Josephus problem

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: $n$ 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 3^rd 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?

Extra

The captors may be especially kind and let $m$ survivors free,
and Josephus might just have   $m-1$   friends to save.

Provide a way to calculate which prisoner is at any given position on the killing sequence.

Notes

1. You can always play the executioner and follow the procedure exactly as described, walking around the circle, counting (and cutting off) heads along the way. This would yield the complete killing sequence and answer the above questions, with a complexity of probably $O(kn)$. However, individually it takes no more than $O(m)$ to find out which prisoner is the $m$-th to die.
2. If it's more convenient, you can number prisoners from   $1$ to $n$   instead.   If you choose to do so, please state it clearly.
3. An alternative description has the people committing assisted suicide instead of being executed, and the last person simply walks away. These details are not relevant, at least not mathematically.

6502 Assembly

This subroutine expects to be called with the value of n in the accumulator and the value of k in index register X. It returns with the index of the survivor in the accumulator, and also leaves an array beginning at address 1000 hex giving the order in which the prisoners died. For example, in the case where n = 5 and k = 2, the values stored in the array are 2, 0, 4, 1, 3. From this we see that prisoner 1 was the first to die, then prisoner 3, and so on. (Note that prisoner 2 in this instance is the survivor.) <lang 6502asm>JSEPHS: STA $D0  ; n

       STX  $D1        ; k
       LDA  #$FF
       LDX  #$00

SETUP: STA $1000,X  ; populate array with hex FF

       INX
       CPX  $D0
       BEQ  KILL
       JMP  SETUP

KILL: LDA #$00  ; number killed so far

       STA  $D2
       LDX  #$00       ; position within array
       LDY  #$01       ; counting up to k

FIND: INY SCAN: INX

       CPX  $D0
       BMI  TEST
       LDX  #$00       ; circle back around

TEST: LDA $1000,X

       CMP  #$FF
       BNE  SCAN       ; already been killed
       CPY  $D1
       BMI  FIND       ; if y < k keep going round
       LDA  $D2
       STA  $1000,X    ; mark as dead
       CLC
       ADC  #$01
       STA  $D2
       CMP  $D0        ; have we killed all but 1?
       BPL  RETURN
       LDY  #$00
       JMP  FIND

RETURN: TXA  ; a <- index of survivor

       RTS</lang>

Ada

The procedure reads up to 4 parameters from the command line: the number N of prisoners, the step size K, the number M of survivors, and an indicator whether the executions shall be printed ("1") or only surviving prisoners (any other input). The defaults are 41, 3, 1, 1. The prison cells are numbered from 0 to N-1. <lang Ada>with Ada.Command_Line, Ada.Text_IO;

procedure Josephus is

  function Arg(Idx, Default: Positive) return Positive is -- read Argument(Idx)
     (if Ada.Command_Line.Argument_Count >= Index
        then Positive'Value(Ada.Command_Line.Argument(Index)) else Default);

  Prisoners:  constant Positive := Arg(Idx => 1, Default => 41);
  Steps:      constant Positive := Arg(Idx => 2, Default =>  3);
  Survivors:  constant Positive := Arg(Idx => 3, Default =>  1);
  Print:               Boolean := (Arg(Idx => 4, Default =>  1) = 1);

  subtype Index_Type is Natural range 0 .. Prisoners-1;
  Next: array(Index_Type) of Index_Type;
  X: Index_Type := (Steps-2) mod Prisoners;

begin

  Ada.Text_IO.Put_Line
    ("N =" & Positive'Image(Prisoners) & ",  K =" & Positive'Image(Steps) &
       (if Survivors > 1 then ",  #survivors =" & Positive'Image(Survivors)
       else ""));
  for Idx in Next'Range loop -- initialize Next
     Next(Idx) := (Idx+1) mod Prisoners;
  end loop;
  if Print then
     Ada.Text_IO.Put("Executed: ");
  end if;
  for Execution in reverse 1 .. Prisoners loop
     if Execution = Survivors then
        Ada.Text_IO.New_Line;
        Ada.Text_IO.Put("Surviving: ");
        Print := True;
     end if;
     if Print then
        Ada.Text_IO.Put(Positive'Image(Next(X)));
     end if;
     Next(X) := Next(Next(X)); -- "delete" a prisoner
     for Prisoner in 1 .. Steps-1 loop
        X := Next(X);
     end loop;
  end loop;

end Josephus;</lang>

Output:

$ ./josephus
N = 41,  K = 3
Executed:  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
Surviving:  30

$ ./josephus 23482 3343 3 0
N = 23482,  K = 3343,  #survivors = 3

Surviving:  13317 1087 1335

ALGOL 68

Translated from the C <lang algol68>BEGIN

  PROC josephus = (INT n, k, m) INT :
  CO Return m-th on the reversed kill list; m=0 is final survivor. CO
  BEGIN
     INT lm := m;			CO Local copy of m CO
     FOR a FROM m+1 WHILE a <= n DO lm := (lm+k) %* a OD;
     lm
  END;
  INT n = 41, k=3;
  printf (($"n = ", g(0), ", k = ", g(0), ", final survivor: ", g(0)l$,

n, k, josephus (n, k, 0))) END</lang>

Output:

n = 41, k = 3, final survivor: 30

AutoHotkey

<lang AHK>; Since AutoHotkey is 1-based, we're numbering prisoners 1-41. nPrisoners := 41 kth  := 3

Build a list, purposefully ending with a separator

Loop % nPrisoners list .= A_Index . "|"

iterate and remove from list

i := 1 Loop { ; Step by 2; the third step was done by removing the previous prisoner i += kth - 1 if (i > nPrisoners) i := Mod(i, nPrisoners) ; Remove from list end := InStr(list, "|", 0, 1, i) bgn := InStr(list, "|", 0, 1, i-1) list := SubStr(list, 1, bgn) . SubStr(list, end+1) nPrisoners-- } Until (nPrisoners = 1) MsgBox % RegExReplace(list, "\|") ; remove the final separator</lang>

Output:

31

Note that since this is one-based, the answer is correct, though it differs with many other examples.

Using Objects

<lang AHK>nPrisoners := 41 kth  := 3 list  := []

Build a list of 41 items

Loop % nPrisoners list.insert(A_Index)

iterate and remove from list

i := 1 Loop { ; Step by 3 i += kth - 1 if (i > list.MaxIndex()) i := Mod(i, list.MaxIndex()) list.remove(i) } Until (list.MaxIndex() = 1) MsgBox % list.1 ; there is only 1 element left</lang>

AWK

<lang AWK>

1. syntax: GAWK -f JOSEPHUS_PROBLEM.AWK
2. converted from PL/I

BEGIN {

   main(5,2,1)
   main(41,3,1)
   main(41,3,3)
   exit(0)

} function main(n,k,s, dead,errors,found,i,killed,nn,p,survived) {

1. n - number of prisoners
2. k - kill every k'th prisoner
3. s - number of survivors

   printf("\nn=%d k=%d s=%d\n",n,k,s) # show arguments
   if (s > n) { print("s>n"); errors++ }
   if (k <= 0) { print("k<=0"); errors++ }
   if (errors > 0) { return(0) }
   nn = n                             # wrap around boundary
   p = -1                             # start here
   while (n != s) {                   # until survivor count is met
     found = 0                        # start looking
     while (found != k) {             # until we have the k-th prisoner
       if (++p == nn) { p = 0 }       # wrap around
       if (dead[p] != 1) { found++ }  # if prisoner is alive increment found
     }
     dead[p] = 1                      # kill the unlucky one
     killed = killed p " "            # build killed list
     n--                              # reduce size of circle
   }
   for (i=0; i<=nn-1; i++) {
     if (dead[i] != 1) {
       survived = survived i " "      # build survivor list
     }
   }
   printf("killed: %s\n",killed)
   printf("survived: %s\n",survived)
   return(1)

} </lang>

Output:

n=5 k=2 s=1
killed: 1 3 0 4
survived: 2

n=41 k=3 s=1
killed: 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
survived: 30

n=41 k=3 s=3
killed: 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
survived: 15 30 34

Batch File

Uses C's jos() function.

