Josephus Problem formula

In computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. A drawing for the Josephus problem sequence for 500 people and skipping value of 6. The horizontal axis is the number of the person. The vertical axis (top to bottom) is time (the number of cycle). A live person is drawn as green, a dead one is. The original Josephus problem consisted of a circle of 41 men with every third man killed (,), illustrated above, where the outer number indicates the order in which a given man is killed. In order for the lives of the last two men to be spared, they must be placed at positions 31 (last) and 16 (second-to-last) The problem has following recursive structure. josephus(n, k) = (josephus(n - 1, k) + k-1) % n + 1 josephus(1, k) = 1. After the first person (kth from beginning) is killed, n-1 persons are left. So we call josephus(n - 1, k) to get the position with n-1 persons. But the position returned by josephus(n - 1, k) will consider the position starting from k%n + 1. So, we must make adjustments to the position returned by josephus(n - 1, k) Der jüdische Historiker Flavius Josephus (37-95) berichtete davon, dass er mit 40 anderen Juden vor den Römern in einen Keller flüchtete. Um dem Feind nicht in die Hände zu geraten, beschlossen sie, sich gegenseitig umzubringen - nur Josephus war dagegen. Deshalb schlug er vor, sich in einem Kreis aufzustellen und jeweils jeden Dritten auf der Stelle zu erschlagen. Da er sich geschickt in den Kreis stellte, blieb er als Letzter übrig und überlebte. In Auswertung der. Josephus Problem Statement. We are given the natural numbers n and k . All natural numbers from 1 to n are written in a circle. First,... Modeling a O ( n) solution. We will try to find a pattern expressing the answer for the problem J n, k through the... Implementation. This formula can also be.

Description (from Rosetta Code) Josephus problem is a math puzzle with a grim description: prisoners are standing on a circle, sequentially numbered from to . An executioner walks along the circle, starting from prisoner , removing every -th prisoner and killing him. As the process goes on, the circle becomes smaller and smaller, until only one. def josephus(n,k): l = list(range(1,n+1)) josephus_permutation = [] m=0 while len(l)>0: for i in range(k): p = (i+m) % len(l) if p == len(l)-1: m=0 else: m=p josephus_permutation.append(l[p]) del l[p] return josephus_permutatio An alternating elimination Josephus problem has a deep connection to the powers of two, a connection reflected in the formula we derived to find the winning spot. The formula requires a few simple calculations, and is a function of the number of participants n: find the largest power of two in n, subtract it from n, double the result, and add 1. The person in that spot will be the winner

The Josephus Problem asks where to start taking out every kth person in the circle consisted of n people, such that you are the last survivor. The following recursive formula is given: f (1, k) = 1, f (n, k) = ((f (n − 1, k) + k − 1) mod n) + 1. But this is not enough explanation, so I don't get where does it come from use zero-indexed counting, will simply the recursion formula. if (n==0), return 0; else { return josephus(n-1, k) + k) % n; the insight behind zero-indexed and one-indexed counting are the same, but the process is much easier - Yossarian42 Sep 10 '20 at 13:5 $\begingroup$ Just that you can eliminate the dependence on k by renumbering the people, then add back in the k after you have solved the problem. So if there are 10 people and you start counting with 1, the sixth one eliminated is number 3 and the seventh is 7. So if you first eliminate 7 (instead of 2) the sixth one eliminated will be $3+5 \pmod {10}=8$ and the seventh will be $7+5 \pmod {10. Das Josephus-Problem Wir stellen wieder ein klassisches Problem vor, das uns auf die zu betrach-tenden Rekursionen fuhren wird: Flavius Josephus, bekannter judischer Historiker { der insbesondere eine der wenigen Quellen zum Leben Jesu liefert { soll im r omisch-j udischen Krieg mit 41 Kameraden den Selbstmord der Gefangenschaft vorgezogen haben. Dazu stellten sie sich in einem Kreis auf.

