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Flat share share

Puzzling Asked on August 2, 2021

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.

Task

Place a complete (black and white) set of proper pieces (no pawns) on a standard chess board such that

  1. the bishops are pairs, i.e. light- and dark-squared
  2. no black piece threatens any white piece and vice versa
  3. 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

Find

  1. a symmetric solution
  2. a non symmetric solution
  3. a pair of solutions that differs only by the placement of a single piece
  4. 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
  5. 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.
  6. a solution with a queen in the centre
  7. a solution with a king in the centre
  8. a solution with another piece in the centre
  9. a solution with the centre empty
  10. 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
  11. a solution where every square is attacked

Note: solutions (not necessarily distinct) for each of the above properties exist.

Hints:

1/(4)

2/(4)

3/(4)

4/(4)

One Answer

With the new hints this has become much simpler. In fact, OP has (unintentionally?) changed the character of the puzzle. At least some solutions can be step-by-step (like a sudoku for want of a better simile) reconstructed from the hints.

For example number 8 (I'll only do one, so people still have a chance to earn the bounty by solving one of the others):

Let us assume that in each non rectangular area every convex corner not touching the boundary is actually occupied and see how far we can get:

Fully reconstructed position:

Nicer picture (thanks @justhalf !)

Answered by Albert.Lang on August 2, 2021

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