Answers.Body: <p>Sopel hasn't been around for a couple of years, but his solution is pretty and worth posting.</p>
  <p><a href="https://i.sstatic.net/NRTzr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/NRTzr.png" alt="enter image description here" /></a></p>
  <p>The pink squares contain the 16 roses. Each rose attacks all the squares whose distance away is sqrt(5), 4, 3 x sqrt(2), sqrt(29), 4 x sqrt(2) or 6. A rose does not attack the square it occupies.</p>
  <p>All the squares are attacked from 1-3 times, including the squares which are actually occupied by the roses.</p>
  <p>Addressing the original question: 100 squares are attacked more than once, of which just 8 are attacked three times.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-28T17:11:58.527Z
Answers.LastActivityDate: 2021-01-28T17:17:08.667Z
Answers.LastEditDate: 2021-01-28T17:17:08.667Z
Answers.UserDisplayName: Laska
Body: <p>The <a href="https://en.wikipedia.org/wiki/Chess_on_a_really_big_board" rel="nofollow noreferrer">rose</a> is a piece that is best described as a "circular <a href="https://en.wikipedia.org/wiki/Nightrider_(chess)" rel="nofollow noreferrer">nightrider</a>", able to reach up to 32 other squares from the center of a 13x13 area by making consecutive knight jumps in a rotating fashion in a single move as long as no landing square en route is occupied. Incidentally, this rotating movement allows it to essentially pass a turn by returning to the square which it started its move from if it's completely unobstructed along at least 1 circle; though it obviously does not protect itself this way. I'll provide a diagram to illustrate its movement capabilities:</p>
  
  <p><a href="https://i.sstatic.net/9uB8r.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9uB8r.png" alt="A diagram illustrating the movement capabilities of a rose"></a></p>
  
  <p>With 16 roses, including but still obstructed by squares occupied by other roses except the square occupied by an observed rose itself; what is the maximum number of squares on an otherwise clear 16x16 board that can be double-attacked while all 256 squares are attacked at least once each? For the purposes of this question, a square that is attacked 3 or more times does not count for anything more than a square that is attacked twice.</p>
  
  <p>I feel I should note that I personally have yet to find an arrangement that attacks all 256 squares at least once by my own efforts; the very corners of the board in particular appear to require poor positioning in order to reach. However, seeing as 16 roses times 32 squares a single rose can theoretically attack creates a theoretical max of 512 attacks; I will be very surprised if an arrangement that meets the "attack each square at least once" prerequisite actually doesn't exist. If it should seem that this is the case, I would like an explanation as to why.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-11-16T22:55:43.113Z
Comments.Text: does a rose attack it's own square when a cycle is possible, in a sense that counts towards your goals?
Comments.UserDisplayName: Sopel
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-11-16T23:13:34.280Z
Comments.Text: If a rose counted as attacking its own square, that would imply that it could "capture" or "protect" itself; which it can't. Think of it this way; if an enemy piece were to capture a rose, that rose wouldn't be there to recapture the piece that took it. At the same time, if it's moving it doesn't make sense that it would be there to impede its own movement; which is why returning to the square it was on is a valid move for it even though it can never capture an enemy piece by doing so, since an enemy piece can never be occupying that square. Does this make sense? Should I edit the explanation?
Comments.UserDisplayName: RadiantDarkBlaze
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-11-17T01:35:34.970Z
Comments.Text: I'm afraid https://pastebin.com/tVHgkN3p may be the only one that attacks all squares (up to rotation). I did millions of semi-random searches and nothing else popped up.
Comments.UserDisplayName: Sopel
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-11-17T19:47:42.370Z
Comments.Text: If that's likely to be the only arrangement of 16 roses that even attacks all 256 squares of a 16x16 board to begin with, then you can post it as an answer and update your answer if you discover otherwise; the only arrangement that attacks all 256 squares is by definition the arrangement that double-attacks the most of those 256 squares while it's at it. I have to say, the fact that 8 of the roses have to be right up against the edge of the board in order to attack all 256 squares is quite surprising to me; in my personal attempts I'd been trying to stay far from the edges and near the center
Comments.UserDisplayName: RadiantDarkBlaze
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-15T11:51:57.823Z
Comments.Text: The diameter of the rose pattern is 12 squares. Therefore the corner 4 squares in each of the 4 corners must be attacked by dedicated rose knights which do not interfere in other corners. This seems to constrain the possibilities enormously and if the existing solution works, it is probably unique - hope Sopel can post it as solution.
Comments.UserDisplayName: Laska
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-08-14T15:03:42.583Z
Comments.Text: Although Sopel basically already answered this question in a comment (I'm still hoping they post it as an answer so they can get the answer upvotes + I can accept their answer), I do have another question here that got barely any attention which may get better reception on Math SE; so I might go ahead and give it a try. Thank you for the suggestion~
Comments.UserDisplayName: RadiantDarkBlaze
Tags: <chess-variants><mathematics><unorthodox-pieces>
Title: 16 roses, 16x16 board; maximum double-attacked squares while all 256 squares are attacked at least once?
ThreadText: # 16 roses, 16x16 board; maximum double-attacked squares while all 256 squares are attacked at least once?
  
