Josephus Problem Using Bit Magic

1.8

The Problem

This problem is named after Flavius Josephus a Jewish historian who fought against the Romans. According to Josephus him and his group of Jewish soldiers were cornered & surrounded by the Romans inside a cave and they chose murder and suicide inside of surrender and capture. They decided that all the soldiers will sit in a circle and starting from the soldier sitting at the first position every soldier will kill the soldier to their in a sequential way. So if there are 5 soldiers sitting in a circle with positions numbered as 1, 2, 3, 4, 5. The soldier 1 kills 2, then 3 kills 4, then 5 kills 1, then 3 kills 5 and since 3 is the only one left then 3 commits suicide.

Now Josephus doesn’t want to get murdered or commit suicide. He would rather be captured by the Romans and is presented with a problem. He has to figure out at which position should he sit in a circle (provided there are n men in total and the man sitting at position 1 gets the first chance to murder) so that he is the last man standing and instead of committing suicide he will surrender to the Romans.

The Pattern

If you work this out for different values of n the you will find a pattern here. If n is a true power of 2 then the answer is always 1. For every n greater than that power of 2 the answer is incremented by 2.

Image Source: Numberphile
Now for every n the right position for Josephus can be found out by deducting the biggest possible power of 2 from the number and we get the answer (provided that value of n is not a pure power of 2 otherwise the answer is 1)

Image Source: Numberphile

The Trick

Whenever someone talks about the powers of 2 the first word that comes to mind is “binary”. The solution to this problem is much is easier and shorter in binary than in decimal. There is a trick to this. Since we need to deduct the biggest possible power of in binary that number is the Most Significant Bit. In the original Josephus problem there were 40 other soldiers along with Josephus which makes n = 41. 41 in binary is 101001. If we shift the MSB i.e. the leftmost 1 to the rightmost place we get 010011 which is 19 (in decimal) which is the answer. This is true for all cases. This can be done easily using bit manipulation.

C

// C program for josephus problem

#include <stdio.h>

// function to find the position of the Most
// Significant Bit
int msbPos(int n)
{
    int pos = 0;
    while (n != 0) {
        pos++;

        // keeps shifting bits to the right
        // until we are left with 0
        n = n >> 1;
    }
    return pos;
}

// function to return at which place Josephus
// should sit to avoid being killed
int josephify(int n)
{
    /*  Getting the position of the Most Significant
        Bit(MSB). The leftmost '1'. If the number is 
       '41' then its binary is '101001'.
        So msbPos(41) = 6   */
    int position = msbPos(n);

    /* 'j' stores the number with which to XOR the 
        number 'n'. Since we need '100000'
       We will do 1<<6-1 to get '100000' */
    int j = 1 << (position - 1);

    /* Toggling the Most Significant Bit. Changing
       the leftmost '1' to '0'.
       101001 ^ 100000 = 001001 (9) */
    n = n ^ j;

    /*  Left-shifting once to add an extra '0' to 
        the right end of the binary number 
        001001 = 010010 (18) */
    n = n << 1;

    /* Toggling the '0' at the end to '1' which 
       is essentially the same as putting the 
       MSB at the rightmost place. 010010 | 1
       = 010011 (19) */
    n = n | 1;

    return n;
}

// hard coded driver main function to run the program
int main()
{
    int n = 41;
    printf("%d\n", josephify(n));
    return 0;
}

Java

// Java program for josephus problem

public class GFG 
{
	// method to find the position of the Most
	// Significant Bit
	static int msbPos(int n)
	{
	    int pos = 0;
	    while (n != 0) {
	        pos++;
	 
	        // keeps shifting bits to the right
	        // until we are left with 0
	        n = n >> 1;
	    }
	    return pos;
	}
	 
	// method to return at which place Josephus
	// should sit to avoid being killed
	static int josephify(int n)
	{
	    /*  Getting the position of the Most Significant
	        Bit(MSB). The leftmost '1'. If the number is 
	       '41' then its binary is '101001'.
	        So msbPos(41) = 6   */
	    int position = msbPos(n);
	 
	    /* 'j' stores the number with which to XOR the 
	        number 'n'. Since we need '100000'
	       We will do 1<<6-1 to get '100000' */
	    int j = 1 << (position - 1);
	 
	    /* Toggling the Most Significant Bit. Changing
	       the leftmost '1' to '0'.
	       101001 ^ 100000 = 001001 (9) */
	    n = n ^ j;
	 
	    /*  Left-shifting once to add an extra '0' to 
	        the right end of the binary number 
	        001001 = 010010 (18) */
	    n = n << 1;
	 
	    /* Toggling the '0' at the end to '1' which 
	       is essentially the same as putting the 
	       MSB at the rightmost place. 010010 | 1
	       = 010011 (19) */
	    n = n | 1;
	 
	    return n;
	}
	
	// Driver Method
	public static void main(String[] args)
	{
		int n = 41;
	    System.out.println(josephify(n));
	}
}


Output:

19

References:

  1. Numberphile
  2. Wikipedia

Previous articles on the same topic:

  1. Josephus problem | Set 1 (A O(n) Solution)
  2. Josephus problem | Set 2 (A Simple Solution when k = 2)

This article is contributed by Palash Nigam . If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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