History
The problem is named after Flavius Josephus, a Jewish historian living in the 1st century. According to Josephus' 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''. 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. If the initial number of people was even, then the person in position during the second time around the circle was originally in position (for every choice of ). Let . The person at who will now survive was originally in position . This yields the recurrence : If the initial number of people was odd, then person 1 can be thought of as dying at the end of the first time around the circle. Again, during the second time around the circle, the new 2nd person dies, then the new 4th person, etc. In this case, the person in position was originally in position . This yields the recurrence : When the values are tabulated of and a pattern emerges (, also the leftmost column of blue numbers in the figure above): This suggests that is an increasing odd sequence that restarts with whenever the index ''n'' is a power of 2. Therefore, if ''m'' and is chosen so that and , then . It is clear that values in the table satisfy this equation. Or it can be thought that after people are dead there are only people and it goes to the th person. This person must be the survivor. So . Below, a proof is given by induction. Theorem: If and , then . Proof: The strong induction is used on . The base case is true. The cases are considered separately when is even and when is odd. If is even, then choose and such that and . Note that . is had where the second equality follows from the induction hypothesis. If is odd, then choose and such that and . Note that . is had where the second equality follows from the induction hypothesis. This completes the proof. can be solved to get an explicit expression for : : The most elegant form of the answer involves the binary representation of size : can be obtained by a one-bit left cyclic shift of itself. If is represented in binary as , then the solution is given by . The proof of this follows from the representation of as or from the above expression for . Implementation: If n denotes the number of people, the safe position is given by the function , where and . Now if the number is represented in binary format, the first bit denotes and remaining bits will denote . For example, when , its binary representation is n = 1 0 1 0 0 1 2m = 1 0 0 0 0 0 l = 0 1 0 0 1Bitwise
The easiest way to find the safe position is by using''k''=3
In 1997, Lorenz Halbeisen and Norbert Hungerbühler discovered a closed-form for the case . They showed that there is a certain constant : that can be computed to arbitrary precision. Given this constant, choose to be the greatest integer such that (this will be either or ). Then, the final survivor is : if is rounded up else for all . As an example computation, Halbeisen and Hungerbühler give (which is actually the original formulation of Josephus' problem). They compute: : : and therefore : (note that this has been rounded down) : This can be verified by looking at each successive pass on the numbers through : : : : : : : : :The general case
References
Citations
Sources
* * * * * * * * *Further reading
* * * * * * * FUN2010 * * * * *External links