Exercise 2.42 queens.rkt

#lang racket

; 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 kth 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 kth column. Now filter these, keeping only the positions for which the queen in
; the kth 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.

; 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 kth 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 kth
; 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.)

; SOLUTION

(define (queens board-size)

	; Note: The way we represent queen positions in this procedure is as follows:
	; (1 4 8 7) means a congiguration in which
	; queen 1 is in the 1st row of the 1st column
	; queen 2 is in the 4th row of the 2nd column
	; queen 3 is in the 8th row of the 3rd column
	; queen 4 is in the 7th row of the 4th column
	; In this way of representing queen positions, the row position is explicit and
	; the column position is implicit. This changes the signature of some of the
	; procedures below. 

 	(define (queen-cols k)  
		(if (= k 0)
			; If k is 0, produce a list which contains a single element which is an empty list
			; This will be the starting point of the process of building up a list of lists
			; that contain valid queen positions

			; 'empty-board' is just an empty list
			(list (list))

			(filter
				(lambda (queen-positions) (safe? k queen-positions))
				(flatmap
					(lambda (rest-of-queens)
						(map
							(lambda (new-row)
								; Here I don't need to use the 'adjoin-position'
								; procedure because all I need to do is append the
								; position of the next queen to the existing list
								; 'rest-of-queens' is an existing position of queens
								; 'new-row' is the position of a new queen
								(append rest-of-queens (list new-row))
							)
							(enumerate-interval 1 board-size)
						)
					)
					(queen-cols (- k 1))
				)
			)
		)
	)
	(queen-cols board-size)
)

(define (safe? k queen-positions)
	; Tests whether the queen in the kth column in this configuration is safe
	; with respect to all the other queens
	; Example: (2 5 8 4 1)

	; The process here is to 'car' through the positions one by one and test.
	; So safe-internal? needs to know which column of the board the first element
	; of the list is in

	(define queen-column k)

	; The following procedure evaluates the position of a queen in a given column
	(define (position-of-queen-in-column x)
		(define (position-of-queen-in-column-internal positions count)
			(if (= count 1)
				(car positions)
				(position-of-queen-in-column-internal (cdr positions) (- count 1))
			)
		)
		(position-of-queen-in-column-internal queen-positions x)
	)

	(define queen-row (position-of-queen-in-column k))

	(define (safe-internal? starting-column positions)
		(cond
			((null? positions) true)
			(else
				(if (check? starting-column (car positions) queen-column queen-row)
					false
					(safe-internal? (+ starting-column 1) (cdr positions))
				)
			)
		)
	)

	(define (check? q1-column q1-row q2-column q2-row)
		; The way we test is:
		; 1. The two queens should not be in the same row
		; 2. The two queens should not be in the same column. (This will never happen because of
		;		the way this program works. But is included here just for completeness.)
		; 3. The two queens should not be in the same diagonal
		(cond
			; if both queens are the same then it it not a check
			((and (= q1-row q2-row) (= q1-column q2-column)) false)
			; row and column check
			((or (= q1-row q2-row) (= q1-column q2-column)) true)
			; diagonal check
			((= (abs (- q1-column q2-column)) (abs (- q1-row q2-row))) true)
			(else
				false
			)
		)
	)

	(safe-internal? 1 queen-positions)
)

(define (flatmap proc seq)
	(accumulate append null (map proc seq))
)

(define (enumerate-interval low high)
	(if (> low high)
		null
		(cons low (enumerate-interval (+ low 1) high))
	)
)

(define (accumulate op initial sequence)
	(if (null? sequence)
		initial
		(op (car sequence) (accumulate op initial (cdr sequence)))
	)
)

