You have to calculate the total distance along all the roads. With a simple x that should be easy enough with Pythag.

With the more complex Stefy-x it will help to know the angle between the lines is 120 degrees.

Bob

]]>The solution looks something like in the picture below.

]]>Close, but you can get shorter.

Oh yes, in case you have misunderstood: it is not required that every city is connected independently to each of the others. You jsut have to be able to get from one to another by road(s)

Bob

]]>Unfortunately, the soap will get you only a path which connects all the roads and that has minimal length and which probably doesn't go from any city to any other with a direct road, ie. the cities are not the only nodes int the "graph" formed by the soap film.

]]>Now if we could just get soap bubbles to solve traveling salesman problems ... hang on, is this remotely possible? Maybe being only 3-dimensional is a limiting factor.

]]>This result can also be proved rigorously, but I thought it might be interesting to mention this.

]]>