Translation of: C

<lang dos>@echo off setlocal enabledelayedexpansion

set "prison=41" %== Number of prisoners ==% set "step=3" %== The step... ==% set "survive=1" %== Number of survivors ==% call :josephus

set "prison=41" set "step=3" set "survive=3" call :josephus pause exit /b 0

%== The Procedure ==%

josephus

set "surv_list=" for /l %%S in (!survive!,-1,1) do (

set /a "m = %%S - 1" for /l %%X in (%%S,1,!prison!) do ( set /a "m = (m + step) %% %%X" ) if defined surv_list ( set "surv_list=!surv_list! !m!" ) else ( set "surv_list=!m!" ) ) echo !surv_list! goto :EOF</lang>

Output:

30
34 15 30
Press any key to continue . . .

Befunge

The number of prisoners and step size are read from stdin.

<lang befunge>>0" :srenosirP">:#,_&>>00p>>v v0p01<&_,#!>#:<"Step size: "< >1+:20p00g`!#v_0" :rovivru"v ^g02%g02+g01<<@.$_,#!>#:<"S"<</lang>

Output:

Prisoners: 41
Step size: 3
Survivor:  30

C

<lang c>#include <stdio.h>

// m-th on the reversed kill list; m = 0 is final survivor int jos(int n, int k, int m) { int a; for (a = m + 1; a <= n; a++) m = (m + k) % a; return m; }

typedef unsigned long long xint;

// same as jos(), useful if n is large and k is not xint jos_large(xint n, xint k, xint m) { if (k <= 1) return n - m - 1;

xint a = m; while (a < n) { xint q = (a - m + k - 2) / (k - 1);

if (a + q > n) q = n - a; else if (!q) q = 1;

m = (m + q * k) % (a += q); }

return m; }

int main(void) { xint n, k, i;

n = 41; k = 3; printf("n = %llu, k = %llu, final survivor: %d\n", n, k, jos(n, k, 0));

n = 9876543210987654321ULL; k = 12031; printf("n = %llu, k = %llu, three survivors:", n, k);

for (i = 3; i--; ) printf(" %llu", jos_large(n, k, i)); putchar('\n');

return 0; }</lang>

Output:

n = 41, k = 3, final survivor: 30
n = 9876543210987654321, k = 12031, three survivors: 6892710366467541051 1946357796579138992 3554846299321782413

C++

<lang cpp>

1. include <iostream>
2. 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;

} //-------------------------------------------------------------------------------------------------- </lang>

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,

Clojure

<lang clojure>(defn rotate [n s] (lazy-cat (drop n s) (take n s)))

(defn josephus [n k]

  (letfn [(survivor [[ h & r :as l] k]
            (cond (empty? r) h
                  :else      (survivor (rest (rotate (dec k) l)) k)))]
    (survivor (range n) k)))

(let [n 41 k 3]

  (println (str "Given " n " prisoners in a circle numbered 1.." n 
                ", an executioner moving around the"))
  (println (str "circle " k " at a time will leave prisoner number " 
                (inc (josephus n k)) " as the last survivor.")))</lang>

Output:

Given 41 prisoners in a circle numbered 1..41, an executioner moving around the
circle 3 at a time will leave prisoner number 31 as the last survivor.

Common Lisp

Using a loop: <lang lisp>(defun kill (n k &aux (m 0))

 (loop for a from (1+ m) upto n do
       (setf m (mod (+ m k) a)))
 m)</lang>

Using a circular list. <lang lisp>(defun make-circular-list (n)

 (let* ((list (loop for i below n
                    collect i))
        (last (last list)))
   (setf (cdr last) list)
   list))

(defun kill (n d)

 (let ((list (make-circular-list n)))
   (flet ((one-element-clist-p (list)
            (eq list (cdr list)))
          (move-forward ()
            (loop repeat (1- d)
                  until (eq list (cdr list))
                  do (setf list (cdr list))))
          (kill-item ()
            (setf (car list) (cadr list)
                  (cdr list) (cddr list))))
     (loop until (one-element-clist-p list) do
           (move-forward)
           (kill-item))
     (first list))))</lang>

Example:

CL-USER > (kill 41 3)
30

D

Translation of: Python

<lang d>import std.stdio, std.algorithm, std.array, std.string, std.range;

T pop(T)(ref T[] items, in size_t i) pure /*nothrow*/ @safe /*@nogc*/ {

   auto aux = items[i];
   items = items.remove(i);
   return aux;

}

string josephus(in int n, in int k) pure /*nothrow*/ @safe {

   auto p = n.iota.array;
   int i;
   immutable(int)[] seq;
   while (!p.empty) {
       i = (i + k - 1) % p.length;
       seq ~= p.pop(i);
   }

   return format("Prisoner killing order:\n%(%(%d %)\n%)." ~
                 "\nSurvivor: %d",
                 seq[0 .. $ - 1].chunks(20), seq[$ - 1]);

}

void main() /*@safe*/ {

   josephus(5, 2).writeln;
   writeln;
   josephus(41, 3).writeln;

}</lang>

Output:

Prisoner killing order:
1 3 0 4.
Survivor: 2

Prisoner killing order:
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.
Survivor: 30

Translation of: Javascript

<lang d>import std.stdio, std.algorithm, std.range;

int[][] Josephus(in int n, int k, int s=1) {

   int[] ks, ps = n.iota.array;
   for (int i=--k; ps.length>s; i=(i+k)%ps.length) {
       ks ~= ps[i];
       ps = remove(ps, i);
   }
   writefln("Josephus(%d,%d,%d) -> %(%d %) / %(%d %)%s", n, k, s, ps, ks[0..min($,45)], ks.length<45 ? "" : " ..." );
   return [ps, ks];

}

void main() {

   Josephus(5, 2);
   Josephus(41, 3);
   Josephus(23482, 3343, 3);

}}</lang>

Output:

Josephus(5,1,1) -> 2 / 1 3 0 4
Josephus(41,2,1) -> 30 / 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
Josephus(23482,3342,3) -> 1087 1335 13317 / 3342 6685 10028 13371 16714 20057 23400 3261 6605 9949 13293 16637 19981 23325 3187 6532 9877 13222 16567 19912 23257 3120 6466 9812 13158 16504 19850 23196 3060 6407 9754 13101 16448 19795 23142 3007 6355 9703 13051 16399 19747 23095 2961 6310 9659 ...

EchoLisp

We use a circular list and apply the 'process'. Successive rests are marked 🔫 (killed) or 😥 (remaining). NB: the (mark) function marks lists and sub-lists, not items in lists. The printed mark appears before the first item in the list. <lang lisp>

input

(define N 41) (define K 3) (define prisoners (apply circular-list (iota N))) (define last-one prisoners) ; current position

kill returns current position = last killed

(define (kill lst skip) (cond

   ((eq? (mark? lst) '🔫 )(kill (cdr lst) skip)) ;; dead ? goto next
   ((zero? skip) (mark lst '🔫)) ;; all skipped ? kill
   (else (mark lst '😥 )  ;; relieved face
          (kill (cdr lst ) (1- skip))))) ;; skip 1 and goto next

</lang>

Output:

<lang lisp>

kill N-1

   (for ((i (1- N) )) (set! last-one (kill last-one  (1- K))))

look at prisoners

prisoners → ( 🔄 🔫 0 🔫 1 🔫 2 🔫 3 🔫 4 🔫 5 🔫 6 🔫 7 🔫 8 🔫 9 🔫 10 🔫 11 🔫 12 🔫 13 🔫 14 🔫 15 🔫 16

🔫 17 🔫 18 🔫 19 🔫 20 🔫 21 🔫 22 🔫 23 🔫 24 🔫 25 🔫 26 🔫 27 🔫 28 🔫 29 😥 30 🔫 31 🔫 32 
🔫 33 🔫 34 🔫 35 🔫 36 🔫 37 🔫 38 🔫 39 🔫 40 🔫 0 🔫 1  … ∞) 

#30 seems happy

kill last

(set! last-one (kill last-one (1- K))) last-one

 → ( 🔫 30 🔫 31 🔫 32 …🔃 ) ;; #30 was the last

extra

we want more survivors

(define SURVIVORS 3) (for ((i (- N SURVIVORS) )) (set! last-one (kill last-one (1- K))))

prisoners → ( 🔄 🔫 0 🔫 1 🔫 2 🔫 3 🔫 4 🔫 5 🔫 6 🔫 7 🔫 8 🔫 9 🔫 10 🔫 11 🔫 12 🔫 13 🔫 14 😥 15 🔫 16

