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 can't prove that the given grid is untraversable, but I haven't found a way.

If you move 'D' (that's the capital D) one square to the right you can connect all the pairs in this manner:
A-a: straight down
B-b: left 1, down 6, right 3.
C-c: left 3, down 1, left 1, down 4, right 3, down 1.
D-d: right 1, up 1.
E-e: right 2.
F-f: right 6, up 3, left 3.

I can't prove that this is the only solution.

Posted by Erik O. on 2005-04-22 15:33:07