; Tests

Welcome to DrRacket, version 6.11 [3m].
Language: racket, with debugging; memory limit: 128 MB.
> (queens 0)
'(())
> (queens 1)
'((1))
> (queens 2)
'()
> (queens 3)
'()
> (queens 4)
'((2 4 1 3) (3 1 4 2))
> (queens 5)
'((1 3 5 2 4)
  (1 4 2 5 3)
  (2 4 1 3 5)
  (2 5 3 1 4)
  (3 1 4 2 5)
  (3 5 2 4 1)
  (4 1 3 5 2)
  (4 2 5 3 1)
  (5 2 4 1 3)
  (5 3 1 4 2))
> (queens 6)
'((2 4 6 1 3 5) (3 6 2 5 1 4) (4 1 5 2 6 3) (5 3 1 6 4 2))
> (queens 7)
'((1 3 5 7 2 4 6)
  (1 4 7 3 6 2 5)
  (1 5 2 6 3 7 4)
  (1 6 4 2 7 5 3)
  (2 4 1 7 5 3 6)
  (2 4 6 1 3 5 7)
  (2 5 1 4 7 3 6)
  (2 5 3 1 7 4 6)
  (2 5 7 4 1 3 6)
  (2 6 3 7 4 1 5)
  (2 7 5 3 1 6 4)
  (3 1 6 2 5 7 4)
  (3 1 6 4 2 7 5)
  (3 5 7 2 4 6 1)
  (3 6 2 5 1 4 7)
  (3 7 2 4 6 1 5)
  (3 7 4 1 5 2 6)
  (4 1 3 6 2 7 5)
  (4 1 5 2 6 3 7)
  (4 2 7 5 3 1 6)
  (4 6 1 3 5 7 2)
  (4 7 3 6 2 5 1)
  (4 7 5 2 6 1 3)
  (5 1 4 7 3 6 2)
  (5 1 6 4 2 7 3)
  (5 2 6 3 7 4 1)
  (5 3 1 6 4 2 7)
  (5 7 2 4 6 1 3)
  (5 7 2 6 3 1 4)
  (6 1 3 5 7 2 4)
  (6 2 5 1 4 7 3)
  (6 3 1 4 7 5 2)
  (6 3 5 7 1 4 2)
  (6 3 7 4 1 5 2)
  (6 4 2 7 5 3 1)
  (6 4 7 1 3 5 2)
  (7 2 4 6 1 3 5)
  (7 3 6 2 5 1 4)
  (7 4 1 5 2 6 3)
  (7 5 3 1 6 4 2))
> (queens 8)
'((1 5 8 6 3 7 2 4)
  (1 6 8 3 7 4 2 5)
  (1 7 4 6 8 2 5 3)
  (1 7 5 8 2 4 6 3)
  (2 4 6 8 3 1 7 5)
  (2 5 7 1 3 8 6 4)
  (2 5 7 4 1 8 6 3)
  (2 6 1 7 4 8 3 5)
  (2 6 8 3 1 4 7 5)
  (2 7 3 6 8 5 1 4)
  (2 7 5 8 1 4 6 3)
  (2 8 6 1 3 5 7 4)
  (3 1 7 5 8 2 4 6)
  (3 5 2 8 1 7 4 6)
  (3 5 2 8 6 4 7 1)
  (3 5 7 1 4 2 8 6)
  (3 5 8 4 1 7 2 6)
  (3 6 2 5 8 1 7 4)
  (3 6 2 7 1 4 8 5)
  (3 6 2 7 5 1 8 4)
  (3 6 4 1 8 5 7 2)
  (3 6 4 2 8 5 7 1)
  (3 6 8 1 4 7 5 2)
  (3 6 8 1 5 7 2 4)
  (3 6 8 2 4 1 7 5)
  (3 7 2 8 5 1 4 6)
  (3 7 2 8 6 4 1 5)
  (3 8 4 7 1 6 2 5)
  (4 1 5 8 2 7 3 6)
  (4 1 5 8 6 3 7 2)
  (4 2 5 8 6 1 3 7)
  (4 2 7 3 6 8 1 5)
  (4 2 7 3 6 8 5 1)
  (4 2 7 5 1 8 6 3)
  (4 2 8 5 7 1 3 6)
  (4 2 8 6 1 3 5 7)
  (4 6 1 5 2 8 3 7)
  (4 6 8 2 7 1 3 5)
  (4 6 8 3 1 7 5 2)
