You want to conduct a tour of this museum:

A-B-C
| | |
D-E-F
| | |
G-H-I
| | |
J-K-L

You want to walk through all the hallways once. What would be the least amount of walking you would need to do (each hallway is a kilometer long) to cover all the hallways at least once? (You can start and stop anywhere.)

(In reply to Solution by Jer)

As has already been pointed out, a network is traversable if it has either zero or two ODD nodes.  This network has six odd nodes:  B,D,F,G,I, and K.  If none the odd nodes were connected to another odd node by a hallway, then you would need 4 extra moves, or 21 total.  But since DG is a path that connects one odd node with another odd node, making DG one of the extra moves knocks out 2 odd nodes with one move.  Same for FI.  This is why Jer's solution requires only 2 extra moves, or 19 total.  I think this qualifies as a proof that 19 is the minimum.

Posted by Larry on 2004-06-04 10:14:09