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MatheMUSEments
Defending the Roman Empire
By Ivars Peterson
Muse, October 2001, p. 34.
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 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.
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