In computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. Such games are used to pick out a person from a group, e.g. eeny, meeny, miny, moe.
In the particular counting-out game that gives rise to the Josephus problem, a number of people are standing in a circle waiting to be executed. Counting begins at a specified point in the circle and proceeds around the circle in a specified direction. After a specified number of people are skipped, the next person is executed. The procedure is repeated with the remaining people, starting with the next person, going in the same direction and skipping the same number of people, until only one person remains, and is freed.
The problem—given the number of people, starting point, direction, and number to be skipped—is to choose the position in the initial circle to avoid execution.
History
The problem is named after Flavius Josephus, a Jewish historian living in the 1st century. According to Josephus' firsthand account of the siege of Yodfat, he and his 40 soldiers were trapped in a cave by Roman soldiers. They chose suicide over capture, and settled on a serial method of committing suicide by drawing lots. Josephus states that by luck or possibly by the hand of God, he and another man remained until the end and surrendered to the Romans rather than killing themselves. This is the story given in Book 3, Chapter 8, part 7 of Josephus' The Jewish War (writing of himself in the third person):
The details of the mechanism used in this feat are rather vague. According to James Dowdy and Michael Mays, in 1612 Claude Gaspard Bachet de Méziriac suggested the specific mechanism of arranging the men in a circle and counting by threes to determine the order of elimination. This story has been often repeated and the specific details vary considerably from source to source. For instance, Israel Nathan Herstein and Irving Kaplansky (1974) have Josephus and 39 comrades stand in a circle with every seventh man eliminated. A history of the problem can be found in S. L. Zabell's Letter to the editor of the Fibonacci Quarterly.
As to intentionality, Josephus asked: “shall we put it down to divine providence or just to luck?” But the surviving Slavonic manuscript of Josephus tells a different story: that he “counted the numbers cunningly and so managed to deceive all the others”. Josephus had an accomplice; the problem was then to find the places of the two last remaining survivors (whose conspiracy would ensure their survival). It is alleged that he placed himself and the other man in the 31st and 16th place respectively (for = 3 below).
Variants and generalizations
A medieval version of the Josephus problem involves 15 Turks and 15 Christians aboard a ship in a storm which will sink unless half the passengers are thrown overboard. All 30 stand in a circle and every ninth person is to be tossed into the sea. The Christians need to determine where to stand to ensure that only the Turks are tossed. In other versions the roles of Turks and Christians are interchanged.
describe and study a "standard" variant: Determine where the last survivor stands if there are people to start and every second person ( = 2 below) is eliminated.
A generalization of this problem is as follows. It is supposed that every th person will be executed from a group of size , in which the th person is the survivor. If there is an addition of people to the circle, then the survivor is in the -th position if this is less than or equal to . If is the smallest value for which , then the survivor is in position .
Solution
In the following, denotes the number of people in the initial circle, and denotes the count for each step, that is, people are skipped and the -th is executed. The people in the circle are numbered from to , the starting position being and the counting being inclusive.
k=2
The problem is explicitly solved when every second person will be killed (every person kills the person on their left or right), i.e. . (For the more general case , a solution is outlined below.) The solution is expressed recursively. Let denote the position of the survivor when there are initially people (and ). The first time around the circle, all of the even-numbered people die. The second time around the circle, the new 2nd person dies, then the new 4th person, etc.; it is as though there were no first time around the circle.