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.




 
begin
 
comment - 100 doors problem in ALGOL-60;
 
boolean array doors[1:100];
integer i, j;
boolean open, closed;
 
open := true;
closed := not true;
 
outstring(1,"100 Doors Problem\n");
 
comment - all doors are initially closed;
for i := 1 step 1 until 100 do
  doors[i] := closed;
 
comment
  cycle through at increasing intervals
  and flip each door encountered;
for i := 1 step 1 until 100 do
  for j := i step i until 100 do
    doors[j] := not doors[j];
 
comment - show which doors are open; 
outstring(1,"The open doors are:");
for i := 1 step 1 until 100 do
  if doors[i] then
     outinteger(1,i);
 
end
 