# 16 roses, 16x16 board; maximum double-attacked squares while all 256 squares are attacked at least once?

The rose is a piece that is best described as a "circular nightrider", 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:

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.

• does a rose attack it's own square when a cycle is possible, in a sense that counts towards your goals? Nov 16, 2019 at 22:55
• 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? Nov 16, 2019 at 23:13
• I'm afraid 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. Nov 17, 2019 at 1:35
• 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. Dec 15, 2019 at 11:51
• You should ask this on Math Stack Exchange - this is more maths than chess.
– user24344
Aug 13, 2020 at 2:20