There are 100 doors in a row that are all initially closed.

You make 100 passes by the doors.

The first time through, visit every door and  toggle  the door  (if the door is closed,  open it;   if it is open,  close it).

The second time, only visit every 2nd door   (door #2, #4, #6, ...),   and toggle it.

The third time, visit every 3rd door   (door #3, #6, #9, ...), etc,   until you only visit the 100th door.



Task

Answer the question:   what state are the doors in after the last pass?   Which are open, which are closed?


Alternate:
As noted in this page's   discussion page,   the only doors that remain open are those whose numbers are perfect squares.

Opening only those doors is an   optimization   that may also be expressed;
however, as should be obvious, this defeats the intent of comparing implementations across programming languages.




(defun visit-door (doors doornum value1 value2)
  "visits a door, swapping the value1 to value2 or vice-versa"
  (let ((d (copy-list doors))
        (n (- doornum 1)))
    (if (eql  (nth n d) value1)
        (setf (nth n d) value2)
      (setf (nth n d) value1))
    d))
 
(defun visit-every (doors num iter value1 value2)
  "visits every 'num' door in the list"
  (if (> (* iter num) (length doors))
      doors
    (visit-every (visit-door doors (* num iter) value1 value2)
                 num
                 (+ 1 iter)
                 value1
                 value2)))
 
(defun do-all-visits (doors cnt value1 value2)
  "Visits all doors changing the values accordingly"
  (if (< cnt 1)
      doors
    (do-all-visits (visit-every doors cnt 1 value1 value2)
                   (- cnt 1)
                   value1
                   value2)))
 
(defun print-doors (doors)
  "Pretty prints the doors list"
  (format T "~{~A ~A ~A ~A ~A ~A ~A ~A ~A ~A~%~}~%" doors))
 
(defun start (&optional (size 100))
  "Start the program"
  (let* ((open "_")
         (shut "#")
         (doors (make-list size :initial-element shut)))
    (print-doors (do-all-visits doors size open shut))))