The Josephus Problem, named after the first$-$century historian Flavius Josephus, goes as follows: During the Jewish uprising against the Romans, Josephus was part of a group of $41$ Jewish soldiers who were trapped in a cave by the Romans. Rather than surrender, the group determined that they would commit suicide, but in a rather orderly way: they formed a circle, and agreed that every third person would be killed until no one was left $($the final person was trusted to kill themselves$).$ However, Josephus and a friend were more willing to accept capture than to die, so they quickly determined the two spots to stand in the circle such that they would be the last two standing at the end$\mathrm{.}1$ First StepsQuestion $1$: Determine the final position in Josephuss account.$($The second$-$to$-$last posi$-$ tion is important to Josephus unnamed accomplice, but we dont need it for our purposes.$)$ Number the soldiers with the numbers $1$ through $41,$ and, beginning with $1,$ count off every three people and remove them from the circle. That is$,$ the first soldier to be killed is number $3,$ then $6,$ etc. Upon reaching the end of the circle, youll continue counting off by threes: so$,$ after soldier $39$ is killed, the next will be soldier $1,$ then soldier $5($three spots down from Soldier $1,$ since soldier $3$ has already been removed.$)$