Josephus problem is a math puzzle with a grim description: *n* prisoners are standing on a circle, sequentially numbered from 0 to *n* − 1.

An executioner walks along the circle, starting from prisoner 0, removing every *k*-th prisoner and killing him.

As the process goes on, the circle becomes smaller and smaller, until only one prisoner remains, who is then freed.

For example, if there are *n* = 5 prisoners and *k* = 2, the order the prisoners are killed in (let’s call it the “killing sequence”) will be 1, 3, 0, and 4, and the survivor will be #2.

**Task** Given any *n*,*k* > 0, find out which prisoner will be the final survivor. In one such incident, there were 41 prisoners and every 3rd prisoner was being killed (*k*= 3). Among them was a clever chap name Josephus who worked out the problem, stood at the surviving position, and lived on to tell the tale. Which number was he?

#include <iostream> #include <vector> //-------------------------------------------------------------------------------------------------- using namespace std; typedef unsigned long long bigint; //-------------------------------------------------------------------------------------------------- class josephus { public: bigint findSurvivors( bigint n, bigint k, bigint s = 0 ) { bigint i = s + 1; for( bigint x = i; x <= n; x++, i++ ) s = ( s + k ) % i; return s; } void getExecutionList( bigint n, bigint k, bigint s = 1 ) { cout << endl << endl << "Execution list: " << endl; prisoners.clear(); for( bigint x = 0; x < n; x++ ) prisoners.push_back( x ); bigint index = 0; while( prisoners.size() > s ) { index += k - 1; if( index >= prisoners.size() ) index %= prisoners.size(); cout << prisoners[static_cast<unsigned int>( index )] << ", "; vector<bigint>::iterator it = prisoners.begin() + static_cast<unsigned int>( index ); prisoners.erase( it ); } } private: vector<bigint> prisoners; }; //-------------------------------------------------------------------------------------------------- int main( int argc, char* argv[] ) { josephus jo; bigint n, k, s; while( true ) { system( "cls" ); cout << "Number of prisoners( 0 to QUIT ): "; cin >> n; if( !n ) return 0; cout << "Execution step: "; cin >> k; cout << "How many survivors: "; cin >> s; cout << endl << "Survivor"; if( s == 1 ) { cout << ": " << jo.findSurvivors( n, k ); jo.getExecutionList( n, k ); } else { cout << "s: "; for( bigint x = 0; x < s; x++ ) cout << jo.findSurvivors( n, k, x ) << ", "; jo.getExecutionList( n, k, s ); } cout << endl << endl; system( "pause" ); } return 0; } //--------------------------------------------------------------------------------------------------

- Output:

Number of prisoners( 0 to QUIT ): 41 Execution step: 3 How many survivors: 1 Survivor: 30 Execution list: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 0, 4, 9, 13, 18, 22, 27, 31, 36 , 40, 6, 12, 19, 25, 33, 39, 7, 16, 28, 37, 10, 24, 1, 21, 3, 34, 15, Number of prisoners( 0 to QUIT ): 41 Execution step: 3 How many survivors: 3 Survivors: 30, 15, 34, Execution list: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 0, 4, 9, 13, 18, 22, 27, 31, 36 , 40, 6, 12, 19, 25, 33, 39, 7, 16, 28, 37, 10, 24, 1, 21, 3, Number of prisoners( 0 to QUIT ): 71 Execution step: 47 How many survivors: 11 Survivors: 29, 58, 41, 14, 39, 28, 35, 45, 64, 49, 27, Execution list: 46, 22, 70, 48, 26, 5, 56, 36, 17, 0, 54, 38, 23, 9, 66, 55, 43, 33, 25, 16, 11, 6, 2, 69, 68, 1, 4, 10, 15, 24, 32, 42, 53, 65, 20, 40, 60, 19, 47, 8, 44, 13, 52, 31, 12, 62, 57, 50, 51, 61, 7, 30, 59, 34, 18, 3, 21, 37, 67, 63,

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