The FedEx Problem

Federal Express is a company that ships packages from one city to another by airplane. In the United States it uses a system that might at first glance seem very odd and inefficient. The challenge of this puzzle is to understand why the company uses these routes.

US map

If you are sending a package from New York City, the blue dot, to Boston, the green dot, it first goes by plane to Memphis, the red dot, and then on to Boston. Similarly, a package going from Los Angeles, yellow dot, to San Francisco, light blue dot, also goes through Memphis. Packages from cities all over the USA are brought to Memphis, where they are sorted according to their destinations. Then each plane returns to the city it came from loaded with packages for delivery there.

But it’s shorter to go directly from New York to Boston and from Los Angeles to San Francisco than going through Memphis. How can all this extra air travel make sense?

Course:

Math [1]
Algebra [2]

Result/Solution(s)

Solution: The FedEx Problem Math Puzzle

If you look at any one shipment, the system seems silly and wasteful, but let’s look at the big picture. For now imagine that FedEx serves only Boston, New York, Los Angeles, and San Francisco. People need to send packages from each city to each of the others. Here are the flights we need if we ship directly from each city to each of the others:

New York Boston
New York Los Angeles
New York San Francisco
Boston New York
Boston San Francisco
Boston Los Angeles
San Francisco New York
San Francisco Boston
San Francisco Los Angeles
Los Angeles New York
Los Angeles Boston
Los Angeles San Francisco

That’s 12 flights. Now how does this work with everything going through Memphis? Here are the flights we need:

New York Memphis
Boston Memphis
Los Angeles Memphis
San Francisco Memphis
Memphis New York
Memphis Boston
Memphis Los Angeles
Memphis San Francisco

That’s only 8 flights. And, we get a 5^th city thrown in as well: Memphis

If only 3 cities were served, it would require 6 flights by either method, but with 4 or more, the FedEx “hub” system requires fewer flights. And the difference gets much greater very quickly. With 10 cities, we’d need 90 direct flights: a flight from each of the 10 cities to the 9 others. 10 x 9 = 90.

With a central hub only 20 flights would be needed: 10 flights from the cities to the hub and 10 return flights.

What happens when there are 100 cities?