  🔫 17 🔫 18 🔫 19 🔫 20 🔫 21 🔫 22 🔫 23 🔫 24 🔫 25 🔫 26 🔫 27 🔫 28 🔫 29 😥 30 🔫 31 🔫 32 
  🔫 33 😥 34 🔫 35 🔫 36 🔫 37 🔫 38 🔫 39 🔫 40 🔫 0 🔫 1  🔫 0 … ∞) 

</lang>

Eiffel

<lang Eiffel> class APPLICATION

create make

feature

make do io.put_string ("Survivor is prisoner: " + execute (12, 4).out) end

execute (n, k: INTEGER): INTEGER -- Survivor of 'n' prisoners, when every 'k'th is executed. require n_positive: n > 0 k_positive: k > 0 n_larger: n > k local killidx: INTEGER prisoners: LINKED_LIST [INTEGER] do create prisoners.make across 0 |..| (n - 1) as c loop prisoners.extend (c.item) end io.put_string ("Prisoners are executed in the order:%N") killidx := 1 from until prisoners.count <= 1 loop killidx := killidx + k - 1 from until killidx <= prisoners.count loop killidx := killidx - prisoners.count end io.put_string (prisoners.at (killidx).out + "%N") prisoners.go_i_th (killidx) prisoners.remove end Result := prisoners.at (1) ensure Result_in_range: Result >= 0 and Result < n end

end </lang>

Prisoners are executed in the order:
3 
7 
11 
4 
9 
2 
10 
6 
5 
8 
1
Survivor is prisoner: 0

Elixir

<lang Elixir> defmodule Josephus do

 def find(n,k) do
   find(Enum.to_list(0..n-1),0..k-2,k..n)
 end

 def find([_|[r|_]],_,_..d) when d < 3 do
   IO.inspect r
 end

 def find(arr,a..c,b..d) when length(arr) >= 3 do
   find(Enum.slice(arr,b..d) ++ Enum.slice(arr,a..c),a..c,b..d-1)
 end

end

Josephus.find(41,3) </lang>

Output:

30

Erlang

<lang Erlang> -module( josephus_problem ).

-export( [general_solution/3, task/0] ).

general_solution( Prisoners, Kill, Survive ) -> general_solution( Prisoners, Kill, Survive, erlang:length(Prisoners), [] ).

task() -> general_solution( lists:seq(0, 40), 3, 1 ).

general_solution( Prisoners, _Kill, Survive, Survive, Kills ) ->

       {Prisoners, lists:reverse(Kills)};

general_solution( Prisoners, Kill, Survive, Prisoners_length, Kills ) ->

       {Skipped, [Killed | Rest]} = kill( Kill, Prisoners, Prisoners_length ),
       general_solution( Rest ++ Skipped, Kill, Survive, Prisoners_length - 1, [Killed | Kills] ).

kill( Kill, Prisoners, Prisoners_length ) when Kill < Prisoners_length ->

   lists:split( Kill - 1, Prisoners );

kill( Kill, Prisoners, Prisoners_length ) ->

   kill_few( Kill rem Prisoners_length, Prisoners ).

kill_few( 0, Prisoners ) ->

   [Last | Rest] = lists:reverse( Prisoners ),
   {lists:reverse( Rest ), [Last]};

kill_few( Kill, Prisoners ) ->

   lists:split( Kill - 1, Prisoners ).

</lang>

Output:

11> josephus_problem:task().        
{[30],
 [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|...]}

The general solution can handle other items than numbers.

12> josephus_problem:general_solution( [joe, jack, william, averell, ratata], 2, 1 ).
{[william],[jack,averell,joe,ratata]}

ERRE

<lang ERRE> PROGRAM JOSEPHUS

! ! for rosettacode.org !

!$INTEGER

DIM DEAD[100]

PROCEDURE MAIN(N,K,S->ERRORS) ! n - number of prisoners ! k - kill every k'th prisoner ! s - number of survivors

   LOCAL KILLED$,SURVIVED$,FOUND,P,NN,I
   ERRORS=0
   FOR I=0 TO 100 DO
       DEAD[I]=0
   END FOR   ! prepare array
   PRINT("N=";N,"K=";K,"S=";S)        ! show arguments
   IF S>N THEN PRINT("S>N";) ERRORS+=1 END IF
   IF K<=0 THEN PRINT("K<=0";) ERRORS+=1 END IF
   IF ERRORS>0 THEN EXIT PROCEDURE END IF
   NN=N                               ! wrap around boundary
   P=-1                               ! start here
   WHILE N<>S DO                      ! until survivor count is met
     FOUND=0                          ! start looking
     WHILE FOUND<>K DO                ! until we have the k-th prisoner
       P+=1
       IF P=NN THEN P=0 END IF        ! wrap around
       IF DEAD[P]<>1 THEN
           FOUND+=1
       END IF                         ! if prisoner is alive increment found
     END WHILE
     DEAD[P]=1                        ! kill the unlucky one
     KILLED$=KILLED$+STR$(P)          ! build killed list
     N-=1                             ! reduce size of circle
   END WHILE
   FOR I=0 TO NN-1 DO
     IF DEAD[I]<>1 THEN
       SURVIVED$=SURVIVED$+STR$(I)    ! build survivor list
     END IF
   END FOR
   PRINT("Killed:";KILLED$)
   PRINT("Survived:";SURVIVED$)

END PROCEDURE

BEGIN

   ERRORS=0
   MAIN(5,2,1->ERRORS)
   MAIN(41,3,1->ERRORS)
   MAIN(41,3,3->ERRORS)

END PROGRAM </lang> Note: Adapted from AWK version! Output is the same.

Factor

<lang factor>USING: kernel locals math math.ranges sequences ; IN: josephus

josephus ( k n -- m )

   n [1,b] 0 [ [ k + ] dip mod ] reduce ;</lang>

IN: scratchpad 3 41 josephus .
30

Forth

<lang forth>: josephus 0 1 begin dup 41 <= while swap 3 + over mod swap 1+ repeat drop ;</lang>

josephus . 
30

Fortran

Naive approach: prisonners are put in a "linked buffer" (implemented as an array giving number of "next living prisonner"). Then we iterate, killing one after each loop, until there is only one left. <lang fortran>program josephus

  implicit none
  integer :: n, i, k, p
  integer, allocatable :: next(:)
  read *, n, k
  allocate(next(0:n - 1))
  do i = 0, n - 2
     next(i) = i + 1
  end do
  next(n - 1) = 0
  p = 0
  do while(next(p) /= p)
     do i = 1, k - 2
        p = next(p)
     end do
     print *, "Kill", next(p)
     next(p) = next(next(p))
     p = next(p)
  end do
  print *, "Alive", p
  deallocate(next)

end program</lang>

friendly interactive shell

<lang fishshell>function execute

   # If the list is empty, don't do anything.
   test (count $argv) -ge 2; or return
   # If the list has only one element, return it
   if test (count $argv) -eq 2
       echo $argv[2]
       return
   end
   # Rotate prisoners
   for i in (seq 2 $argv[1])
       set argv $argv[1 3..-1 2]
   end
   # Mention killed prisoner
   echo $argv[2]
   # Kill rest recursively
   execute $argv[1 3..-1]

end

echo Prisoner (execute 3 (seq 0 40))[-1] survived.</lang>

Output:

Prisoner 30 survived.

It's also possible to calculate more than one survivor. <lang fishshell>echo Prisoners (execute 3 (seq 0 40))[-3..-1] survived.</lang>

Output:

Prisoners 34 15 30 survived.

Prisoners don't have to be numbers. <lang fishshell>echo Prisoner (execute 2 Joe Jack William Averell Rantanplan)[-1] survived.</lang>

Output:

Prisoner William survived.

