Challenge question

Its impossible.

http://en.wikipedia.org/wiki/Walls_&_Lines

thanks! Now I have the prove that is impossible :)

For reference, label the "rooms" as A and B in the top row; C, D, and E in the bottom row. Note that A, B, and D each has (5) "wall" segments that the path must cross. No matter how you do it, a path going through every segment will have one end in the "room" and the other end out. Thus your path must have (3) ends, one each in "room": A, B, and D. Impossible.

But your diagram is not the stated challenge. Your diagram is one of the topological shapes that this is not possible. The challenge is :

It is possible to chop up every line of a lower character with one uninterrupted line.

The lower character here is the lower character shapes of text, I believe. So this finite set of characters are " a b c d e f g h i j k l m n o p q r s t u v w x y z ". Since none of these characters create any more than two regions this is a trivial task. Also the way the question is asked, one considers each character individually. To do the entire alphabet with one red line then the inner region of e, p, and q must be bound by two black lines and not one. This will mandate that one must start or end at the character o.

(Maybe the lower character was played by George Spelvin. But who'd want to cut him?)

Do the rules say that you can't fold the paper over? milo

Perhaps not explicitly, but why not take the challenge "straight" rather than devise evasive trickery? It might be almost impossible for the challenge poser to anticipate every weird idea. So, no, you can't go up to a wall and immediately retreat, or go through one of the corners, or go twice through the same wall segment, etc.

Well "out of the box thinking" could be considered "evasive trickery."

Or itcould be a creative approach to solving the problem.

milo

One can draw this diagram on a torus. If one of the (5)-wall-segment "rooms" surrounds the hole, the puzzle can be solved. However, I don't think these variations represent the intent of the OP. Like the Konigsberg bridges, the problem is interesting and instructive in its own right, without getting "weird." Of course, tastes vary as to what is "weird."

This is identical to the Seven Bridges of Konigsberg problem. Each enclosed area is a node and the number of exits plus entrances must be even (black lines) for every one. Note that there are three with five sides, and thus it is impossible. A non-closed path can start at an odd number (doesn't need to enter) and end at an odd number (doesn't need to exit).