This Math Problem Will Make Your Brain Hurt - But It Could Save Your Life


In a suicide circle, where would you need to stand in order to be the last person alive?

This is the question Flavius Josephus, a Jewish-Roman historian from the first century, asked himself while standing in a circle with 40 other Jewish soldiers, who all chose to die rather than turn themselves over to the enemy.



A company of 40 soldiers, along with Josephus himself, were trapped in a cave by Roman soldiers during the Siege of Yodfat in 67 A.D. The Jewish soldiers chose to die rather than surrender, so they devised a system to kill off each other until only one person remained. (That last person would be the only one required to die by their own hand.)


All 41 people stood in a circle. The first soldier killed the man to his left, the next surviving soldier killed the man to his left, and so on. 



However, Josephus did not want to commit suicide and was worried that if he went against the plan, the others would turn on him.


Instead, Josephus determined where to sit in the circle that would allow him to be the last man standing and then he could hand himself over to the Romans.


Flavius Josephus (pictured) had to determine where to stand in the circle that would leave him the last man alive.


This tale may be apocryphal and fantastic, but it gives rise to a fascinating math problem.

That is: If you're in a similar situation to Josephus, how do you know where to stand so you will be the last man standing?


"Let's say there are seven people in the circle. Person number one kills number two.
Then person number three kills person number four and five kills six.
So seven kills one, then three kills five and seven kills three, which leaves seven as the last person alive.", said Daniel Erman from the University of Wisconsin-Madison, who also spoke in the  fantastic video at the end of the article, featured on the Youtube channel Numberphile.


Erman explains that when you start using this sequence with different numbers of people in the starting circle, you will begin to see distinct patterns emerge throughout the process.



If you start running through this sequence with different numbers of people in the starting circle, you will see a few patterns emerge. First of all, the final survivor is never someone in an even-numbered position because all of the people standing in even-numbered positions are killed first (1 kills 2, 3 kills 4, and so on).


Do enough trial and error and you might notice that any time the starting number of people is a power of 2, the final person standing is the same as the person who started the sequence (position number 1). This is the key to figuring out where you should stand. When the number of people left standing is equal to a power of 2, then you want it to be your turn to kill your neighbor.


As Numberphile demonstrates, you can use math to determine the winning spot beforehand. You just need to figure out what the highest power of 2 is that is smaller than the starting number of people. For Josephus, the starting number is 41, and the highest power of 2 that is fewer than 41 is 32 (2 to the power of 5). You want it to be your turn when there are exactly 32 people left. Because of the way the problem works, with every other person dying, the position you want to stand in is 2 times the difference between 41 and 32 (41 - 32 = 9), plus 1. So, 2 x 9 + 1 = 19.


There's the magic number: Josephus must have been standing in position 19 of the circle (or his fellow survivor was, in which case he was in position 35, second-to-last standing).


The video does a amazing job of explaining this part visually, so give it a watch.




Source: YouTube Numberphile, Popularmechanics.com, Dailymail.co.uk









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