Groovy

<lang groovy>int[] Josephus (int size, int kill, int survivors) {

   // init user pool
   def users = new int[size];
   
   // give initial values such that [0] = 1 (first person) [1] = 2 (second person) etc
   users.eachWithIndex() {obj, i -> users[i] = i + 1};
   
   // keep track of which person we are on (ranging from 1 to kill)
   def person = 1;
   
   // keep going until we have the desired number of survivors
   while (users.size() > survivors)
   {
       // for each person, if they are the kill'th person, set them to -1 to show eliminated
       users.eachWithIndex() {obj, i ->
           if (person++ % kill == 0) {
               users[i] = -1;
           }
           
           // if person overflowed kill then reset back to 1
           if (person > kill) {person = 1;}
       }
       
       // clear out all eliminated persons
       users = users.findAll{w -> w >= 0};
   }
   
   // resulting set is the safe positions
   return users;

}

// Run some test cases

println "Final survivor for n = 10201 and k = 17: " + Josephus(10201,17,1)[0];

println "4 safe spots for n = 10201 and k = 17: " + Josephus(10201,17,4); </lang>

Output:

Final survivor for n = 10201 and k = 17: 7450
4 safe spots for n = 10201 and k = 17: [3413, 7244, 7450, 7605]

Go

<lang go>package main

import "fmt"

// basic task function func finalSurvivor(n, k int) int {

   // argument validation omitted
   circle := make([]int, n)
   for i := range circle {
       circle[i] = i
   }
   k--
   exPos := 0
   for len(circle) > 1 {
       exPos = (exPos + k) % len(circle)
       circle = append(circle[:exPos], circle[exPos+1:]...)
   }
   return circle[0]

}

// extra func position(n, k, pos int) int {

   // argument validation omitted
   circle := make([]int, n)
   for i := range circle {
       circle[i] = i
   }
   k--
   exPos := 0
   for len(circle) > 1 {
       exPos = (exPos + k) % len(circle)
       if pos == 0 {
           return circle[exPos]
       }
       pos--
       circle = append(circle[:exPos], circle[exPos+1:]...)
   }
   return circle[0]

}

func main() {

   // show basic task function on given test case
   fmt.Println(finalSurvivor(41, 3))
   // show extra function on all positions of given test case
   fmt.Println("Position  Prisoner")
   for i := 0; i < 41; i++ {
       fmt.Printf("%5d%10d\n", i, position(41, 3, i))
   }

}</lang>

Output:

30
Position  Prisoner
    0         2
    1         5
    2         8
    3        11
    4        14
    5        17
    6        20
    7        23
    8        26
    9        29
   10        32
   11        35
   12        38
   13         0
   14         4
   15         9
   16        13
   17        18
   18        22
   19        27
   20        31
   21        36
   22        40
   23         6
   24        12
   25        19
   26        25
   27        33
   28        39
   29         7
   30        16
   31        28
   32        37
   33        10
   34        24
   35         1
   36        21
   37         3
   38        34
   39        15
   40        30

Haskell

Shows only the surviving prisoners. Change "print $ snd" to just "print" to show the killed prisoners, too. The arguments to the "main" function are: n = number of prisoners, k = kill every kth prisoner, m = show at most m survivors <lang Haskell>import Data.List ((\\)) import System.Environment (getArgs)

prisoners :: Int -> [Int] prisoners n = [0 .. n - 1]

counter :: Int -> [Int] counter k = cycle [k, k-1 .. 1]

killList :: [Int] -> [Int] -> ([Int], [Int], [Int]) killList xs cs = (killed, survivors, newCs)

   where
       (killed, newCs) = kill xs cs []
       survivors = xs \\ killed
       kill [] cs rs = (rs, cs)
       kill (x:xs) (c:cs) rs
           | c == 1 =
               let ts = rs ++ [x]
               in  kill xs cs ts
           | otherwise =
               kill xs cs rs

killRecursive :: [Int] -> [Int] -> Int -> ([Int], [Int]) killRecursive xs cs m = killR ([], xs, cs)

   where
       killR (killed, remaining, counter)
           | length remaining <= m = (killed, remaining)
           | otherwise =
               let (newKilled, newRemaining, newCounter) =
                       killList remaining counter
                   allKilled = killed ++ newKilled
               in  killR (allKilled, newRemaining, newCounter)

main :: IO () main = do

   args <- getArgs
   case args of
       [n, k, m] -> print $ snd $ killRecursive (prisoners (read n))
                       (counter (read k)) (read m)
       _         -> print $ snd $ killRecursive (prisoners 41) (counter 3) 1

</lang>

Using modulo and list split, indices are 1-based. This is much faster than cycled list for larger numbers: <lang Haskell>jseq n k = f n [1 .. n] where

   f 0 _ = []
   f m s = x:f (m-1) (right ++ left) where
       (left,x:right) = splitAt ((k-1) `mod` m) s

-- the final survivor is ((k + ...((k + ((k + 0)`mod` 1)) `mod` 2) ... ) `mod` n) jos n k = 1 + foldl (\x->((k+x)`mod`)) 0 [2..n]

main = do

   print $ jseq 41 3
   print $ jos 10000 100</lang>

Icon and Unicon

The following works in both languages.

<lang unicon>procedure main(A)

  m := integer(A[1]) | 41
  c := integer(A[2]) | 3
  write("With ",m," men, counting to ",c," last position is: ", j(m,c))

end

procedure j(m,c)

  return if m==1 then 0 else (j(m-1,c)+c)%m

end</lang>

Output:

->josephus
With 41 men, counting to 3 last position is: 30
->

Extra 'credit' version:

This is done awkwardly, but I've had this laying around since the late 1980's...

<lang unicon>procedure main(args)

  n := total := integer(args[1]) | 41		# Number of people
  k := count := integer(args[2]) | 3		# Count
  s := integer(args[3])-1 | 0                  # Number to save
  write("With ",n," people, counting by ",k,", the ",s+1," safe places are:")
  every write("\t",j(n,k,(n-s) to n))

end

procedure j(n,k,s)

  a := k*(n-s) + 1
  q := k/(k-1.0)
  nk := n*k
  olda := a
  while a <= nk do {
     olda := a
     a := ceil(a,q)
     }
  t := nk - olda
  return t

end

procedure ceil(a,q)

 n := a*q
 if n = integer(n) then return integer(n)
 n ?:= integer(tab(upto('.'))) + 1
 return n

end</lang>

Sample run:

->josephus2 41 3 4
With 41 people, counting by 3, the 4 safe places are:
	3
        34
        15
        30
->

J

Using the executioner's algorithm.

Tacit version

<lang J> 3 ([ (1 }. <:@[ |. ])^:(1 < #@])^:_ i.@]) 41 30</lang> Structured derivation of the fixed tacit code <lang J> DropNext=. 1 }. <:@[ |. ]

  MoreThanOne=. 1 < #@]
  WhileMoreThanOne=. (^:MoreThanOne f.) (^:_)
  prisoners=. i.@]
  
  [ DropNext WhileMoreThanOne prisoners f.

[ (1 }. <:@[ |. ])^:(1 < #@])^:_ i.@]</lang>

Explicit version

<lang J>Josephus =: dyad define NB. explicit form, assume executioner starts at position 0

NB. use:  SKIP josephus NUMBER_OF_PRISONERS
N =: y
K =: N | x
EXECUTIONER =: 0
PRISONERS =: i. N
kill =: ] #~ (~: ([: i. #))
while. 1 (< #) PRISONERS do.
 EXECUTIONER =: (# PRISONERS) | <: K + EXECUTIONER
 PRISONERS =: EXECUTIONER kill PRISONERS
end.

)

  3 Josephus 41

30</lang>

Explicit version 2

<lang J> NB. this is a direct translation of the algo from C code above.