  (4 7 1 8 5 2 6 3)
  (4 7 3 8 2 5 1 6)
  (4 7 5 2 6 1 3 8)
  (4 7 5 3 1 6 8 2)
  (4 8 1 3 6 2 7 5)
  (4 8 1 5 7 2 6 3)
  (4 8 5 3 1 7 2 6)
  (5 1 4 6 8 2 7 3)
  (5 1 8 4 2 7 3 6)
  (5 1 8 6 3 7 2 4)
  (5 2 4 6 8 3 1 7)
  (5 2 4 7 3 8 6 1)
  (5 2 6 1 7 4 8 3)
  (5 2 8 1 4 7 3 6)
  (5 3 1 6 8 2 4 7)
  (5 3 1 7 2 8 6 4)
  (5 3 8 4 7 1 6 2)
  (5 7 1 3 8 6 4 2)
  (5 7 1 4 2 8 6 3)
  (5 7 2 4 8 1 3 6)
  (5 7 2 6 3 1 4 8)
  (5 7 2 6 3 1 8 4)
  (5 7 4 1 3 8 6 2)
  (5 8 4 1 3 6 2 7)
  (5 8 4 1 7 2 6 3)
  (6 1 5 2 8 3 7 4)
  (6 2 7 1 3 5 8 4)
  (6 2 7 1 4 8 5 3)
  (6 3 1 7 5 8 2 4)
  (6 3 1 8 4 2 7 5)
  (6 3 1 8 5 2 4 7)
  (6 3 5 7 1 4 2 8)
  (6 3 5 8 1 4 2 7)
  (6 3 7 2 4 8 1 5)
  (6 3 7 2 8 5 1 4)
  (6 3 7 4 1 8 2 5)
  (6 4 1 5 8 2 7 3)
  (6 4 2 8 5 7 1 3)
  (6 4 7 1 3 5 2 8)
  (6 4 7 1 8 2 5 3)
  (6 8 2 4 1 7 5 3)
  (7 1 3 8 6 4 2 5)
  (7 2 4 1 8 5 3 6)
  (7 2 6 3 1 4 8 5)
  (7 3 1 6 8 5 2 4)
  (7 3 8 2 5 1 6 4)
  (7 4 2 5 8 1 3 6)
  (7 4 2 8 6 1 3 5)
  (7 5 3 1 6 8 2 4)
  (8 2 4 1 7 5 3 6)
  (8 2 5 3 1 7 4 6)
  (8 3 1 6 2 5 7 4)
  (8 4 1 3 6 2 7 5))
> (queens 9)
'((1 3 6 8 2 4 9 7 5)
  (1 3 7 2 8 5 9 4 6)
  (1 3 8 6 9 2 5 7 4)
  (1 4 2 8 6 9 3 5 7)
  (1 4 6 3 9 2 8 5 7)
  (1 4 6 8 2 5 3 9 7)
  (1 4 7 3 8 2 5 9 6)
  (1 4 7 9 2 5 8 6 3)
  (1 4 8 3 9 7 5 2 6)
  (1 5 2 6 9 3 8 4 7)
  (1 5 7 2 6 3 9 4 8)
  (1 5 7 9 3 8 2 4 6)
  (1 5 7 9 4 2 8 6 3)
  (1 5 9 2 6 8 3 7 4)
  (1 5 9 6 4 2 8 3 7)
  (1 6 2 9 7 4 8 3 5)
  (1 6 4 2 7 9 3 5 8)
  (1 6 4 2 8 3 9 7 5)
  (1 6 8 3 7 4 2 9 5)
  (1 6 8 5 2 4 9 7 3)
  (1 6 9 5 2 8 3 7 4)
  (1 7 4 6 9 2 5 3 8)
  (1 7 4 8 3 5 9 2 6)
  (1 7 4 8 3 9 6 2 5)
  (1 7 5 8 2 9 3 6 4)
  (1 8 4 2 7 9 6 3 5)
  (1 8 5 3 6 9 2 4 7)
  (1 8 5 3 9 7 2 4 6)
  (2 4 1 7 9 6 3 5 8)
  (2 4 7 1 3 9 6 8 5)
  (2 4 8 3 9 6 1 5 7)
  (2 4 9 7 3 1 6 8 5)
  (2 4 9 7 5 3 1 6 8)
  (2 5 7 1 3 8 6 4 9)
  (2 5 7 4 1 3 9 6 8)
  (2 5 7 9 3 6 4 1 8)
  (2 5 7 9 4 8 1 3 6)
  (2 5 8 1 3 6 9 7 4)
  (2 5 8 1 9 6 3 7 4)
  (2 5 8 6 9 3 1 4 7)
  (2 5 8 6 9 3 1 7 4)
