Twelve bugs – two of six different species – want to find their respective mates. Your job is to connect A to a, B to b, …, and so forth using an unbroken path between each bug pair. Your lines must travel through the center of each square in the array, and you can only travel up, down, left and right (not diagonally).

Because the bugs leave behind poison trails, no path can cross another, and no path can cross itself. When you are finished, every square must have been traversed once and only once.

 _ _ _ _ _ _ _
|A|_|B|_|_|_|C|
|_|_|_|_|d|_|_|
|_|_|D|_|_|_|_|
|_|_|_|E|_|_|e|
|_|_|_|F|_|_|_|
|_|_|_|_|_|_|_|
|a|_|_|_|b|c|_|
|f|_|_|_|_|_|_|

Prove whether or not there is a solution.

If you think there is a bug in the problem, can you move EXACTLY one bug EXACTLY one square from its original position (not diagonally) and find a solution? Would it be unique?

The bulk of this problem was created by Clifford Pickover

I don't know if this can be considered as proof (There is some uncertainty).

Here's my reasoning:
1) The line connecting A-a must go left of D:
If that line was to travel inbetween B D, then at d it cannot go above d (B is isolated) and it cannot go under d because then A-a, B-b, D-d and C-c must all pass between E-e and there is only room for two lines.
2) The line connecting B-b must go left of D:
If that line was to travel right from D above d, then C is isolated.  And if the line went under d, then B-b, D-d and C-c must all pass between E-e an,d there is only room for two lines
3) With the line B-b going left of D, forces the F-f line to go under and to the right of c.  Let's cal this point f2
4) The lines connecting F-f2 and C-c must cross as the end points are situated at opposite sites of the other line.
5) To fulfill the condition in 4, you must go around one end point, thus avoiding the crossing condition.
6) I is not possible to go around F (Left of D is blocked) or around C (sits in a corner) or around f2 (sits at a side) or around c (sits between b and f2 and under c the square is occupied by F-f2)
  
Edited for the unique route part of the question, Mark has already given the answer

Edited on April 22, 2005, 4:12 pm

Posted by Hugo on 2005-04-22 16:11:19