Note that Mathematica will not always be able to find the best possible route, so it is conceivable that you will be able to find a shorter route than Mathematica's best. While this is not likely, especially when there are only a small number of cities, if you play enough (and are skillful) you will eventually encounter this phenomenon.

Tips for game play: The scoreboard will reset when the number of cities is changed. The scoreboard is updated when the "new game" button is pushed after you have created a valid tour. You may undo an edge or clear your entire path without affecting your score. The radius slider can be used to find the closest city to your current location. Just crank it up until the red disk encounters its first dot, then slide it back. You may safely ignore this feature, although occasionally it can be quite useful (if you wish to pursue a "nearest neighbor" strategy, for instance). Serious gamers should hide both the distance tally and Mathematica's best tour during play.