  (2 5 9 4 1 8 6 3 7)
  (2 6 1 3 7 9 4 8 5)
  (2 6 1 7 4 8 3 5 9)
  (2 6 1 7 5 3 9 4 8)
  (2 6 1 9 5 8 4 7 3)
  (2 6 3 1 8 4 9 7 5)
  (2 6 9 3 5 8 4 1 7)
  (2 7 5 1 9 4 6 8 3)
  (2 7 5 8 1 4 6 3 9)
  (2 7 9 6 3 1 4 8 5)
  (2 8 1 4 7 9 6 3 5)
  (2 8 5 3 9 6 4 1 7)
  (2 8 6 9 3 1 4 7 5)
  (2 9 5 3 8 4 7 1 6)
  (2 9 6 3 5 8 1 4 7)
  (2 9 6 3 7 4 1 8 5)
  (2 9 6 4 7 1 3 5 8)
  (3 1 4 7 9 2 5 8 6)
  (3 1 6 8 5 2 4 9 7)
  (3 1 7 2 8 6 4 9 5)
  (3 1 7 5 8 2 4 6 9)
  (3 1 8 4 9 7 5 2 6)
  (3 1 9 7 5 2 8 6 4)
  (3 5 2 8 1 4 7 9 6)
  (3 5 2 8 1 7 4 6 9)
  (3 5 7 1 4 2 8 6 9)
  (3 5 8 2 9 6 1 7 4)
  (3 5 8 2 9 7 1 4 6)
  (3 5 9 2 4 7 1 8 6)
  (3 5 9 4 1 7 2 6 8)
  (3 6 2 7 1 4 8 5 9)
  (3 6 2 9 5 1 8 4 7)
  (3 6 8 1 4 7 5 2 9)
  (3 6 8 1 5 9 2 4 7)
  (3 6 8 2 4 9 7 5 1)
  (3 6 8 5 1 9 7 2 4)
  (3 6 8 5 2 9 7 4 1)
  (3 6 9 1 8 4 2 7 5)
  (3 6 9 2 5 7 4 1 8)
  (3 6 9 2 8 1 4 7 5)
  (3 6 9 5 8 1 4 2 7)
  (3 6 9 7 1 4 2 5 8)
  (3 6 9 7 2 4 8 1 5)
  (3 6 9 7 4 1 8 2 5)
  (3 7 2 4 8 1 5 9 6)
  (3 7 2 8 5 9 1 6 4)
  (3 7 2 8 6 4 1 5 9)
  (3 7 4 2 9 5 1 8 6)
  (3 7 4 2 9 6 1 5 8)
  (3 7 4 8 5 9 1 6 2)
  (3 7 9 1 5 2 8 6 4)
  (3 7 9 4 2 5 8 6 1)
  (3 8 2 4 9 7 5 1 6)
  (3 8 4 7 9 2 5 1 6)
  (3 8 6 1 9 2 5 7 4)
  (3 8 6 4 9 1 5 7 2)
  (3 8 6 9 2 5 1 4 7)
  (3 9 2 5 8 1 7 4 6)
  (3 9 4 1 8 6 2 7 5)
  (3 9 4 2 8 6 1 7 5)
  (3 9 4 8 5 2 6 1 7)
  (3 9 6 2 5 7 1 4 8)
  (3 9 6 4 1 7 5 2 8)
  (3 9 6 8 2 4 1 7 5)
  (4 1 3 6 9 2 8 5 7)
  (4 1 5 2 9 7 3 8 6)
  (4 1 5 8 2 7 3 6 9)
  (4 1 5 9 2 6 8 3 7)
  (4 1 7 9 2 6 8 3 5)
  (4 1 9 6 3 7 2 8 5)
  (4 2 5 8 1 3 6 9 7)
  (4 2 7 3 1 8 5 9 6)
  (4 2 7 9 1 5 8 6 3)
  (4 2 7 9 1 8 5 3 6)
  (4 2 8 3 9 7 5 1 6)
  (4 2 9 3 6 8 1 5 7)
  (4 2 9 5 1 8 6 3 7)
  (4 6 1 5 2 8 3 7 9)
  (4 6 1 9 5 8 2 7 3)
  (4 6 1 9 7 3 8 2 5)
  (4 6 3 9 2 5 8 1 7)
  (4 6 3 9 2 8 5 7 1)
  (4 6 3 9 7 1 8 2 5)
  (4 6 8 2 5 1 9 7 3)
  (4 6 8 2 5 7 9 1 3)
  (4 6 8 2 7 1 3 5 9)
  (4 6 8 3 1 7 5 2 9)
  (4 6 9 3 1 8 2 5 7)
  (4 7 1 3 9 6 8 5 2)
