This is a sequel to Gentrification in Chessshire.
Due to the febrile state of the Chesster housing market you and your flat mates are forced to rent out half your place to another group of sharers.
Keeping every one (including punters BTL) happy was not easy before and will only become more tricky in the crammed new conditions.
You quickly find that the new lot are not at all compatible with you and your friends and to prevent open warfare arrangements have to be made that minimise contact between you and them while still maintaining the delicate dynamics within each group.
Place a complete (black and white) set of proper pieces (no pawns) on a standard chess board such that
- the bishops are pairs, i.e. light- and dark-squared
- no black piece threatens any white piece and vice versa
- each white piece attacks exactly one other white piece and every white piece is attacked by exactly one other white piece. This results in a single daisy chain running through all white pieces. The same holds for the black pieces
Bonus / Tie breaker
- a symmetric solution
- a non symmetric solution
- a pair of solutions that differs only by the placement of a single piece
- a solution that permits drawing a straight line such that all squares occupied by white pieces are entirely on one side and all squares occupied by black pieces are entirely on the other side
- a solution with alternating corner occupancy, i.e. a white piece on at least one of a1,a2,b1 and of g8,h7,h8 and a black piece on at least one of a7,a8,b8 and of g1,h1,h2.
- a solution with a queen in the centre
- a solution with a king in the centre
- a solution with another piece in the centre
- a solution with the centre empty
- a solution with four squares that can be connected by a straight line that is entirely covered by the squares such that the first and third square are occupied by white and the second and fourth by black pieces
- a solution where every square is attacked
Note: solutions (not necessarily distinct) for each of the above properties exist.
There are a total of 9 solutions up to rotation, mirroring and swapping colours. As a hint I'm giving rough territorial maps for all of them:
BBBBB.WW BBBB.BBB BBBBBBBB BBBBB.WW BBBB.BBB BBBBBBBB BBBBB.WW BBBB.BBB BBBWWWWW BBBBB.WW .....WWW BBBWWWWW BBBBB.WW BBB..... BBBWWWWW ......WW WWW.WWWW BBBWWWWW ......WW WWW.WWWW BBBWWWWW WWWWWWWW WWW.WWWW BBBWWWWW WBBBBBBB BBBBBBWW BBBBBBWW WWBBBBBB BBBBBBWW BBBBBBWW WWWBBBBB BBBBBBWW BBBBBBWW WWWWBBBB BBBBBBWW BBBBBBWW WWWWBBBB WWWWBBBB WWWWBBBB WWWWWBBB WWWWBBBB WWWWBBBB WWWWWWBB WWWWBBBB WWWWBBBB WWWWWWWB WWWWBBBB WWWWBBBB XXXXXXXX BBBBBBBB BBBBBWWW XXXXXXXX BBBBBBBB BBBBBWWW WWWWWWW. WBBBBBBB BBBBBWWW WWWWWW.B WWWWWBBB BBB..WWW WWWWW.BB WWWWWBBB BBB..WWW WWWW.BBB WWWWWBBB BBBWWWWW WWW.BBBB WWWWWBBB BBBWWWWW WW.BBBBB WWWWWWBB BBBWWWWW W: white B: black X: both .: neither
Please note that rectanglese of B,W,X are not necessarily minimal. They may contain empty ranks or files, possibly at the boundary.
The first three maps at better resolution:
BBBBB... BBBBBBB. BBBBBBB. BBBB.... BBBBBBB. BBBBBBB. BBBB..WW BBBBBB.. BBBWWWWW BBBB..WW B......W BBBWWWWW BB....WW B......W B..WWWWW ......WW ..WWWWWW B..WWWWW ......WW .WWWWWWW B..WWWWW .WWWWWWW .WWWWWWW ...WWWWW
And maps 4-6:
....BBBB BBBBBB.. BBBBB... WW..BBBB BBBBBBWW BBBBB.WW WW...BBB BBBBB.WW BBBBB.WW WWWW.BBB ......WW ......WW WWWW.BBB WW...... WW...... WWWW.BBB WW...BBB WW...BBB WWWWWW.B WWWW.BBB WWWW.BBB WWWWWW.. WWWW.... WWWW....
And the last lot:
..XXXX.. .BBBBBB. .....WWW ..XXXX.. .BBBBBB. B.....WW WW...... W....... BBBB..WW WW.....B W......B BB.....W WWWW...B WWWW.BBB BB.....W WW.....B WWWWWBBB BB.WWWWW WW....BB WWWWWBBB B...WWWW WW.BBBBB WWWWWW.B B.......