The 4 compound leapers I'd like to look at are:
(1,2), (0,3), (2,2); knight+threeleaper+alfil
(1,2), (0,2), (1,1); knight+kirin
(1,2), (1,3); knight+camel
(1,2), (2,3); knight+zebra
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; are the "strongest coverage properties" I'm defining already a thing which there is another name for? I also should note that I first tried asking this question over on the Chess Stack Exchange a couple months back, but it got close to zero attention there. Please let me know if the tags I'm using here are good or if I should add or remove any tags.)
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.