- Ward Sanford

# The Josephus Problem: Counting in Circles

Updated: Apr 23

“The lot is cast into the lap, but its every decision is from the Lord.”

Proverbs 16:33

When the Romans captured the town after the siege at Yodfat in 67 CE (see last blog), Josephus and a band of his Jewish soldiers took to a hiding place in a cave beneath the town (see photos below). They held out for many days but were eventually discovered by the Romans. The Romans made an offer to Josephus that if he surrendered, he would not be harmed. But his soldiers would have none of that. They knew they may end up being crucified, and so they threatened to kill themselves and Josephus if he tried to surrender. So, Josephus had a problem. He thought he could help the Jews and Jerusalem if he was taken captive, for he could negotiate with the Romans, but his soldiers would not permit it.

At present day Yodfat in Galilee one can see caves beneath the hill on which the town stood.

Josephus tried to persuade his soldiers that suicide was against God’s law, but they were not convinced. So, his solution was for them to cast lots to see who would slit the throat of the next one in line. The last two would remain alive to bear witness to what had occurred. In this way God would choose who would live or die, and also no one would die by their own hand. The soldiers agreed. As fate would have it, Josephus was the next to the last chosen by lot and survived. Many critics and enemies of Josephus later claimed his story to be totally invented to cover for his own cowardice at surrendering to the Romans when his soldiers perished. But, in his defense, the second survivor was there to bear witness to the story. Emperor Titus was a witness as well when the account was told to Titus’s father (General Vespasian) right after Josephus was captured.

A number of airholes still exist around the top of the hill that led down to caves. They are currently covered by grates to prevent visitors from being injured.

Josephus did not give a lot of details as to how the lots were cast, so different individuals have suggested different scenarios. This story has, in fact, led to a formal problem in mathematics known as the “Josephus Problem”. Rather than trying to placate suicidal soldiers, the mathematical problem is how to avoid death yourself in this situation. The assumption behind the problem is that a group of people are standing in a circle. Everyone counts off by three where every third person is to be killed. Given there are many persons in the circle, when you reach around to the beginning of the circle, the counting continues around a second time, and then as many times as is needed until only one survivor remains.

Inside one of the larger caves one can look up through the air hole.

The problem to solve is this: If you are a very smart person, where should you stand in the circle to be the survivor? This implies that Josephus might have been a mathematical genius and figured out where to stand so the lot would never fall to him. Even though Josephus does not describe choosing every third or fourth person, or them standing in a circle, this mathematical problem has still intrigued mathematicians for centuries. It might not be too hard to figure out the answer by counting if you knew how many were in the circle, and at what interval there were to be culled. But the true challenge has been to come up with a general formula that can predict the safe position in which to stand given any number of persons in a circle, and for different culling intervals. For more on this mathematical puzzle, I recommend the reader visit https://en.wikipedia.org/wiki/Josephus_problem.

We discovered a well inside one of the larger caves.