  (4 7 1 6 9 2 8 5 3)
  (4 7 1 8 5 2 9 3 6)
  (4 7 3 6 9 1 8 5 2)
  (4 7 3 8 2 5 9 6 1)
  (4 7 3 8 6 1 9 2 5)
  (4 7 3 8 6 2 9 5 1)
  (4 7 5 2 9 1 3 8 6)
  (4 7 5 2 9 1 6 8 3)
  (4 7 5 2 9 6 8 3 1)
  (4 7 9 2 5 8 1 3 6)
  (4 7 9 2 6 1 3 5 8)
  (4 7 9 6 3 1 8 5 2)
  (4 8 1 5 7 2 6 3 9)
  (4 8 5 3 1 6 2 9 7)
  (4 8 5 3 1 7 2 6 9)
  (4 9 3 6 2 7 5 1 8)
  (4 9 5 3 1 6 8 2 7)
  (4 9 5 3 1 7 2 8 6)
  (4 9 5 8 1 3 6 2 7)
  (5 1 6 4 2 8 3 9 7)
  (5 1 8 4 2 7 9 6 3)
  (5 1 8 6 3 7 2 4 9)
  (5 2 4 1 7 9 3 6 8)
  (5 2 4 9 7 3 1 6 8)
  (5 2 6 1 3 7 9 4 8)
  (5 2 6 9 3 8 4 7 1)
  (5 2 6 9 7 4 1 3 8)
  (5 2 8 1 4 7 9 6 3)
  (5 2 8 1 7 9 3 6 4)
  (5 2 8 3 7 4 1 9 6)
  (5 2 8 3 7 9 1 6 4)
  (5 2 9 1 6 8 3 7 4)
  (5 2 9 6 3 7 4 1 8)
  (5 3 1 6 2 9 7 4 8)
  (5 3 1 6 8 2 4 7 9)
  (5 3 1 7 2 8 6 4 9)
  (5 3 6 9 2 8 1 4 7)
  (5 3 6 9 7 1 4 2 8)
  (5 3 6 9 7 2 4 8 1)
  (5 3 6 9 7 4 1 8 2)
  (5 3 8 4 2 9 6 1 7)
  (5 3 8 4 7 9 2 6 1)
  (5 3 8 6 2 9 1 4 7)
  (5 3 8 6 2 9 7 1 4)
  (5 3 9 4 2 8 6 1 7)
  (5 3 9 6 8 2 4 1 7)
  (5 7 1 4 2 8 6 9 3)
  (5 7 1 6 8 2 4 9 3)
  (5 7 2 4 8 1 3 9 6)
  (5 7 2 4 8 1 9 6 3)
  (5 7 2 6 3 1 8 4 9)
  (5 7 2 6 8 1 4 9 3)
  (5 7 4 1 3 6 9 2 8)
  (5 7 4 1 3 8 6 2 9)
  (5 7 4 1 3 9 6 8 2)
  (5 7 4 1 8 2 9 6 3)
  (5 7 9 3 8 2 4 6 1)
  (5 7 9 4 2 8 6 3 1)
  (5 7 9 4 8 1 3 6 2)
  (5 8 1 4 7 3 6 9 2)
  (5 8 1 9 4 2 7 3 6)
  (5 8 2 7 3 1 9 4 6)
  (5 8 2 7 3 6 9 1 4)
  (5 8 2 9 3 1 7 4 6)
  (5 8 2 9 6 3 1 4 7)
  (5 8 4 1 3 6 9 7 2)
  (5 8 4 1 7 2 6 3 9)
  (5 8 4 9 7 3 1 6 2)
  (5 8 6 1 3 7 9 4 2)
  (5 8 6 9 3 1 7 4 2)
  (5 9 2 4 7 3 8 6 1)
  (5 9 2 6 8 3 1 4 7)
  (5 9 4 6 8 2 7 1 3)
  (6 1 5 2 9 7 4 8 3)
  (6 1 5 7 9 3 8 2 4)
  (6 1 5 7 9 4 2 8 3)
  (6 1 7 4 8 3 5 9 2)
  (6 2 5 7 9 3 8 4 1)
  (6 2 5 7 9 4 8 1 3)
  (6 2 9 5 3 8 4 7 1)
  (6 3 1 4 7 9 2 5 8)
  (6 3 1 8 4 9 7 5 2)
  (6 3 1 8 5 2 9 7 4)
  (6 3 5 8 1 4 2 7 9)
  (6 3 5 8 1 9 4 2 7)
  (6 3 5 8 1 9 7 2 4)
  (6 3 7 2 4 8 1 5 9)
  (6 3 7 2 4 9 1 8 5)
  (6 3 7 2 8 5 1 4 9)
  (6 3 7 4 1 9 2 5 8)
