Hi, I'm stuck on the extended exercise 28.2 of How to Design Programs.

Here is the link to the question: http://www.htdp.org/2003-09-26/Book/curriculum-Z-H-35.html#node_chap_28

I used a vector of true or false values to represent the board instead of using a list.

This is what I've got which doesn't work:

#lang Scheme

(define-struct posn (i j))

;takes in a position in i, j form and a board and returns a natural number that represents the position in index form
;example for board xxx
;                  xxx
;                  xxx
;(0, 1) -> 1
;(2, 1) -> 7
(define (board-ref a-posn a-board)
  (+ (* (sqrt (vector-length a-board)) (posn-i a-posn))
     (posn-j a-posn)))

;reverse of the above function
;1 -> (0, 1)
;7 -> (2, 1)
(define (get-posn n a-board)
  (local ((define board-length (sqrt (vector-length a-board))))
    (make-posn (floor (/ n board-length)) 
               (remainder n board-length))))

;determines if posn1 threatens posn2
;true if they are on the same row/column/diagonal
(define (threatened? posn1 posn2)
  (cond
    ((= (posn-i posn1) (posn-i posn2)) #t)
    ((= (posn-j posn1) (posn-j posn2)) #t)
    ((= (abs (- (posn-i posn1)
                (posn-i posn2)))
        (abs (- (posn-j posn1)
                (posn-j posn2)))) #t)
    (else #f)))

;returns a list of positions that are not threatened or occupied by queens
;basically any position with the value true
(define (get-available-posn a-board)
  (local ((define (get-ava index)
            (cond
              ((= index (vector-length a-board)) '())
              ((vector-ref a-board index)
               (cons index (get-ava (add1 index))))
              (else (get-ava (add1 index))))))
    (get-ava 0)))

;consume a position in the form of a natural number and a board
;returns a board after placing a queen on the position of the board
(define (place n a-board)
  (local ((define (foo x)
            (cond
              ((not (board-ref (get-posn x a-board) a-board)) #f)
              ((threatened? (get-posn x a-board) (get-posn n a-board)) #f)
              (else #t))))
    (build-vector (vector-length a-board) foo)))

;consume a list of positions in the form of natural number and consumes a board
;returns a list of boards after placing queens on each of the positions on the board
(define (place/list alop a-board)
  (cond
    ((empty? alop) '())
    (else (cons (place (first alop) a-board)
                (place/list (rest alop) a-board)))))

;returns a possible board after placing n queens on a-board
;returns false if impossible
(define (placement n a-board)
  (cond
    ((zero? n) a-board)
    (else (local ((define available-posn (get-available-posn a-board)))
            (cond
              ((empty? available-posn) #f)
              (else (or (placement (sub1 n) (place (first available-posn) a-board))
                        (placement/list (sub1 n) (place/list (rest available-posn) a-board)))))))))

;returns a possible board after placing n queens on a list of boards
;returns false if all the boards are not valid
(define (placement/list n boards)
  (cond
    ((empty? boards) #f)
    ((zero? n) (first boards))
    ((not (boolean? (placement n (first boards)))) (first boards))
    (else (placement/list n (rest boards)))))