Josephus problem - Wikipedi

In computer science and mathematics, the Josephus Problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. There are people standing in a circle waiting to be executed. The counting out begins at some point in the circle and proceeds around the circle in a fixed direction. In each step, a certain number of people are skipped and the next person is executed. The elimination proceeds around the circle (which is becoming smaller and smaller as the. Josephus problem | Set 1 (A O(n) Solution) In this post, a special case is discussed when k = 2. Examples : Input : n = 5 Output : The person at position 3 survives Explanation : Firstly, the person at position 2 is killed, then at 4, then at 1 is killed. Finally, the person at position 5 is killed. So the person at position 3 survives. Input : n = 14 Output : The person at position 13 survive Recursive Formulas for the Josephus Problem. Ask Question Asked 4 years, 3 months ago. I've recently been looking at sites trying to prove the Josephus Problem lately, such as the Wikipedia page, or this cut-the-knot site but I'm confused as to how they came up with these relationships: f(2j) = 2f(j) - 1, if the number of people is even. f(2j+1) = 2f(j) + 1, if the number of people is odd. Das Josephus-Problem. Posted by Erich Neuwirth on 8. Januar 2018 in Allgemein | ∞. Eine genaue Beschreibung des Hintergrunds des Problems findet sich in der Wikipedia. Problemstellung: Eine bestimmte Zahl von Personen (n) steht im Kreis. Es gibt eine Startperson. Geben wir ihr Nummer 1 und nummerieren wir fortlaufend im Kreis herum. Dann wird jede zweite Person aus dem Spiel genommen. The Josephus Problem To find the general formula, we can use the fact that any number can be written as a power of 2 plus a remainder (r). Note: after r steps, whoever's turn it is will be the winner as we will be left with a power of 2. When there are 5 soldiers, the winning position is 3..

To solve this problem we use the following recursive formula: Josephus (n, k)= (JosephusProblem (n-1, k)+k-1)%n +1 Josephus (n-1, k)+k-1)%n will give an answer between 0 to n-1 and then finally we add 1 to it Josephus problem You are encouraged to solve this task according to the task description, using any language you may know. Josephus problem is a math puzzle with a grim description: prisoners are standing on a circle, sequentially numbered from . to . An executioner walks along. The mathematical formula to the problem is provided in the video and can be summarised as follows. If n = p + L, where n is the total number of people and p (referred to as 2 a in the video) is the greatest power of 2 that is less than n, then the winning position is 2L + 1 return 1 else: (k-1 + josephus (n - 1, k)) % n + 1. By starting with the final index and working backwards, we were able to break down the simplicity of our equation. At the start, we were given. 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.

Formula for Josephus Problem: The answer is obtained by circular shifting of binary form of n to left [1]. Following formula for the answer, was derived from a mor In simple terms Josephus problem is all about finding a position in a circular arrangement which would be safe if executions were handled out using a skip parameter which is known beforehand

Josephus Problem -- from Wolfram MathWorl

  1. http://christianbender.bplaced.net/https://de-de.facebook.com/LehrvideosAUFGABEN ZUM VIDEO:1. Spielen Sie das Josephus Problem mal auf dem Notizblock durch...
  2. If you've ever played the game Ninja, you can think about the Josephus problem in terms of the game if you only needed one hit to get out, the game only proceeded in one direction, let's say clockwise, and the players were perfect, always striking each other out successfully. Your goal is to start in the position where you will win the game
  3. That is just about it for the Josephus Problem which, I think, is a pretty interesting logic problem. I hope you enjoyed learning how to create this and that you will also have fun running the simulation and looking at the patterns in the graphs as well. Download the Excel file and other images here: josephus problem
  4. imum number of moves needed T n =2 n 1 : Let's look at the example borrowed fromMartin Hofmann and Berteun Damman. The.

Josephus problem Set 1 (A O(n) Solution) - GeeksforGeek

  1. By inspection, the solution to the Josephus Problem is a sequence of odd numbers (from 1) starting at each power of 2. This yields a direct closed form expression (see formula below). - Gregory Pat Scandalis, Oct 15 2013. Also zero together with a triangle read by rows in which row n lists the first 2^(n-1) odd numbers (see A005408), n >= 1
  2. Der folgende Algorithmus realisiert das Problem rekursiv nach der obigen Rekursionsformel für den Fall k=2 und besitzt eine Laufzeit von O(log(n)). int josephus(int n) { if(n == 1) return 1; if((n%2) == 0) return 2 * josephus(n / 2) - 1; if((n%2) == 1) return 2 * josephus((n - 1) / 2) + 1;
  3. We give explicit non-recursive formulas to compute the Josephus-numbers j(n, 2, i') and j(n,3,i) and explicit upper and lower bounds for j(n, k, i) (where k > 4) which differ by 2k - 2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j(n, k, i ) which is based upon the mentioned bounds. 1. Introduction The Josephus problem in its original form.
  4. ate the dependence on k by renumbering the people, then add back in the k after you have solved the problem. So if there are 10 people and you start counting with 1, the sixth one eli
  5. g? reference-request.