  (6 3 9 2 5 8 1 7 4)
  (6 3 9 4 1 8 2 5 7)
  (6 3 9 7 1 4 2 5 8)
  (6 4 1 7 9 2 8 5 3)
  (6 4 2 7 9 3 5 8 1)
  (6 4 2 8 3 9 7 5 1)
  (6 4 2 8 5 3 1 9 7)
  (6 4 2 8 5 9 1 3 7)
  (6 4 7 1 3 9 2 8 5)
  (6 4 7 1 8 2 5 3 9)
  (6 4 7 1 8 5 2 9 3)
  (6 4 9 1 3 7 2 8 5)
  (6 4 9 1 5 2 8 3 7)
  (6 4 9 5 8 2 7 3 1)
  (6 8 1 5 9 2 4 7 3)
  (6 8 1 7 4 2 9 5 3)
  (6 8 2 7 1 3 5 9 4)
  (6 8 3 1 9 2 5 7 4)
  (6 8 3 1 9 5 2 4 7)
  (6 8 3 7 9 2 5 1 4)
  (6 8 5 2 9 7 4 1 3)
  (6 9 1 4 7 3 8 2 5)
  (6 9 3 1 8 4 2 7 5)
  (6 9 5 1 8 4 2 7 3)
  (6 9 5 2 8 3 7 4 1)
  (6 9 5 8 1 3 7 2 4)
  (6 9 7 4 1 8 2 5 3)
  (7 1 4 2 8 6 9 3 5)
  (7 1 4 6 9 3 5 8 2)
  (7 1 4 8 5 3 9 6 2)
  (7 1 6 2 5 8 4 9 3)
  (7 1 6 8 2 4 9 3 5)
  (7 1 6 9 2 4 8 3 5)
  (7 1 8 5 2 9 3 6 4)
  (7 2 4 1 8 5 9 6 3)
  (7 2 4 6 1 9 5 3 8)
  (7 2 4 9 1 8 5 3 6)
  (7 2 6 3 1 8 5 9 4)
  (7 2 8 6 1 3 5 9 4)
  (7 3 1 6 8 5 2 4 9)
  (7 3 1 9 5 8 2 4 6)
  (7 3 6 2 5 1 9 4 8)
  (7 3 6 8 1 4 9 5 2)
  (7 3 6 8 1 5 9 2 4)
  (7 3 8 2 4 6 9 5 1)
  (7 3 8 2 5 1 9 4 6)
  (7 3 8 6 2 9 5 1 4)
  (7 4 1 3 6 9 2 8 5)
  (7 4 1 3 8 6 2 9 5)
  (7 4 1 3 9 6 8 5 2)
  (7 4 1 5 2 9 6 8 3)
  (7 4 1 8 2 9 6 3 5)
  (7 4 1 8 5 3 6 9 2)
  (7 4 1 9 2 6 8 3 5)
  (7 4 2 5 8 1 3 6 9)
  (7 4 2 5 9 1 3 8 6)
  (7 4 2 8 6 1 3 5 9)
  (7 4 2 9 5 1 8 6 3)
  (7 4 2 9 6 3 5 8 1)
  (7 4 8 1 5 9 2 6 3)
  (7 4 8 3 9 6 2 5 1)
  (7 5 1 6 9 3 8 4 2)
  (7 5 1 8 6 3 9 2 4)
  (7 5 2 8 1 3 9 6 4)
  (7 5 2 8 1 4 9 3 6)
  (7 5 3 9 6 8 2 4 1)
  (7 5 8 2 9 3 6 4 1)
  (7 5 8 2 9 6 3 1 4)
  (7 9 1 3 5 8 2 4 6)
  (7 9 2 6 1 3 5 8 4)
  (7 9 3 5 2 8 6 4 1)
  (7 9 3 8 2 4 6 1 5)
  (7 9 4 2 5 8 6 1 3)
  (7 9 6 3 1 8 5 2 4)
  (8 1 4 6 3 9 7 5 2)
  (8 1 4 7 3 6 9 2 5)
  (8 1 4 7 5 2 9 6 3)
  (8 1 5 7 2 6 3 9 4)
  (8 2 4 1 7 9 6 3 5)
  (8 2 5 7 1 4 6 9 3)
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> (queens 10)
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  (7 4 1 8 6 3 10 2 9 5)
  (7 4 1 9 6 3 10 8 5 2)
