On the generalized Josephus problem

Uchiyama, Saburô

doi:10.21099/tkbjm/1496164652

Cited by 3 publications

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“…1. This is a continuation, with supplementary notes, of our previous paper [3] in which some solutions have been provided for the generalized Josephus problem. If we are given two integers $m\geqq 2$ and $n\geqq 1$ , supposing that $n$ objects, numbered from 1 to $n$ , are arranged in a circle, starting then with object number 1 and counting each object in tum around the circle, clockwise or anticlockwise in the same direction fixed once for all, we eliminate every mth object until all the objects are removed.…”

mentioning

confidence: 85%

“…Now, let $ J_{1}=\langle 1,2, \ldots, n\rangle$ be as before the initial Josephus array. With the cyclic permutations $w_{r}=w_{r,n}(1\leqq r\leqq n)$ specified in (3) we have PROPOSITION 9. The Josephus array $J_{(m)}$ of $n$ objects 1, 2, .…”

Section: We Have (4)mentioning

confidence: 99%

“…In the article [3] we have formulated a hypothesis on the infinitude of "limitative numbers" for every fixed $m$ , suggested by Seki Takakazu (cf. [3]).…”

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confidence: 99%

“…In the article [3] we have formulated a hypothesis on the infinitude of "limitative numbers" for every fixed $m$ , suggested by Seki Takakazu (cf. [3]). A limitative number with respect to a given $m\geqq 2$ is by definition a positive integer $n$ satisfying the condition $d_{m}(n+1)$ $:=a_{m}(n+1, n+1)=1$ .…”

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confidence: 99%

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A note on the generalized Josephus problem

Uchiyama¹

2005

Tsukuba J. Math.

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confidence: 85%

Section: We Have (4)mentioning

confidence: 99%

“…In the article [3] we have formulated a hypothesis on the infinitude of "limitative numbers" for every fixed $m$ , suggested by Seki Takakazu (cf. [3]).…”

mentioning

confidence: 99%

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confidence: 99%

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A note on the generalized Josephus problem

Uchiyama¹

2005

Tsukuba J. Math.

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“…The Josephus Problem is to identify the position of the surviving player in this game. Although the problem has been solved in various ways [3] [4] [5] [6], many variations and generalizations of the game have since been introduced.…”

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confidence: 99%

Generalizations of the Feline and Texas Chainsaw Josephus Problems

Ariyibi¹,

Chang²,

Harris³

2019

OJDM

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We define and study the Extended Feline Josephus Game, a game in which n players, each with ℓ lives, stand in a circle. The game proceeds by alternating between hitting k consecutive players-each of whom will consequently lose a life-and skipping s consecutive players. This cycle continues until every player except one loses all of their lives. Given the nonnegative integer parameters n, k, s and ℓ, the goal of the game is to identify the surviving player. In this paper, we show how the defining parameters n, k, s, and ℓ affect the survivor of games with specific constraints on those parameters and our main results provide new closed formulas to determine the survivor of these Extended Feline Josephus Games. Moreover, for cases where these formulas do not apply, we provide recursive formulas for reducing the initial game to other games with smaller parameter values. For the interested reader, we present a variety of directions for future work in this area, including an extension which considers players lying on a general graph, rather than on a circle.

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Analytical Study and Efficient Evaluation of the Josephus Function

Bello-Cruz¹,

Quintero-Contreras²

2023

Preprint

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A new approach to analyzing intrinsic properties of the Josephus function, J k , is presented in this paper. The linear structure between extreme points of J k is fully revealed, leading to the design of an efficient algorithm for evaluating J k (n). Algebraic expressions that describe how recursively compute extreme points, including fixed points, are derived. The existence of consecutive extreme and also fixed points for all k ≥ 2 is proven as a consequence, which generalizes Knuth result for k = 2. Moreover, an extensive comparative numerical experiment is conducted to illustrate the performance of the proposed algorithm for evaluating the Josephus function compared to established algorithms. The results show that the proposed scheme is highly effective in computing J k (n) for large inputs.

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On the generalized Josephus problem

Cited by 3 publications

References 0 publications

A note on the generalized Josephus problem

A note on the generalized Josephus problem

Generalizations of the Feline and Texas Chainsaw Josephus Problems

Analytical Study and Efficient Evaluation of the Josephus Function

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