Eric Weisstein's World of Mathematics, Josephus Problem. Wikipedia, Josephus problem. Index entries for sequences related to the Josephus Problem; FORMULA: To get a(n), write n in binary, rotate left 1 place. a(n) = 2*A053645(n) + 1 = 2(n-msb(n))+1. - Marc LeBrun, Jul 11 2001. Here msb = most significant bit, A053644 From the table above we may surmise the formula. J(2 a + t) = 2t + 1. We prove the formula by induction on a. For a = 1 the only admissible value of t is 0 and we only have to verify that J(1) = 1 which we know to be true. For the induction step, we assume that the formula holds for all a up to a certain value The General Algorithm for Josephus Problem In a matter of fact, we can use the same justifications we used to show that the survivor for ℎ () (where every ℎ person is killed) is computed in the following way: This is the general algorithm. Let us note that for = 2 we'd get in every phase = 2, so for the case of = 2 we get: ℎ2 = 2 ⋅ − 2 log2 + Das Problem wurde nach dem jüdischen Historiker Flavius Josephus benannt, welcher sich 67 n. Chr. beim Kampf um die galiläische Stadt Jotapata mit 40 weiteren Männern in einer Höhle vor den Römern versteckt hielt (insgesamt also 41 Personen). Er berichtet darüber in seinem Buch Jüdischer Krieg (Buch 3, Kapitel 8). Als das Versteck verraten wurde, forderten die Römer sie auf sich in.

Here, 1-indexing makes for a somewhat messy formula; if you instead number the positions from 0, you get a very elegant formula: $$J_ {n, k} = (J _ {(n-1), k} + k) \ \bmod n$$ So, we found a solution to the problem of Josephus, working in $O(n)$ operations Josephus problem illustration and formula test. GitHub Gist: instantly share code, notes, and snippets

Josephus-Problem - Mathematik alph

Josephus Problem - Competitive Programming Algorithm

The Josephus problem has been named after Flavius Josephus, who with other 40 members was blocked by Romans in a cave and they all decided to kill themselves by forming a circle and killing themselves in step of three over capture by the Romans. Josephus states that by luck or possibly by the hand of God, he and another ma Computer Science: A Formula For Generalized Josephus problemHelpful? Please support me on Patreon: https://www.patreon.com/roelvandepaarWith thanks & praise.. FIGURE 1. The Josephus sequence for n = 5, k = 3. Numerous aspects of the Josephus problem and properties of the function j are treated in the literature: In [5] the structure of the permutation is investigated. In [7] a recursion formula for j(n, k, n) is derived (ther The Josephus Problem Close Formula 41 = 2 +9 = 2x9+15 = 19 41. The Josephus Problem Thank You Recommended Explore professional development books with Scribd. Scribd - Free 30 day trial. ICT in Bangladesh Kamrul Hasan. A* Search or Algorithm in Artificial Intelligence | A* Algorithm with Example Kamrul Hasan. What to Upload to SlideShare SlideShare. Customer Code: Creating a Company Customers.

Josephus problem - Fōrmulæ wik

The Algorithm Our algorithm represents a straightforward approach to the Josephus Problem, in that the elements of the n : m Josephus Permutation are generated in permutation order using a counting approach roughly corresponding to the problem description. That is, after an integer is generated 264 ERROL L. LLOYD (i.e., placed into the permutation and thus removed from the circle) we count around the circle m places to locate the next integer to be generated. The time bound is. Josephus-Permutation — Das Josephus Problem oder die Josephus Permutation ist ein theoretisches Problem aus der Informatik oder Mathematik. Es werden n nummerierte Objekte im Kreis angeordnet; dann wird beginnend mit der Nummer s, jedes s te Objekt entfernt, wobei der N = 22 + 2 = 2x + L {where x is the highest power of 2 with 2x <= N} Answer = 2 * L + 1 = 2 * 2 + 1 = 4 + 1 = 5 => [ 1 0 1 ] Now if we take a look at the survivors for every index we can notice that they are in the order [ 1 0 1 ] = 5. Hence this verifies our implementation with the formula In this paper we will study an alternative row version of Josephus problem. Suppose that $n$numbers 1, 2, $¥cdot$ . . , $n$ are arranged in a line from left to right in this order PDF | On Jan 1, 2012, Shaun Sullivan and others published Structured Shuffles and the Josephus Problem | Find, read and cite all the research you need on ResearchGat

