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The Josephus Problem

For our first article in the series, I want to introduce a topic that not only includes high-level math concepts but some fascinating Jewish History as well. Yosef ben Matityahu, as Josephus was known to the Jews, was born in the Rome-occupied Jerusalem in 37 CE. He became a community leader and an army commander during the time of the Roman invasion of Israel which ultimately resulted in the destruction of the 2nd Beit Hamikdash. During his defense of the Galilee region, Josephus and his group of 40 Jewish soldiers were trapped in a fortress surrounded by the Roman army. They knew that capture was not an option, as they would be tortured and their bodies mutilated, so they decided that death was the only option. But, there was another problem. In Jewish law, it is better to kill someone than to commit suicide. So, together they devised a system where all the soldiers got in a circle and each soldier killed his comrade to the left of him (in succession) until there would be one soldier left who would commit suicide. This plan would minimize the number of suicides in an organized manner thereby solving both of their initial problems. However, Josephus had a different plan in mind: he wanted to survive and turn himself into the Romans. So, knowing that he couldn’t run away since his soldiers would turn on him, he had to calculate which place in the circle he had to be in order to survive this “circle of death.”

There were 41 people in the circle, and Josephus correctly calculated that in order to survive he would have to be number 19. Josephus’ plan worked and after turning himself into the Romans, he became a famous historian by the name of Josephus Flavius and lived to tell this incredible story. Since then, this story has become a famous math problem. Many have wondered how Josephus knew which spot in the circle to be in, and whether or not there is a formula to consistently determine which spot will survive given any number of people. When approaching this problem, it initially seems daunting, and most people don’t know where to start. In my experience, when doing math problems that you don’t know how to solve, you start by collecting data. In this case, we start off by setting up a T chart, where in one column we write the number of people in the circle, and on the other side write the position that survived (which we calculate manually). After collecting a little bit of data, our table should look like this:

People (n) Position
1 1
2 1
3 3
4 1
5 3
6 5
7 7
8 1
9 3
10 5
11 7
12 9
13 11
14 13
15 15
16 1

 

At this point, we may be noticing a pattern. The winning position keeps going up by two but resets at every square of two.  Looking at this pattern we can come up with the formula,  n = 2A + L, where 2A is the power of 2 closest to n, and L is the difference between 2A and n. We can now see that 2L+ 1 is the winning spot. To test this hypothesis we can try it on a few examples. Let’s say there are seven soldiers. The closest power of two is two squared, which equals 4. L equals seven minus four, which is three. To get the winning spot we now multiply L by two, which gives us six; and then we add one, giving us seven as the winning spot. If we look back at our table, we see that seven is indeed the winning spot. Now let’s try this for the original problem. In the original problem we had forty-one soldiers, so the closest power of two is 25, which is thirty-two. We then have to find L so we subtract 32 from 41 and get nine. To find the winning spot we now have to multiply 9 by 2 which equals 18. From there you add one spot and your final answer is 19.  

 

For the full proof, please watch this video- https://www.youtube.com/watch?v=uCsD3ZGzMgE. This is from Numberphile, a very valuable Youtube channel which discusses a number of interesting math concepts

 

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