Exercise 2.42: The “eight-queens puzzle” asks
how to place eight queens on a chessboard so that no queen is in check from any
other (i.e., no two queens are in the same row, column, or diagonal).  One
possible solution is shown in Figure 2.8.  One way to solve the puzzle is
to work across the board, placing a queen in each column.  Once we have placed

  
    k
    −
    1
  
 queens, we must place the 
  
    k
    
      th
    
  
 queen in a position where it does
not check any of the queens already on the board.  We can formulate this
approach recursively: Assume that we have already generated the sequence of all
possible ways to place 
  
    k
    −
    1
  
 queens in the first 
  
    k
    −
    1
  
 columns of the
board.  For each of these ways, generate an extended set of positions by
placing a queen in each row of the 
  
    k
    
      th
    
  
 column.  Now filter these, keeping
only the positions for which the queen in the 
  
    k
    
      th
    
  
 column is safe with
respect to the other queens.  This produces the sequence of all ways to place

  k
 queens in the first 
  k
 columns.  By continuing this process, we will
produce not only one solution, but all solutions to the puzzle.



SVG


Figure 2.8: A solution to the eight-queens puzzle.



We implement this solution as a procedure queens, which returns a
sequence of all solutions to the problem of placing 
  n
 queens on an

  
    n
    ×
    n
  
 chessboard.  Queens has an internal procedure
queen-cols that returns the sequence of all ways to place queens in the
first 
  k
 columns of the board.


(define (queens board-size)
  (define (queen-cols k)
    (if (= k 0)
        (list empty-board)
        (filter
         (lambda (positions) 
           (safe? k positions))
         (flatmap
          (lambda (rest-of-queens)
            (map (lambda (new-row)
                   (adjoin-position 
                    new-row 
                    k 
                    rest-of-queens))
                 (enumerate-interval 
                  1 
                  board-size)))
          (queen-cols (- k 1))))))
  (queen-cols board-size))

In this procedure rest-of-queens is a way to place 
  
    k
    −
    1
  
 queens in
the first 
  
    k
    −
    1
  
 columns, and new-row is a proposed row in which to
place the queen for the 
  
    k
    
      th
    
  
 column.  Complete the program by implementing
the representation for sets of board positions, including the procedure
adjoin-position, which adjoins a new row-column position to a set of
positions, and empty-board, which represents an empty set of positions.
You must also write the procedure safe?, which determines for a set of
positions, whether the queen in the 
  
    k
    
      th
    
  
 column is safe with respect to the
others.  (Note that we need only check whether the new queen is safe—the
other queens are already guaranteed safe with respect to each other.)