  Josephus2 =: 4 : '(| x&+)/i. - 1+y' 

  3 Josephus2 41

30</lang>

Java

Works with: Java version 1.5+

<lang java5>import java.util.ArrayList;

public class Josephus {

   public static int execute(int n, int k){
       int killIdx = 0;
       ArrayList<Integer> prisoners = new ArrayList<Integer>(n);
       for(int i = 0;i < n;i++){
           prisoners.add(i);
       }
       System.out.println("Prisoners executed in order:");
       while(prisoners.size() > 1){
           killIdx = (killIdx + k - 1) % prisoners.size();
           System.out.print(prisoners.get(killIdx) + " ");
           prisoners.remove(killIdx);
       }
       System.out.println();
       return prisoners.get(0);
   }
   
   public static ArrayList<Integer> executeAllButM(int n, int k, int m){
       int killIdx = 0;
       ArrayList<Integer> prisoners = new ArrayList<Integer>(n);
       for(int i = 0;i < n;i++){
           prisoners.add(i);
       }
       System.out.println("Prisoners executed in order:");
       while(prisoners.size() > m){
           killIdx = (killIdx + k - 1) % prisoners.size();
           System.out.print(prisoners.get(killIdx) + " ");
           prisoners.remove(killIdx);
       }
       System.out.println();
       return prisoners;
   }
   
   public static void main(String[] args){
       System.out.println("Survivor: " + execute(41, 3));
       System.out.println("Survivors: " + executeAllButM(41, 3, 3));
   }

}</lang>

Output:

Prisoners executed in order:
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 
Survivor: 30
Prisoners executed in order:
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 
Survivors: [15, 30, 34]

Translation of: Javascript

<lang java5>import java.util.ArrayList; import java.util.List;

public class Josephus {

public static void main(String[] args) { execute(5, 1); execute(41, 2); execute(23482, 3342, 3); }

public static int[][] execute(int n, int k) { return execute(n, k, 1); }

public static int[][] execute(int n, int k, int s) { List<Integer> ps = new ArrayList<Integer>(n); for (int i=0; i<n; i+=1) ps.add(i); List<Integer> ks = new ArrayList<Integer>(n-s); for (int i=k; ps.size()>s; i=(i+k)%ps.size()) ks.add(ps.remove(i)); System.out.printf("Josephus(%d,%d,%d) -> %s / %s\n", n, k, s, toString(ps), toString(ks)); return new int[][] { ps.stream().mapToInt(Integer::intValue).toArray(), ks.stream().mapToInt(Integer::intValue).toArray() }; }

private static String toString(List <Integer> ls) { String dot = ""; if (ls.size() >= 45) { dot = ", ..."; ls = ls.subList(0, 45); } String s = ls.toString(); return s.substring(1, s.length()-1) + dot; } }</lang>

Output:

Josephus(5,1,1) -> 2 / 1, 3, 0, 4
Josephus(41,2,1) -> 30 / 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
Josephus(23482,3342,3) -> 1087, 1335, 13317 / 3342, 6685, 10028, 13371, 16714, 20057, 23400, 3261, 6605, 9949, 13293, 16637, 19981, 23325, 3187, 6532, 9877, 13222, 16567, 19912, 23257, 3120, 6466, 9812, 13158, 16504, 19850, 23196, 3060, 6407, 9754, 13101, 16448, 19795, 23142, 3007, 6355, 9703, 13051, 16399, 19747, 23095, 2961, 6310, 9659, ...

JavaScript

Labels are 1-based, executioner's solution: <lang javascript>var Josephus = {

 init: function(n) {
   this.head = {};
   var current = this.head;
   for (var i = 0; i < n-1; i++) {
     current.label = i+1;
     current.next = {prev: current};
     current = current.next;
   }
   current.label = n;
   current.next = this.head;
   this.head.prev = current;
   return this;
 },
 kill: function(spacing) {
   var current = this.head;
   while (current.next !== current) {
     for (var i = 0; i < spacing-1; i++) {
       current = current.next;
     }
     current.prev.next = current.next;
     current.next.prev = current.prev;
     current = current.next;
   }
   return current.label;
 }

}</lang>

Output:

> Josephus.init(30).kill(2)
29

With Array methods: <lang javascript>function Josephus(n, k, s) { s = s | 1 for (var ps=[], i=n; i--; ) ps[i]=i for (var ks=[], i=--k; ps.length>s; i=(i+k)%ps.length) ks.push(ps.splice(i, 1)) document.write((arguments.callee+).split(/\s|\(/)[1], '(', [].slice.call(arguments, 0), ') -> ', ps, ' / ', ks.length<45?ks:ks.slice(0,45)+',...' , '
') return [ps, ks] }</lang>

Output:

Josephus(5,1) -> 2 / 1,3,0,4
Josephus(41,2) -> 30 / 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
Josephus(23482,3342,3) -> 1087,1335,13317 / 3342,6685,10028,13371,16714,20057,23400,3261,6605,9949,13293,16637,19981,23325,3187,6532,9877,13222,16567,19912,23257,3120,6466,9812,13158,16504,19850,23196,3060,6407,9754,13101,16448,19795,23142,3007,6355,9703,13051,16399,19747,23095,2961,6310,9659,...

Julia

Recursive: <lang julia> josephus(n, k, m=1) = n == m ? collect(0:m-1) : mod(josephus(n-1, k, m) + k, n) </lang>

Output:

<lang julia> julia> print(josephus(41,3)) [30] julia> print(josephus(41,3,5)) [3,15,21,30,34] </lang> Translated from python <lang julia> function j(n,k)

   p, i, seq=[0:n-1], 0, Int[]
   while !isempty(p)
       i=(i+k-1)%length(p)
       push!(seq,splice!(p,i+1))
   end
   @sprintf("Prisoner killing order: %s.\nSurvivor: %i",replace(chomp(string(seq[1:end-1])),"\n",", "),seq[end])

end</lang>

Output:

<lang julia>julia> print(j(41,3)) Prisoner killing order: 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. Survivor: 30</lang> Getting the remaining m survivors <lang julia> function j2(n,k,m)

   p, i, seq=[0:n-1], 0, Int[]
   while length(p)>m
       i=(i+k-1)%length(p)
       push!(seq,splice!(p,i+1))
   end
   prt_array(x)=replace(chomp(string(x)),"\n",", ")
   @sprintf("Prisoner killing order: %s.\nSurvivors: %s",prt_array(seq),"["*prt_array(p)*"]")

end </lang>

Output:

<lang julia> julia> print(j2(41,3,3)) Prisoner killing order: 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. Survivors: [15, 30, 34] </lang>

jq

Works with: jq version 1.4

This section illustrates how a simulation can be directly modeled in jq while being fast enough to solve problems such as [n,k,m] = [23482, 3343, 3].

The prisoners are numbered from 0 to (n-1) in keeping with jq's array index origin of 0, but the nature of their labeling is immaterial to the algorithm. <lang jq># A control structure, for convenience:

1. as soon as "condition" is true, then emit . and stop:

def do_until(condition; next):

 def u: if condition then . else (next|u) end;
 u;

1. n is the initial number; every k-th prisoner is removed until m remain.
2. Solution by simulation

def josephus(n;k;m):

   reduce range(0;n) as $i ([]; . + [$i])    # Number the prisoners from 0 to (n-1)
   | do_until( length < k or length <= m; .[k:] + .[0:k-1] )
   | do_until( length <= m; (k % length) as $i | .[$i:] + .[0:$i-1] );</lang>

Examples: <lang jq>def task(n;k;m):

  "Survivors for n=\(n), k=\(k), m=\(m): \( josephus(n;k;m) )";

task(41;3;1), task(23482; 3343; 3)</lang>

Output:

$ jq -M -r -n -f josephus.jq
Survivors for n=41, k=3, m=1: [30]
Survivors for n=23482, k=3343, m=3: [13317,1087,1335]

Lua

Lua indexes tables starting at 1. Positions are stored from 0,n-1. <lang lua>function josephus(n, k, m)

   local positions={}
   for i=1,n do
       table.insert(positions, i-1)
   end
   local i,j=1,1
   local s='Execution order: '
   while #positions>m do
       if j==k then
           s=s .. positions[i] .. ', '
           table.remove(positions, i)
           i=i-1
       end
       i=i+1
       j=j+1
       if i>#positions then i=1 end
       if j>k then j=1 end
   end
   print(s:sub(1,#s-2) .. '.')
   local s='Survivors: '
   for _,v in pairs(positions) do s=s .. v .. ', ' end
   print(s:sub(1,#s-2) .. '.')

end josephus(41,3, 1) </lang>

Output:

Execution order: 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.
Survivors: 30.