Rekursionsformel, f rus. рекуррентная формула, f pranc. formule de récurrence, f Fizikos terminų žodynas . Josephus-Problem — Das Josephus Problem oder die Josephus Permutation ist ein theoretisches Problem aus der Informatik oder Mathematik. Es werden n nummerierte Objekte im Kreis angeordnet; dann wird beginnend mit der Nummer s, jedes s te Objekt entfernt, wobei. In this paper, an image encryption algorithm based on a hyperchaotic system and variable-step Josephus problem is proposed. Based on an in-depth analysis of the classic Josephus problem, a new variable-step Josephus problem that combines the pseudorandom sequence with the Josephus problem is proposed. Firstly, the hash value of the plaintext image is calculated, which is converted to the. Das Josephus-Problem oder die Josephus-Permutation ist ein theoretisches Problem aus der Informatik oder Mathematik.. Es werden nummerierte Objekte im Kreis angeordnet; dann wird, beginnend mit der Nummer , jedes -te Objekt entfernt, wobei der Kreis immer wieder geschlossen wird.Die Reihenfolge der entfernten Objekte wird als Josephus-Permutation bezeichnet

Josephus Problem in Bangla-History-How Josephus Problem Works-Binary Solution-Recursive Format -Mathematical Problem Thanks for watchin We develop a formula for the general case of . N. cards, and then extend that generalization further to cases involving the discard of . k. cards before or after putting one on the bottom of the deck. Finally, we discuss the connection of the Australian Shuffle and its gener- alizations to the famous Josephus problem. Keywords: Josephus; Shuffling . 1. Introduction . A colleague of ours who is. The Orthogonal Josephus Problem Ledah Casburn 1 and Tuyet-Linh Phan2 July 2001 Abstract We consider the Josephus Problem from a new perspective. J(n,k) represents the position of the survivor when n people are eliminated with a skip factor of k. We demonstrate that there exists an explicit formula for J(n,k) when n is fixed. We show that the. This is Josephus Problem. Consider there are 10 persons. They would like to choose a leader. The way they decide is that all 10 sit in a circle. They start a count with person 1 and go in clockwise direction and skip 3. Person 4 reached is eliminated. The count starts with the fifth and the next person to go is the fourth in count. Eventually, a single person remains

This problem is similar to Josephus problem when k=2, the recursive version is easy after referring to the josephus problem on wiki. it is highly recommend to refer to Josephus problem first, because i am chinese, my english is poor, my explanation may not be good, but the wiki explanation is very good.. public int lastRemaining (int n) { return ((Integer.highestOneBit(n) - 1) & (n. The authors have studied variants of the Josephus Problem, and have published our result in , and our article is going to be published in . In this article the authors are going to present new results of our research on the variants of the Josephus Problem. With a proper computer program it becomes very easy to study this variant of the Josephus Problem. Please read the appendix of this. Josephus problem DenisTRYSTRAM LecturenotesMathsforComputerScience-MOSIG1-2018 1 Josephus' problem TheproblemcomesfromanoldstoryreportedbyFlaviusJosephusdurin

math matrix pi fibonacci monte-carlo-simulation factorial prime-numbers golden-ratio tower-of-hanoi prime-factorizations pascals-triangle quadratic-equations josephus-problem greatest-common-divisor euclidean-algorithm perfect-number quadratic-formula maths-problem eratosthenes-prime-numbers automorphic-number The Josephus Problem The Siege of Yodfat took place over the span of forty seven days in 67 CE involving Roman troops who stormed Yodfat, a Jewish town, during the Great revolt, when Jewish civilians staged three major rebellions against the Roman Empire. The fighting took place in Roman-controlled Judea and consequently, the Roman forces overwhelmingly gained victory, prompting Jewish towns. of the traditional Josephus Problem. We are going to use numbers instead of persons in the definition. Definition 2.1. Let n and r be natural numbers. We put n numbers in a circle. We start with the 1st number removing every rth number. We denote by J (n,r) the last number that remains. When. r = 2, the Josephus Problem has a very simple. We give explicit non-recursive formulas to compute the Josephus-numbers j (n, 2, i) and j (n, 3, i) and explicit upper and lower bounds for j (n, k, i) (where k ≥ 4) which differ by 2 k-2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j (n, k, i) which is based upon the mentioned bounds. Détail; BibTeX; Comment citer; MR 1617400 | Zbl 0905. I'm totally hooked on CodeEval, and one of the problems on there caught my attention. Here it is, copied from here: Challenge Description: Flavius Josephus was a famous Jewish historian of the first century, at the time of the destruction of the Second Temple. According to legend, during the Jewish-Roman war he was trapped in a cave with a.