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  (7 4 2 5 9 1 10 8 6 3)
  (7 4 2 5 9 6 10 3 1 8)
  (7 4 2 9 5 10 8 6 3 1)
  (7 4 2 9 6 3 10 8 1 5)
  (7 4 6 1 9 5 3 8 10 2)
  (7 4 6 8 10 1 3 5 2 9)
  (7 4 8 1 5 9 2 6 10 3)
  (7 4 8 1 5 10 2 6 3 9)
  (7 4 8 5 9 1 10 2 6 3)
  (7 4 10 5 2 9 3 6 8 1)
  (7 4 10 5 9 1 3 8 6 2)
  (7 5 1 6 4 10 8 3 9 2)
  (7 5 1 6 9 3 8 4 2 10)
  (7 5 1 8 6 3 9 2 4 10)
  (7 5 1 8 10 3 6 9 2 4)
  (7 5 2 6 1 10 4 9 3 8)
  (7 5 2 6 1 10 8 3 9 4)
  (7 5 2 8 1 3 9 6 4 10)
  (7 5 2 8 1 4 9 3 6 10)
  (7 5 2 8 10 3 9 6 4 1)
  (7 5 3 1 9 4 2 10 8 6)
  (7 5 3 1 9 4 8 10 2 6)
  (7 5 3 9 6 10 2 4 1 8)
  (7 5 8 1 4 6 9 2 10 3)
  (7 5 8 1 4 10 3 6 9 2)
  (7 5 8 1 9 4 2 10 3 6)
  (7 5 8 2 9 3 6 4 1 10)
  (7 5 10 1 4 6 3 9 2 8)
  (7 5 10 1 6 4 2 8 3 9)
  (7 5 10 2 4 8 3 9 6 1)
  (7 9 1 5 2 6 3 10 8 4)
  (7 9 2 5 1 4 10 8 6 3)
  (7 9 2 5 8 1 3 10 6 4)
  (7 9 2 6 1 3 5 8 10 4)
  (7 9 3 1 4 8 5 2 10 6)
  (7 9 3 1 10 8 5 2 4 6)
  (7 9 3 5 2 10 6 4 1 8)
  (7 9 3 6 2 10 5 1 8 4)
  (7 9 3 8 2 4 6 1 10 5)
  (7 9 4 1 5 10 2 6 8 3)
  (7 9 4 2 5 8 10 1 3 6)
  (7 9 4 6 1 10 2 5 3 8)
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  (7 9 6 3 1 8 5 2 4 10)
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  (7 10 4 1 5 9 2 6 8 3)
  (7 10 4 2 9 3 6 8 1 5)
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  (8 1 4 2 10 6 9 3 5 7)
  (8 1 4 6 10 2 5 3 9 7)
  (8 1 4 7 9 2 5 10 6 3)
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  (8 3 5 2 10 1 6 4 9 7)
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  (8 5 2 4 9 7 3 10 6 1)
  (8 5 2 4 10 7 9 6 3 1)
  (8 5 2 6 3 10 7 9 4 1)
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  (8 5 2 9 7 4 1 3 6 10)
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  (8 5 2 10 3 7 9 4 6 1)
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  (8 5 2 10 7 4 1 9 6 3)
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  (8 5 3 10 7 4 1 9 2 6)
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  (10 3 1 4 7 9 2 5 8 6)