Mathematica

<lang mathematica>survivor[n_, k_] := Nest[Most[RotateLeft[#, k]] &, Range[0, n - 1], n - 1] survivor[41, 3]</lang>

Output:

{30}

NetRexx

Translation of: REXX

Hardly any changes at all... <lang NetRexx>/* NetRexx */ options replace format comments java crossref symbols nobinary

/* REXX **************************************************************

• 15.11.2012 Walter Pachl - my own solution
• 16.11.2012 Walter Pachl generalized n prisoners + w killing distance
• and s=number of survivors
  • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /

dead = 0 /* nobody's dead yet */ n = 41 /* number of alive prisoners */ nn = n /* wrap around boundary */ w = 3 /* killing count */ s = 1 /* nuber of survivors */ p = -1 /* start here */ killed = /* output of killings */ Loop until n = s /* until one alive prisoner */

 found = 0                            /* start looking              */
 Loop Until found = w                 /* until we have the third    */
   p = p + 1                          /* next position              */
   If p = nn Then p = 0               /* wrap around                */
   If dead[p] = 0 Then                /* a prisoner who is alive    */
     found = found + 1                /* increment found count      */
   End
 dead[p] = 1
 n = n - 1                            /* shoot the one on this pos. */
 killed = killed p                    /* add to output              */
 End                                  /* End of main loop           */

Say 'killed:'killed.subword(1, 20) /* output killing sequence */ Say ' 'killed.subword(21) /* output killing sequence */ Say 'Survivor(s):' /* show */ Loop i = 0 To 40 /* look for the surviving p's */

 If dead[i] = 0 Then Say i            /* found one                  */
 End</lang>

Output:

killed: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
Survivor(s):
30

Nim

Translation of: Python

<lang nim>import sequtils, strutils, future

proc j(n, k): string =

 var
   p = toSeq(0 .. < n)
   i = 0
   s = newSeq[int]()

 while p.len > 0:
   i = (i + k - 1) mod p.len
   s.add p[i]
   system.delete(p, i)

 result = "Prisoner killing order: "
 result.add s.map((x: int) => $x).join(", ")
 result.add ".\nSurvivor: "
 result.add($s[s.high])

echo j(5,2) echo j(41,3)</lang>

Output:

Prisoner killing order: 1, 3, 0, 4, 2.
Survivor: 2
Prisoner killing order: 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, 30.
Survivor: 30

Objeck

<lang objeck>class Josephus {

 function : Execute(n : Int, k : Int) ~ Int {
   killIdx := 0;
   prisoners := Collection.IntVector->New();
   for(i := 0;i < n;i+=1;){
     prisoners->AddBack(i);
   };
   
   "Prisoners executed in order:"->PrintLine();
   while(prisoners->Size() > 1){
     killIdx := (killIdx + k - 1) % prisoners->Size();
     executed := prisoners->Get(killIdx);
     "{$executed} "->Print();
     prisoners->Remove(killIdx);
   };
   '\n'->Print();    
   return prisoners->Get(0);
 }
 
 function : ExecuteAllButM(n : Int, k : Int, m : Int) ~ Collection.IntVector {
   killIdx := 0;
   prisoners := Collection.IntVector->New();
   for(i := 0;i < n;i+=1;){
     prisoners->AddBack(i);
   };
   "Prisoners executed in order:"->PrintLine();
   while(prisoners->Size() > m){
     killIdx := (killIdx + k - 1) % prisoners->Size();
     executed := prisoners->Get(killIdx);
     "{$executed} "->Print();
     prisoners->Remove(killIdx);
   };
   '\n'->Print();    
   return prisoners;
 }
 
 function : Main(args : String[]) ~ Nil {
   result := Execute(41, 3);
   "Survivor: {$result}"->PrintLine();

   results := ExecuteAllButM(41, 3, 3);
   "Survivors: "->Print();
   each(i : results) {
   results->Get(i)->Print();
     if(i + 1 < results->Size()) {
       ' '->Print();
     };
   };
 }

} </lang>

Oforth

Oforth lists are 1-based : prisoners are numbered from 1 to n.

<lang Oforth>: josephus(n, k) | prisoners killed i |

  n seq asListBuffer ->prisoners
  ListBuffer newSize(n) ->killed

  0 n 1- loop: i [ 
     k 1- + prisoners size mod dup 1+ prisoners removeAt
     killed add 
     ] drop

  System.Out "Killed : " << killed << "\nSurvivor : " << prisoners << cr

</lang>

Output:

>josephus(41, 3)
Killed : [3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 1, 5, 10, 14, 19, 23, 28, 32, 37, 41, 7, 13, 20, 26, 34, 40, 8, 17, 29, 38, 11, 25, 2, 22, 4, 35, 16]
Survivor : [31]

PARI/GP

<lang parigp>Josephus(n, k)=if(n<2, n>0, my(t=(Josephus(n-1, k)+k)%n); if(t, t, n))</lang>

Perl

Translation of: Perl6

<lang Perl>my @prisoner = 0 .. 40; my $k = 3; until (@prisoner == 1) {

   push @prisoner, shift @prisoner for 1 .. $k-1;
   shift @prisoner;

}

print "Prisoner @prisoner survived.\n"</lang>

Output:

Prisoner 30 survived.

Perl 6

Works with: rakudo version 2015-11-12

Straightforward implementation of the executioner's algorithm: <lang Perl6>sub Execute(@prisoner, $k) {

   until @prisoner == 1 {

@prisoner.=rotate($k - 1); @prisoner.shift;

   }

}

my @prisoner = ^41; Execute @prisoner, 3; say "Prisoner {@prisoner} survived.";</lang>

Output:

Prisoner 30 survived.

We don't have to use numbers. Any list will do: <lang Perl6>my @dalton = <Joe Jack William Averell Rantanplan>; Execute @dalton, 2; say "{@dalton} survived.";</lang>

Output:

William survived.

PHP

<lang php><?php //Josephus.php function Jotapata($n=41,$k=3,$m=1){$m--; $prisoners=array_fill(0,$n,false);//make a circle of n prisoners, store false ie: dead=false $deadpool=1;//count to next execution $order=0;//death order and *dead* flag, ie. deadpool while((array_sum(array_count_values($prisoners))<$n)){//while sum of count of unique values dead times < n (they start as all false) foreach($prisoners as $thisPrisoner=>$dead){ if(!$dead){//so yeah...if not dead... if($deadpool==$k){//if their time is up in the deadpool... $order++; //set the deadpool value or enumerate as survivor $prisoners[$thisPrisoner]=((($n-$m)>($order)?$order:(($n)==$order?'Call me *Titus Flavius* Josephus':'Joe\'s friend '.(($order)-($n-$m-1))))); $deadpool=1;//reset count to next execution }else{$duckpool++;} } } } return $prisoners; }

echo '

'.print_r(Jotapata(41,3,5),true).'<pre>';
</lang>

=={{header|PL/I}}==
<lang pli>*process or(!) source attributes xref;
 joseph: Proc Options(main);
 /* REXX **************************************************************
 * 15.11.2012 Walter Pachl - my own solution
 * 16.11.2012 Walter Pachl generalized n prisoners + w killing distance
 *                         and s=number of survivors
 * 03.05.2013 Walter Pachl Translated From REXX Version 1
 **********************************************************************/
 Dcl dead(0:100) Bit(1);
 Dcl (n,nn,w,s,p,found) Bin Fixed(15);
 Dcl pp Pic'99';
 Dcl killed Char(300) Var Init('killed: '); /* output of killings     */
 Dcl survived Char(300) Var Init('Survivor(s): ');
 dead='';                               /* nobody's dead yet          */
 n=41;                                  /* number of alive prisoners  */
 nn=n;                                  /* wrap around boundary       */
 w=3;                                   /* killing count              */
 s=1;                                   /* number of survivors         */
 p=-1;                                  /* start here                 */
 Do Until(n=s);                         /* until one alive prisoner   */
   found=0;                             /* start looking              */
   Do Until(found=w);                   /* until we have the third    */
     p=p+1;                             /* next position              */
     If p=nn Then p=0;                  /* wrap around                */
     If ^dead(p) Then                   /* a prisoner who is alive    */
       found=found+1;                   /* increment found count      */
     End;
   dead(p)='1'b;                        /* shoot the one on this pos. */
   n=n-1;
   pp=p;
   killed=killed!!' '!!pp;              /* add to output              */
   End;                                 /* End of main loop           */
 Call o(killed);
 Do i=0 To nn-1;                        /* look for the surviving p's */
   If ^dead(i) Then Do;                 /* found one                  */
     pp=i;
     survived=survived!!' '!!pp;
     End;
   End;
 Call o(survived);

 o: Proc(s);
 /*********************************************************************
 * Formatted Output of given string:
 * xxxxxxxxxx xxx xx xx xxx ---
 *         xx xxx xxx
 *         xxxxx xxx
 *********************************************************************/
 Dcl s Char(*) Var;
 Dcl p Bin Fixed(15);
 Dcl ll Bin Fixed(15) Init(72);
 Do While(length(s)>ll);
   Do p=ll+1 To 10 By -1;
     If substr(s,p,1)=' ' Then
       Leave;
     End;
   Put Edit(left(s,p))(Skip,a);
   s=repeat(' ',8)!!substr(s,p+1);
   End;
 Put Edit(s)(Skip,a);
 End;

 End;</lang>
{{out}}
<pre>killed:  02 05 08 11 14 17 20 23 26 29 32 35 38 00 04 09 13 18 22 27 31
         36 40 06 12 19 25 33 39 07 16 28 37 10 24 01 21 03 34 15
Survivor(s):  30 

PowerShell

Works with: PowerShell version 2

Adapted from the iterative algorithm in Sidef.