The solution is for Josephus (so-called because of the obvious snickering caused by saying Flavius'' out loud --- try it and see) to stand in the twenty-fourth position. It is yet another historical example of how those with a distaste for mathematics quickly become the chaff of evolution. That point aside, the problem rightfully raises the question of how someone might be able to quickly. The Josephus Triangle offers an interesting twist on the Josephus problem, a long-standing problem in Mathematics and Computer Science. For the purpose of this project, we have changed the context of the problem while retaining its mathematical definition. You are in a group of N people that visit a casino and everyone wins a little bit of money. You suggest it would be better if just one.

Josephus-Problem - Programmieraufgaben

The survivor w(n,3) in a modified Josephus problem, with a step of 3. See A090569 or the reference for the definition of w(n,q). LINKS: Table of n, a(n) for n=1..84. Chris Groër, The Mathematics of Survival: From Antiquity to the Playground, Amer. Math. Monthly, 110 (No. 9, 2003), 812-825. Index entries for sequences related to the Josephus Problem; FORMULA: A recurrence is given in the. #game #josephus-problem-1. Flavius Josephus and 40 fellow rebels were trapped by the Romans. His companions preferred suicide to surrender, so they decided to form a circle and to kill every third person and to proceed around the circle until no one was left. Josephus was not excited by the idea of killing himself, so he calculated the position to be the last man standing (and then he did not. I was going through this really interesting book of mathematical patterns in history, and I came across the Josephus Problem. I tired figuring out the formula of the problem, yet whenever I attempt the formula it never worked. I was wondering if someone could explain how to chose the best place within the Josephus Problem and how does the pattern because I just don't understand it Als Josephus Problem bezeichnet man eine Fragestellung aus einer Abzählproblematik. ( Hi Hermann, ist das etwas für Dich!? ) Dazu eine Sage: Um der Sklaverei zu entgehen, vereinbarten die 40 Belagerten von Masada, sich im Kreis aufzustellen. Dann sollte jeder siebte getötet werden, bis nur noch einer übrig blieb, der Selbstmord begehen sollte. Der spätere Geschichtsschreiber Flavius.

The Josephus Problem The Josephus Problem Introduction The Josephus problem is based around Josephus Flavius; a Jewish soldier and historian who inspired an interesting set of mathematical problems. In 67 C.E., Josephus and 40 fellow soldiers were surrounded by a group of Roman soldiers who were intent on capturing them. Fearing capture, they. The Josephus Problem for n = 12 (link to LaTeX source) As an example, consider the Josephus number J(n) for n = 12, shown in the diagram above. Labels get eliminated in the following sequence: 2, 4, 6, 8, 10, 12, 3, 7, 11, 5, 1 leaving 9 at the very end. J(12)=9. In looking for a general solution to this problem, we first observe that the pattern of elimination is slightly different.

Josephus problem

josephus function j (n; k ; i) fol lowing r e cursion holds j (n +1;k;i +1) = (k + n; k ; i)) mo d (+1); 1;k i) (1) with initial value j (n; k ; 1) = (k 1) mo d n; n 1;k 1): (2) Remark: By \ a mo d b w e mean the non-negativ ein teger remainder of the division of b y. Pro of: The form ula (2) follo ws directly from the de nition. T o see (1) w e pro ceed b y induction. Supp ose, w e kno the v alue of j (n; k ; i Mathematical derivation formula (get the number of the last person out of the circle) - > to learn Resources: solve the Joseph Ring problem with one line of cod