  (10 3 5 2 8 1 7 4 6 9)
  (10 3 5 8 2 9 6 1 7 4)
  (10 3 5 8 2 9 7 1 4 6)
  (10 3 5 9 2 4 7 1 8 6)
  (10 3 5 9 4 1 7 2 6 8)
  (10 3 6 9 2 8 1 4 7 5)
  (10 3 7 2 4 8 1 5 9 6)
  (10 3 7 4 2 9 5 1 8 6)
  (10 3 7 4 2 9 6 1 5 8)
  (10 3 9 2 5 8 1 7 4 6)
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  (10 5 2 6 1 3 7 9 4 8)
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  (10 8 5 3 1 6 2 9 7 4))
> (queens 11)
'((1 3 5 7 9 11 2 4 6 8 10)
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  (1 11 5 7 10 4 2 9 3 6 8)
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  (2 4 8 11 3 6 10 5 1 9 7)
  (2 4 8 11 3 10 7 5 1 9 6)
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  (2 4 8 11 5 10 6 1 3 7 9)
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  (2 4 9 11 8 5 3 1 7 10 6)
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  (2 9 5 1 8 4 11 7 3 10 6)
  (2 9 5 1 11 6 10 7 4 8 3)
  (2 9 5 3 10 6 11 1 7 4 8)
  (2 9 5 8 1 3 6 11 7 10 4)
  (2 9 5 8 1 4 7 11 6 3 10)
  (2 9 5 8 10 3 6 11 1 4 7)
  (2 9 6 1 3 8 11 7 4 10 5)
  (2 9 6 1 5 8 11 4 7 3 10)
  (2 9 6 1 7 4 11 8 5 3 10)
  (2 9 6 3 1 8 11 5 7 10 4)
  (2 9 6 3 1 10 7 5 11 8 4)
  (2 9 6 3 10 4 11 8 5 7 1)
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  (2 9 6 4 1 10 5 7 11 3 8)
  (2 9 6 8 10 1 3 5 7 4 11)
  (2 9 6 10 5 1 11 4 7 3 8)
  (2 9 6 10 5 11 1 4 7 3 8)
  (2 9 7 3 1 6 10 5 11 8 4)
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  (2 9 7 4 10 3 6 11 5 8 1)
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  (2 9 11 6 8 3 1 4 7 5 10)
  (2 10 3 7 4 8 11 5 1 6 9)
  (2 10 3 7 4 11 9 6 1 5 8)
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  (2 10 6 1 5 11 9 7 4 8 3)
  (2 10 6 1 9 11 3 7 4 8 5)
  (2 10 6 3 9 4 1 8 11 5 7)
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  (2 10 6 4 9 1 3 7 11 8 5)
  (2 10 6 9 3 1 11 8 5 7 4)
  (2 10 6 9 3 5 11 8 1 7 4)
  (2 10 6 9 5 1 4 7 11 8 3)
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