Rotating the circle K prisoners is equivalent to the executioner walking around the circle K prisoners. We rotate the circle to bring the next selectee to the "front" of the circle, then "select" him by moving past him to the remaining circle. After repeating through the entire prisoner population, we are left with the prisoners sorted into the order in which they are selected.

The lonely comma in the line where we create the $Prisoners arraylist is to prevent PowerShell from being too helpful. Normally when we present the PowerShell parser with an array within an array, it treats it as a cast, and we end up with the single array of elements. In those cases where we need an array to be treated as a single element of a parent array, we can use the unary comma to force PowerShell to treat it as an element.<lang PowerShell> <lang PowerShell> function Get-JosephusPrisoners ( [int]$N, [int]$K )

   {
   #  Just for convenience
   $End = $N - 1

   #  Create circle of prisoners
   $Prisoners = New-Object System.Collections.ArrayList ( , (0..$End) )

   #  For each starting point of the reducing circle...
   ForEach ( $Start in 0..($End - 1) )
       {
       #  We subtract one from K for the one we advanced by incrementing $Start
       #  Then take K modulus the length of the remaining circle
       $RoundK = ( $K - 1 ) % ( $End - $Start + 1 )
      
       #  Rotate the remaining prisoners K places around the remaining circle
       $Prisoners.SetRange( $Start, $Prisoners[ $Start..$End ][ ( $RoundK + $Start - $End - 1 )..( $RoundK - 1 ) ] )
       }
   return $Prisoners
   }

</lang> <lang PowerShell>

1. Get the prisoner order for a circle of 41 prisoners, selecting every third

$Prisoners = Get-JosephusPrisoners -N 41 -K 3

1. Display the prisoner order

$Prisoners -join " "

1. Display the last remaining prisoner

"Last prisoner remmaining: " + $Prisoners[-1]

1. Display the last three remaining prisoners

$S = 3 "Last $S remaining: " + $Prisoners[-$S..-1] </lang>

Output:

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 30
Last prisoner remmaining: 30
Last 3 remaining: 34 15 30

PureBasic

<lang purebasic>Define.i NewList prisoners.i()

Procedure f2l(List p.i())

 FirstElement(p())    : tmp.i=p()
 DeleteElement(p(),1) : LastElement(p())
 AddElement(p())      : p()=tmp 

EndProcedure

Procedure l2f(List p.i())

 LastElement(p())   : tmp.i=p()
 DeleteElement(p()) : FirstElement(p())
 InsertElement(p()) : p()=tmp  

EndProcedure

OpenConsole() Repeat

 Print(#LF$+#LF$)
 Print("Josephus problem - input prisoners : ") : n=Val(Input())
 If n=0 : Break : EndIf  
 Print("                 - input steps     : ") : k=Val(Input())
 Print("                 - input survivors : ") : s=Val(Input()) : If s<1 : s=1 : EndIf
 ClearList(prisoners()) : For i=0 To n-1 : AddElement(prisoners()) : prisoners()=i : Next
 If n<100 : Print("Executed : ") : EndIf
 While ListSize(prisoners())>s And n>0 And k>0 And k<n    
   For j=1 To k : f2l(prisoners()) : Next    
   l2f(prisoners()) : FirstElement(prisoners()) : If n<100 : Print(Str(prisoners())+Space(2)) : EndIf 
   DeleteElement(prisoners())    
 Wend
 Print(#LF$+"Surviving: ")
 ForEach prisoners()
   Print(Str(prisoners())+Space(2))
 Next      

ForEver End</lang>

Output:

Josephus problem - input prisoners : 5
                 - input steps     : 2
                 - input survivors : 1
Executed : 1  3  0  4
Surviving: 2

Josephus problem - input prisoners : 41
                 - input steps     : 3
                 - input survivors : 1
Executed : 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
Surviving: 30

Josephus problem - input prisoners : 41
                 - input steps     : 3
                 - input survivors : 3
Executed : 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
Surviving: 15  30  34

Josephus problem - input prisoners : 71
                 - input steps     : 47
                 - input survivors : 11
Executed : 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
Surviving: 64  14  27  28  29  35  39  41  45  49  58

Josephus problem - input prisoners :

Python

<lang python>>>> def j(n, k): p, i, seq = list(range(n)), 0, [] while p: i = (i+k-1) % len(p) seq.append(p.pop(i)) return 'Prisoner killing order: %s.\nSurvivor: %i' % (', '.join(str(i) for i in seq[:-1]), seq[-1])

>>> print(j(5, 2)) Prisoner killing order: 1, 3, 0, 4. Survivor: 2 >>> print(j(41, 3)) Prisoner killing order: 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. Survivor: 30 >>> </lang>

Faster way to solve in python, it does not show the killing order. <lang python>>>>def josephus(n, k):

       r = 0
       for i in xrange(1, n+1):
           r = (r+k)%i
       return 'Survivor: %d' %r

>>> print(josephus(5, 2)) Survivor: 2 >>> print(josephus(41, 3)) Survivor: 30 >>> </lang>

Alternate solution with a circular linked list

The function returns the killing order. The last in the list stays alive. Notice that the result is a permutation of [0, 1, ... n - 1]. In the program, a[p] is the index of the next living prisoner after 'p'. The program stops when p = a[p], that is, when there remains only one living prisoner.

<lang python>def josephus(n, k):

   a = list(range(1, n + 1))
   a[n - 1] = 0
   p = 0
   v = []
   while a[p] != p:
       for i in range(k - 2):
           p = a[p]
       v.append(a[p])
       a[p] = a[a[p]]
       p = a[p]
   v.append(p)
   return v

josephus(10, 2) [1, 3, 5, 7, 9, 2, 6, 0, 8, 4]

josephus(41, 3)[-1] 30</lang>

R

<lang R> jose <-function(s, r,n){ y <- 0:(r-1)

for (i in (r+1):n)
 y <- (y + s) %% i 
return(y)

} > jose(3,1,41) # r is the number of remained prisoner. [1] 30 </lang>

Racket

<lang Racket>#lang racket (define (josephus n k (m 0))

 (for/fold ((m (add1 m)))
   ((a (in-range (add1 m) (add1 n))))
   (remainder (+ m k) a)))

(josephus 41 3) ; ->30</lang>

REBOL

Works in Rebol 2 or 3 <lang REBOL>Rebol []

execute: func [death-list [block!] kill [integer!]] [

   assert [not empty? death-list]
   until [
       loop kill - 1 [append death-list take death-list]
       (1 == length? remove death-list)
   ]

]

prisoner: [] for n 0 40 1 [append prisoner n] execute prisoner 3 print ["Prisoner" prisoner "survived"]</lang>

Output:

Prisoner 30 survived

And any kind of list will do: <lang REBOL>for-the-chop: [Joe Jack William Averell Rantanplan] execute for-the-chop 2 print [for-the-chop "survived"]</lang>

Output:

William survived

REXX

version 1

<lang rexx>/* REXX **************************************************************

• 15.11.2012 Walter Pachl - my own solution
• 16.11.2012 Walter Pachl generalized n prisoners + w killing distance
• and s=number of survivors
• 09.05.2013 Walter Pachl accept arguments n w s and fix output
• thanks for the review/test
• I see no need for specifying a start count (actually a start number)
• This program should work on EVERY REXX.
• Pls report if this is not the case and let us know what's a problem.
  • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /

Parse Arg n w s . If n='?' Then Do

 Say 'Invoke the program with the following arguments:'
 Say 'n number of prisoners            (default 41)'
 Say 'w killing count                  (default  3)'
 Say 's number of prisoners to survive (default  1)'
 Exit
 End

If n= Then n=41 /* number of alive prisoners */ If w= Then w=3 /* killing count */ If s= Then s=1 /* nuber of survivors */ dead.=0 /* nobody's dead yet */ nn=n /* wrap around boundary */ p=-1 /* start here */ killed= /* output of killings */ Do until n=s /* until one alive prisoner */

 found=0                              /* start looking              */
 Do Until found=w                     /* until we have the third    */
   p=p+1                              /* next position              */
   If p=nn Then p=0                   /* wrap around                */
   If dead.p=0 Then                   /* a prisoner who is alive    */
     found=found+1                    /* increment found count      */
   End
 dead.p=1
 /*
 Say 'killing' p 'now'
 */
 n=n-1                                /* shoot the one on this pos. */
 killed=killed p                      /* add to output              */
 End                                  /* End of main loop           */

Say 'killed:'killed /* output killing sequence */ s= Do i=0 To nn-1 /* look for the surviving p's */

 If dead.i=0 Then s=s i               /* found one                  */
 End

Say 'Survivor(s):'s /* show */</lang>

Output:

killed: 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
Survivor(s): 30

version 2

This version allows the user to specify:

•   the number of prisoners
•   the count-off   [every K^th prisoner]
•   the start count   [zero or one]
•   the number of survivors
•   the solving of the extra credit task requirement of multiple survivors

The output echoes the choices specified and was made "English" readable.