  (Asked by: username_0 on 2019-11-12)
  
  The [rose](https://en.wikipedia.org/wiki/Chess_on_a_really_big_board) is a piece that is best described as a "circular [nightrider](https://en.wikipedia.org/wiki/Nightrider_(chess))", able to reach up to 32 other squares from the center of a 13x13 area by making consecutive knight jumps in a rotating fashion in a single move as long as no landing square en route is occupied. Incidentally, this rotating movement allows it to essentially pass a turn by returning to the square which it started its move from if it's completely unobstructed along at least 1 circle; though it obviously does not protect itself this way. I'll provide a diagram to illustrate its movement capabilities:
  
  [![A diagram illustrating the movement capabilities of a rose](https://i.sstatic.net/9uB8r.png)](https://i.sstatic.net/9uB8r.png)
  
  With 16 roses, including but still obstructed by squares occupied by other roses except the square occupied by an observed rose itself; what is the maximum number of squares on an otherwise clear 16x16 board that can be double-attacked while all 256 squares are attacked at least once each? For the purposes of this question, a square that is attacked 3 or more times does not count for anything more than a square that is attacked twice.
  
  I feel I should note that I personally have yet to find an arrangement that attacks all 256 squares at least once by my own efforts; the very corners of the board in particular appear to require poor positioning in order to reach. However, seeing as 16 roses times 32 squares a single rose can theoretically attack creates a theoretical max of 512 attacks; I will be very surprised if an arrangement that meets the "attack each square at least once" prerequisite actually doesn't exist. If it should seem that this is the case, I would like an explanation as to why.
  
  **Tags:** chess-variants, mathematics, unorthodox-pieces
  
  > **Comments:**
  > **username_1**: does a rose attack it's own square when a cycle is possible, in a sense that counts towards your goals?  
  > **username_0**: If a rose counted as attacking its own square, that would imply that it could "capture" or "protect" itself; which it can't. Think of it this way; if an enemy piece were to capture a rose, that rose wouldn't be there to recapture the piece that took it. At the same time, if it's moving it doesn't make sense that it would be there to impede its own movement; which is why returning to the square it was on is a valid move for it even though it can never capture an enemy piece by doing so, since an enemy piece can never be occupying that square. Does this make sense? Should I edit the explanation?  
  > **username_1**: I'm afraid https://pastebin.com/tVHgkN3p may be the only one that attacks all squares (up to rotation). I did millions of semi-random searches and nothing else popped up.  
  > **username_0**: If that's likely to be the only arrangement of 16 roses that even attacks all 256 squares of a 16x16 board to begin with, then you can post it as an answer and update your answer if you discover otherwise; the only arrangement that attacks all 256 squares is by definition the arrangement that double-attacks the most of those 256 squares while it's at it. I have to say, the fact that 8 of the roses have to be right up against the edge of the board in order to attack all 256 squares is quite surprising to me; in my personal attempts I'd been trying to stay far from the edges and near the center  
  > **username_2**: The diameter of the rose pattern is 12 squares. Therefore the corner 4 squares in each of the 4 corners must be attacked by dedicated rose knights which do not interfere in other corners. This seems to constrain the possibilities enormously and if the existing solution works, it is probably unique - hope username_1 can post it as solution.  
  > **username_0**: Although username_1 basically already answered this question in a comment (I'm still hoping they post it as an answer so they can get the answer upvotes + I can accept their answer), I do have another question here that got barely any attention which may get better reception on Math SE; so I might go ahead and give it a try. Thank you for the suggestion~  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2021-01-28. Score: 1
  
  username_1 hasn't been around for a couple of years, but his solution is pretty and worth posting.
  
  [![enter image description here](https://i.sstatic.net/NRTzr.png)](https://i.sstatic.net/NRTzr.png)
  
  The pink squares contain the 16 roses. Each rose attacks all the squares whose distance away is sqrt(5), 4, 3 x sqrt(2), sqrt(29), 4 x sqrt(2) or 6. A rose does not attack the square it occupies.
  
  All the squares are attacked from 1-3 times, including the squares which are actually occupied by the roses.
  
  Addressing the original question: 100 squares are attacked more than once, of which just 8 are attacked three times.
  
  
  ---
  
  