The Solving of the Puzzle

I was once faced with the following puzzle.

The object is to draw a single thread through each line in the puzzle (see example 1). The green circle points out the line that was untouched. While the puzzle seems unsolvable I asure you it has a solutions. It is true that you cannot achieve the initial goal but that does not render it unsolvable.

The solution

The solution is a not a simple one but it can be easily understood. First one must start simple think of a single box (see example 2). You can easily draw a single thread through each side just in and out twice.

So what makes the goal impossible to reach? Look at the puzzle some of the boxes have five sides. You'll find it quite easy to draw a thread through five sides out three times and in twice, or vice-versa. (see example 3) You'll even find it easy to draw a line through two five sided boxes, because you can start inside one and end inside the other (see example 4). On a note: In example 3 one end of the thread ends inside and the other outside. The end that exits ouside will go into the second box and since you went inside first you have to complete the second box by going inside and because both ends of the thread are inside boxes there is no way to exit and go into a third box.

But in the puzzle there are three five sided boxes and this renders it impossible to achieve the goal (see example 5).

When you had two boxes you could start by drawing out of one and end drawing into the other. When a third box is added you'll find that when you exit the second box one time you'll have to exit into the third box that leaves four sides on the third box and one on the second box (example 6). You'll have to exit the box then enter it then exit it then enter it to complete it but that leaves the last side on the second box and you can't exit the third box to get to it because you've already filled in all it's sides.

Or if this explanation helps better. Look at example 4 how are you going to be able to exit box 2 to draw into another box.

In short

The solution can simply be put: You cannot achieve the goal of the puzzle because there are more than two boxes with five sides. It doesn't matter how you arrange the boxes (see example 6) and it doesn't matter how many four sided squares (see original puzzle) it will always be impossible.
Example 1Example 2Example 3
Example 4Example 5Example 6