Johns Hopkins Magazine -- June 1997
Johns Hopkins Magazine

JUNE 1997

S C I E N C E    &    T E C H N O L O G Y

Test Your Solution to
"Can You Protect the Roman Empire?"

In April's feature story on location science, professor Charles ReVelle invited readers to match wits with Emperor Constantine in tackling one of the world's oldest recorded location problems ("Can You Protect the Roman Empire?" p.40). Below, Professor ReVelle lays out possible solutions:

To review, a region was secured if it already had a pebble (six legions) within its borders. Alternatively, a region was securable if a pebble could reach the region in just one step, and that could occur only if somewhere, within one step of the region, there were one or more places that had two pebbles garrisoned. That is, a pebble could only be "launched" to its destination by another pebble.

The problem we pictured last time was meant to get you "warmed up" for greater challenges. There is, looking exhaustively at the possibilities for Figure 1, just one alternative that secures or makes securable all the regions of the empire. That single alternative, as shown in Figure 1, places two pebbles at Rome, as well as one pebble in Britain and one in Asia Minor. With this deployment, Britain, Asia Minor, and Rome are secured and all other regions can be reached by a pebble launched from Rome. It is a "Roman Solution"; that is, Rome, the heart of the empire, is not without forces at its disposal.

That alternative, of course, is different from the choice that Constantine made. Constantine placed two pebbles at Rome and two pebbles at Constantinople, securing or making securable all but Britain--which he, not surprisingly, ended up losing. Why, then, did Constantine pursue this strategy?

Probably Constantine excluded the apparently stronger alternative because it was not "robust." By this we mean that, although the deployment protected all regions in the event of a first war, it lacked the capacity to perform well if, elsewhere in the empire, the forces of insurrection or invasion opened a second front. To see why this is so, consider the solution (2-R, 1-B, 1-AM) in place, and imagine a first war breaking out anywhere in the empire that a pebble does not currently reside. This could be in Gaul, Iberia, North Africa, Egypt, or Constantinople. Suppose, for example, the war is in Gaul and a pebble is dispatched from Rome to Gaul, where it now must reside for a time while order is restored. With this pebble deployed, how many regions will now be unprotected? The other four regions previously protected by the two-pebble unit in Rome.

This suggests the need to place two-pebble units in such a way as to double-cover regions--so that they are still protected in the event of a second war. In sum, the described alternative (2-R, 1-B, 1-AM), though it appeared strong at first glance, can now be seen to have a serious weakness: an inability to respond to a second war. To correct for this weakness, we need to expand our thinking to alternatives that take into consideration readiness in the event of a second war.

Before we go on, however, we have to explain why we called this problem a "warm-up." We had to get you thinking about alternatives and show how a deployment that looked good initially really wasn't so strong. Now we come to the tougher problem.

In Figure 2, the Roman Empire looks slightly different than in Figure 1. The same regions are shown, but two further connections are displayed--one between Iberia and Britain, the other between Egypt and Asia Minor. The problem shown in Figure 2 is, in fact, the real Roman Empire. These two further connections complicate the problem since they introduce new possibilities for coverage. The deployment (2-R, 1-B, 1-AM) still protects the entire empire in the event of a first war, so it is still a solution. But now, the new connections make possible other deployments that also protect the entire empire.

What are these other deployments that protect the entire empire? Which of these does the best in the event of a second war? Does Constantine's solution look any better than before?

We'll feature a thorough discussion of the expanded Roman Legions problem in September's web version of the magazine.