofollow noreferrer" title="Wikipedia entry">threeleaper</a>+<a href="https://en.wikipedia.org/wiki/Alfil" rel="nofollow noreferrer" title="Wikipedia entry">alfil</a></p></li>
  <li><p>(1,2), (0,2), (1,1); knight+<a href="https://en.wikipedia.org/wiki/Kirin_(chess)" rel="nofollow noreferrer" title="Wikipedia entry">kirin</a></p></li>
  <li><p>(1,2), (1,3); knight+<a href="https://en.wikipedia.org/wiki/Camel_(chess)" rel="nofollow noreferrer" title="Wikipedia entry">camel</a></p></li>
  <li><p>(1,2), (2,3); knight+<a href="https://en.wikipedia.org/wiki/Zebra_(chess)" rel="nofollow noreferrer" title="Wikipedia entry">zebra</a></p></li>
  </ul>
  
  <p>What arrangement of 8 of each of these 4 compound leapers (a total of 32 pieces) has the "strongest coverage properties" on a 16x16 board, defined as such?:</p>
  
  <ul>
  <li>Any square that a piece can attack is counted as being covered once per piece</li>
  <li>A square occupied by a piece is not covered by the piece occupying that square</li>
  <li>Sort first by # of squares covered once; then by 2x, 3x, up to highest cover count</li>
  </ul>
  
  <p>For example, an arrangement of pieces that covered 256 squares at least once each and covered 96 of those squares at least twice each would by this definition be considered "stronger" than an arrangement that covered 252 squares once and 112 squares twice; the higher at least twice-covered square count doesn't matter if it covers less squares at least once each, and so on.</p>
  
  <p>If any riders were in the mix (rooks and bishops are examples of rider pieces; rooks the (0,1) rider and bishops the (1,1) rider), I would add that "a square that a rider's path to is blocked is not covered by that rider"; my intention is for this definition to be generalized to any combination of pieces on a board of any dimensions.</p>
  
  <p>(I should double-check; is this "strongest coverage arrangement" I'm defining already a thing which there is another name for?)</p>
  
Tags: <chess-variants><mathematics><unorthodox-pieces>
Title: What arrangement of 8 each of these 4 different compound leapers has the "strongest coverage properties" on a 16x16 board?
ThreadText: # What arrangement of 8 each of these 4 different compound leapers has the "strongest coverage properties" on a 16x16 board?
  
  (Asked by: username_0 on 2020-05-30)
  
  The 4 compound leapers I'd like to look at are:
  
  * (1,2), (0,3), (2,2); knight+[threeleaper](https://en.wikipedia.org/wiki/Threeleaper "Wikipedia entry")+[alfil](https://en.wikipedia.org/wiki/Alfil "Wikipedia entry")
  * (1,2), (0,2), (1,1); knight+[kirin](https://en.wikipedia.org/wiki/Kirin_(chess) "Wikipedia entry")
  * (1,2), (1,3); knight+[camel](https://en.wikipedia.org/wiki/Camel_(chess) "Wikipedia entry")
  * (1,2), (2,3); knight+[zebra](https://en.wikipedia.org/wiki/Zebra_(chess) "Wikipedia entry")
  
  What arrangement of 8 of each of these 4 compound leapers (a total of 32 pieces) has the "strongest coverage properties" on a 16x16 board, defined as such?:
  
  * Any square that a piece can attack is counted as being covered once per piece
  * A square occupied by a piece is not covered by the piece occupying that square
  * Sort first by # of squares covered once; then by 2x, 3x, up to highest cover count
  
  For example, an arrangement of pieces that covered 256 squares at least once each and covered 96 of those squares at least twice each would by this definition be considered "stronger" than an arrangement that covered 252 squares once and 112 squares twice; the higher at least twice-covered square count doesn't matter if it covers less squares at least once each, and so on.
  
  If any riders were in the mix (rooks and bishops are examples of rider pieces; rooks the (0,1) rider and bishops the (1,1) rider), I would add that "a square that a rider's path to is blocked is not covered by that rider"; my intention is for this definition to be generalized to any combination of pieces on a board of any dimensions.
  
  (I should double-check; is this "strongest coverage arrangement" I'm defining already a thing which there is another name for?)
  
  **Tags:** chess-variants, mathematics, unorthodox-pieces
  
  
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  ## 1 answers
  
  ### Answer by username_0 on 2020-05-30. Score: 1
  
  This is the best arrangement that I've been able to find by hand. It covers 248 squares at least once; missing the 8 squares which are the 2 knight moves outwards from each of the center 4 squares. If I've counted right, I believe it covers 148 of those 248 squares at least twice each. I don't expect triple-covered squares to be a tiebreaker for this question and it's difficult to do by eye, so I admit to not personally counting which and how many squares are covered at least 3 times each in this arrangement.
  
  [![The best arrangement that I've managed to find by hand](https://i.sstatic.net/9sBOX.png)](https://i.sstatic.net/9sBOX.png)
  
  
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