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

(In reply to Move 1 bug by Bruno)

There is no unique solution by moving a bug one orthogonal space. Though there is only one bug, D, that can be moved one orthogonal space that permits a solution:

4 variations on the solution for D moved right to (4,6):
• • a to A : (2,1), (1,8)
  • b to B : (2,5), (2,2), (2,8), (3,8)
  • c to C : (6,2), (6,3), (3,3), (3,7), (4,7), (4,8), (7,8)
  • d to D : (5,7), (7,7), (7,6), (4,6)
  • e to E : (7,5), (4,5)
    
• • a to A : (2,1), (1,8)
  • b to B : (2,5), (2,2), (2,8), (3,8)
  • c to C : (6,2), (6,3), (3,3), (3,7), (4,7), (4,8), (7,8)
  • d to D : (5,7), (5,6), (4,6)
  • e to E : (7,5), (7,7), (6,7), (6,5), (4,5)
    
• • a to A : (2,1), (1,8)
  • b to B : (2,5), (2,2), (2,8), (3,8)
  • c to C : (6,2), (6,3), (3,3), (3,7), (4,7), (4,8), (6,8), (6,7), (7,7), (7,8)
  • d to D : (5,7), (5,6), (4,6)
  • e to E : (7,5), (7,6), (6,6), (6,5), (4,5)
    
• • a to A : (2,1), (1,8)
  • b to B : (2,5), (2,2), (2,8), (3,8)
  • c to C : (6,2), (6,3), (3,3), (3,7), (4,7), (4,8), (6,8), (6,6), (7,6), (7,8)
  • d to D : (5,7), (5,6), (4,6)
  • e to E : (7,5), (4,5)
    

2 variations for the solution for D moved up to (3,7):
• • a to A : (2,1), (1,8)
  • b to B : (2,5), (2,2), (2,8), (3,8)
  • c to C : (6,2), (6,3), (3,3), (3,6), (7,6), (7,8)
  • d to D : (5,7), (6,7), (6,8), (4,8), (4,7), (3,7)
  • e to E : (7,5), (4,5)
    
• • a to A : (2,1), (1,8)
  • b to B : (2,5), (2,2), (2,8), (3,8)
  • c to C : (6,2), (6,3), (3,3), (3,6), (7,6), (7,7), (6,7), (7,6), (6,8), (7,8)
  • d to D : (5,7), (5,8), (4,8), (4,7), (3,7)
  • e to E : (7,5), (4,5)
    

Posted by Dej Mar on 2008-10-01 00:34:57