## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## April 29, 2007

### Defending the Roman Empire

About 1700 years ago, the Roman Empire was under attack, and Emperor Constantine had to decide where to station his diminished forces. Constantine organized his legions into four field armies. He needed to protect eight regions with these forces. The trick was to place the armies so that each region was either occupied by an army or was only one step away from an army. But an army could be sent onward only if there were another army to stay behind and defend its original position.

Take a look at this map (below) showing the regions and the steps between the regions. Where would you put the four armies?

Constantine chose to place two armies in Rome and two at his new capital, Constantinople. This meant only Britain could not be reached in one step. Defending Britain would require moving an army from Constantinople to Rome, then from Rome to Gaul, and finally to Britain—a total of four steps.

Can you do better than Constantine—either by reducing the number of regions that can’t be reached in one step or by cutting the number of steps it would take to get to the worst-off region? Try it and see.

Charles S. ReVelle, an environmental engineering professor at Johns Hopkins University in Baltimore, used a computer to test various possibilities. He came up with several alternatives. Each beats Constantine, but has its own flaws.

One possibility is to put two armies in Rome, one in Britain, and one in Asia Minor. Then every unoccupied region can be reached in one step from Rome. But the emperor would have trouble responding if a second was should suddenly erupt elsewhere in the empire.

Another solution is to put two armies in Iberia and two in Egypt. Again every unoccupied region could be reached in one step from one of these power centers. There is a political problem with this solution, however: Rome itself does not have an army.

Mathematics can help you figure out the best places to put military units, especially when you have a limited number of units and a lot of territory to defend. The same sort of math is also useful when people want to know the best place in town to put a new hospital, fire station, or fast-food restaurant.

Muse, October 2001, p. 34.