Josephus Problem

The First Algorithm is the solution to the Josephus Problem.

The problem is based on a tale during the Roman Empire, 41 jews were hiding in a cave and were about to be captured by the Romans but the jews preferred suicide over surrender and hence devised a strategy. They all would stand in a circle and every third person would be killed. Josephus did not want to commit suicide and as he was a smart man he calculated the exact position where he should stand so he is the last man standing.

The Modern problem is based on a similar lines but instead of 41, n people are given and instead of every third person to be killed every k person is killed. You have to find the safe position

Say n = 5 and k = 2, then the order of killing is 2,4,1,5

Here we observe that after the first kill i.e the killing of second person we have n-1 people and so the problem can be considered as finding the safe position with n-1 people when person on k position is executed. So, the problem has a nice recursive structure but the only problem is after the first kill the k+1 position person assumes the first position which can be adjusted as k%n+1k \% n + 1

Algorithm:

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function josephus(n, k):
    if n == 1:
        return n
    else:
        return (josephus(n-1,k) + k) % n + 1

For a nice visualisation of the Josephus Problem visit the following link: Josephus

#CombinatorialGames #ComputerScience #Algorithm

Note by Gaurav Pathak
7 years ago

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