From this table, we observe that the safest position can be derived from the formula 2L+1. Hence, using bit manipulation, we will use this formula to calculate the safest position. Firstly, we will make the highest number 2^x less than the number of soldiers to find L Josephus Problem (Auszählen) 29 . Die Herausforderung. Schreiben Sie eine Funktion, die zwei positive ganze Zahlen n und k als Argumente verwendet und die Nummer der letzten von n verbleibenden Person nach dem Auszählen jeder k-ten Person zurückgibt . Dies ist eine Code-Golf-Herausforderung, also gewinnt der kürzeste Code. Das Problem. n Personen (nummeriert von 1 bis n) stehen in einem. For the standard Josephus problem, q = 2, and so Proposition 1 implies that J(n + 1) _ J(n) + 2(modn + 1), an interesting way to interpret the result stated ear- lier [3] als Abzählfolge, d.h. der 18. wird wohl Flavius Josephus gewesen sein! Verallgemeinert, hat man die Parameter n= Anzahl Personen, m= Abzählwert, s= Startstelle und z= Zielstell

The Josephus problem is an election method that works by having a group of people stand in a circle. Starting at a predetermined person, one may count around the circle n times. Once the n th person is reached, one should remove them from the circle and have the members close the circle The Josephus problem Lorenz Halbeisen and Norbert Hungerbühler We give explicit non-recursive formulas to compute the Josephus-numbers j (n,2,i) and j (n,3,i) and explicit upper and lower bounds for j (n,k,i) (where k > 3) which differ by 2k-2 (for k=4 the bounds are even better)

Powers Of Two In The Josephus Problem - Exploring Binar

Josephus problem The Josephus problem is the following game: \(N\) people, numbered \(1\) to \(N\) , are sitting in a circle. Starting at person 1, a hot potato is passed. the \(M\) th person holding the hot potato is eliminated, the circle closes ranks, and the game continues with the person who was sitting after the eliminated person picking up the hot potato We develop a formula for the general case of N cards, and then extend that generalization further to cases involving the discard of k cards before or after putting one on the bottom of the deck. Finally, we discuss the connection of the Australian Shuffle and its generalizations to the famous Josephus problem. The Australian Shuffle consists of placing a deck of cards onto a table according to. This type of problem is called a Josephus problem, after a story about a historian of the first century, Flavius Josephus, who survived the Jewish-Roman war perhaps due to his mathematical talents. In his book The Jewish Wars Flavius tells that he was one out of 41 Jewish rebels trapped by the Romans. His companions preferred suicide to escape, so they decided to draw lots to see who would kill whom so that they could avoid both capture and the sin of suicide. The idea that they decided to. There are n people sitting in a circle, and you number all of them. Starting from the 1st person, eliminate every m person. So who will be the last person? Can someone please explain it in layman terms for a 13-year-old to understand? Extensions: Given that there are n people in a group, where everyone sits in a straight line, and they are counting off every 3rd person, how can we predict the.

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The Josephus problem belongs to the history and is a perfect example that how mathematics can save your life. There were Jewish soldiers who were captured by roman army, but in order to avoid the capture and probably torture they devised a method of suicide which will ensure the death of each individual. They formed a circle and numbered each member starting from 1 to n, n being the number of. Josephus Problem. 17 Aug 2011. Puzzle. There are n persons in a circle, numbered 1 thru n. Going around the circle, every second person is removed from the circle, starting with person number 2, 4, and so on. Show that the number of the last person remaining in the circle can be obtained by writing n in binary, then moving the leftmost 1 to the right. So for example, with n = 13 persons (1101. Find J (40) —the solution to the Josephus problem for n = 40. 14. Prove that the solution to the Josephus problem is 1 for every n that is a power of 2. 15. For the Josephus problem, compute J (n) for n = 1, 2, . . . , 15. discern a pattern in the solutions for the first fifteen values of n and prove its general validity The Josephus Problem can be described as follows: There are n objects arranged in a circle. Beginning with the first object, we move around the circle and remove every m th object. As each object is removed, the circle closes in. Eventually, all n objects will have been removed from the circle We give explicit non-recursive formulas to compute the Josephus-numbers j(n; 2; i) and j(n; 3; i) and explicit upper and lower bounds for j(n; k; i) (where k 4) which differ by 2k \Gamma 2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j(n; k; i) which is based upon the mentioned bounds. 1 Introduction The Josephus problem in its original form goes back to the Roman historian Flavius Josephus (see [3]). In the Romano-Jewish conflict of 67 A.

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