Where in the World is Hermione Granger?

 Hermione has decided to get away from saving the world from He-Who-Must-Not-Be-Named by traveling the world.  In order not to attract attention she decides not to use magic and chooses instead to partake in the muggle way of traveling. Her airline of choice is United Airlines.  She has chosen to go to Japan (J) to learn Ninjutsu, Australia (A) to observe Kangaroos, New Zealand (NZ) to watch the All Blacks Rugby Team, and South Africa (SA) for the World Cup.

Philadelphia- Australia (9,689) Philadelphia- New Zealand (8,739) Philadelphia- Japan (6,771 mi) Philadelphia- South Africa (8,764 mi) New Zealand-Japan (5,555 mi) New Zealand – Australia (1,338 mi) South Africa- New Zealand (7,192 mi) Japan- South Africa (9,134 mi) Japan- Australia (4,345 mi) Australia- South Africa (6,235 mi) P SA J A NZ

Philadelphia (P) Japan (J)New Zealand (NZ) Australia (A) South Africa (SA) Philadelphia (P) - 6,771 mi8,744 mi9,689 mi8,764 mi Japan (J)6,771 mi-5,472 mi4,345 mi9,134 mi New Zealand (NZ) 8,744 mi5,472 mi-1,338 mi7,253 mi Australia (A)9,689 mi4,345 mi1,338 mi-6,235 mi South Africa (SA) 8,764 mi9,134 mi7,253 mi6,235 mi-

 (n-1)! is the formula used to find the total number of Hamilton circuits in the graph.  Since we have 5 vertices the formula would give: (5-1)!  This simplifies to 4! Which means there are 4 x 3 x 2 x 1 = 24 circuits.

 Hermione chooses the shortest distance from each country/state beginning from home. Thus from Philadelphia she went to Japan.  From Japan, she chooses the shortest distance which took her to Australia; continuing to choose the shortest distance from each country/state, she then went to New Zealand, and on to South Africa before returning to Philadelphia.  This gave her a total distance of 28,471 miles.

Beginning and ending at P, the total distance is 28,471 miles. 6,771 4,345 1,338 7,253 8,764 P J A NZ SA

 She chooses the cheapest (shortest) edge avoiding mini – circuits and/or adding edges that result in a vertex with three edges (claw) to obtain the following:  New Zealand -- Australia: 1,338 miles  Japan -- Australia: 4,345 miles  South Africa -- Australia: XXX (Claw)  Japan – Philadelphia: 6,771 miles  South Africa – New Zealand: 7, 253 miles  South Africa – Philadelphia: 8,764 miles  Total = 28, 471 miles

P SA J A NZ 1,338 7,253 6,771 8,764 4,345

 In conclusion Hermione realizes that her best (optimal) route is given by either algorithm since they both produce the same route and total mileage: P – J – A – NZ – SA – P: 28, 471 miles  She did not use the Brute Force Algorithm because she could just use magic for that!  We wish her luck in her new film and her fight against He-Who-Must-Not-Be-Named.He-Who-Must-Not-Be-Named.

 Terry, E., Class Notes, July  Wikipedia, Internet, July 2010.