This solution is an   executor's   solution. <lang rexx>/*REXX program solves Josephus problem: N men standing in a circle, every Kth kilt.*/

                                                /*N=men, K=kilt, Z=start, R=remaining. */

parse arg N K Z R . /*obtain optional arguments from the CL*/ if N== | N=="," then N = 41 /*Not specified? Then use the default.*/ if K== | K=="," then K = 3 /* " " " " " " */ if Z== | Z=="," then Z = 0 /* " " " " " " */ if R== | R=="," then R = 1 /* " " " " " " */ $=; x=; do pop=Z for N; $=$ pop; end /*pop*/ /*populate prisoner's circle (with a #)*/ c=0 /*initial prisoner count─off number. */

   do remove=0 by 0;     p=words($)             /*keep removing until  R  are remaining*/
   c=c+K                                        /*bump the prisoner  count-off  by  K. */
   if c>p then do                               /*   [↓] remove (kill) some prisoner(s)*/
                  do j=1  for words(x);    $=delword($, word(x, j) + 1 - j,   1)
                  if words($)==R  then leave remove             /*is the slaying done?*/
                  end   /*j*/
               c=(c//p) // words($);        x=  /*adjust prisoner count-off and circle.*/
               end
   if c\==0  then x=x c                         /*the list of prisoners to be removed. */
   end   /*remove*/                             /*remove 'til   R   prisoners are left.*/

say 'removing every ' th(K) " prisoner out of " N ' (starting at' Z") with ",

                          R    ' survivor's(R)",  leaving prisoner"s(R)':'   $

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ s: if arg(1)==1 then return arg(3); return word( arg(2) 's', 1) th: y=arg(1); return y || word('th st nd rd', 1+ y // 10 * (y//100%10\==1) * (y//10<4))</lang> Programming note:   the 1^st do loop (remove)   is to enable the leave to function correctly,   and this form of leave requires a

      do loop with an index (variable).   The by of zero is just to help identify what's happening (er ..., or not happening).

output   when using the default input:

removing every  3rd  prisoner out of  41  (starting at 0)  with  1  survivor,  leaving prisoner:  30

output   when using the input of:   41   3   1

removing every  3rd  prisoner out of  41  (starting at 1)  with  1  survivor,  leaving prisoner:  31

output   when using the input of:   41   3   1   2

removing every  3rd  prisoner out of  41  (starting at 1)  with  2  survivors,  leaving prisoners:  16 31

output   when using the input of:   5   2

removing every  2nd  prisoner out of  5  (starting at 0)  with  1  survivor,  leaving prisoner:  2

Ruby

<lang ruby>def main

 n = (ARGV[0] || 41).to_i
 k = (ARGV[1] || 3).to_i
 puts josephus(n,k)

end

def josephus(n, k)

 prisoners = (0...n).to_a
 prisoners.rotate!(k-1).shift  while prisoners.length > 1
 return prisoners.first

end

main</lang>

Scala

Executioner's Solution, not Josephus'

(Prisoners labeled 0 to n-1) <lang scala>def executed( prisonerCount:Int, step:Int ) = {

 val prisoners = ((0 until prisonerCount) map (_.toString)).toList

 def behead( dead:Seq[String], alive:Seq[String] )(countOff:Int) : (Seq[String], Seq[String]) = {
   val group = if( alive.size < countOff ) countOff - alive.size else countOff

   (dead ++ alive.take(group).drop(group-1), alive.drop(group) ++ alive.take(group-1))
 }

 def beheadN( dead:Seq[String], alive:Seq[String] ) : (Seq[String], Seq[String]) =
   behead(dead,alive)(step)

 def execute( t:(Seq[String], Seq[String]) ) : (Seq[String], Seq[String]) = t._2 match {
   case x :: Nil => (t._1, Seq(x))
   case x :: xs => execute(beheadN(t._1,t._2))
 }

 execute((List(),prisoners))

}

val (dead,alive) = executed(41,3)

println( "Prisoners executed in order:" ) print( dead.mkString(" ") )

println( "\n\nJosephus is prisoner " + alive(0) )</lang>

Output:

Prisoners executed in order:
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

Josephus is prisoner 30

Seed7

The main task (find one survivor) is a special case of the extra task (find m survivors). The function executeAllButM solves the extra task and is called with m=1 to solve the main task. The function str converts an array of integer elements to a string. The function enable_output uses str to define everything necessary to write an array of integers. This way the main program can write the survivor array. <lang seed7>$ include "seed7_05.s7i";

const func array integer: executeAllButM (in integer: n, in integer: k, in integer: m) is func

 result
   var array integer: prisoners is [0 .. -1] times 0;
 local
   var integer: killIdx is 0;
   var integer: prisonerNum is 0;
 begin
   for prisonerNum range 0 to pred(n) do
     prisoners &:= prisonerNum;
   end for;
   writeln("Prisoners executed in order:");
   while length(prisoners) > m do
     killIdx := (killIdx + k - 1) rem length(prisoners);
     write(prisoners[killIdx] <& " ");
     ignore(remove(prisoners, killIdx));
   end while;
   writeln;
 end func;

const func string: str (in array integer: intArr) is func

 result
   var string: stri is "";
 local
   var integer: index is 0;
 begin
   for key index range intArr do
     if index <> minIdx(intArr) then
       stri &:= ", ";
     end if;
     stri &:= str(intArr[index]);
   end for;
 end func;

enable_output(array integer);

const proc: main is func

 begin
   writeln("Survivor: " <& executeAllButM(41, 3, 1));
   writeln("Survivors: " <& executeAllButM(41, 3, 3));
 end func;</lang>

Output:

Prisoners executed in order:
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 
Survivor: 30
Prisoners executed in order:
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 
Survivors: 15, 30, 34

Sidef

Iterative: <lang ruby>func josephus(n, k) {

   var prisoners = @^n
   while (prisoners.len > 1) {
       prisoners.rotate!(k - 1).shift
   }
   return prisoners[0]

}</lang>

Recursive: <lang ruby>func josephus(n, k) {

   n == 1 ? 0 : ((__FUNC__(n-1, k) + k) % n)

};</lang>

Calling the function: <lang ruby>var survivor = josephus(41, 3); say "Prisoner #{survivor} survived.";</lang>

Output:

Prisoner 30 survived.

Swift

<lang Swift>class Josephus {

   class func lineUp(#numberOfPeople:Int) -> [Int] {
       var people = [Int]()
       for (var i = 0; i < numberOfPeople; i++) {
           people.append(i)
       }
       return people
   }
   
   class func execute(#numberOfPeople:Int, spacing:Int) -> Int {
       var killIndex = 0
       var people = self.lineUp(numberOfPeople: numberOfPeople)
       
       println("Prisoners executed in order:")
       while (people.count > 1) {
           killIndex = (killIndex + spacing - 1) % people.count
           executeAndRemove(&people, killIndex: killIndex)
       }
       println()
       return people[0]
   }
   
   class func executeAndRemove(inout people:[Int], killIndex:Int) {
       print("\(people[killIndex]) ")
       people.removeAtIndex(killIndex)
   }

   class func execucteAllButM(#numberOfPeople:Int, spacing:Int, save:Int) -> [Int] {
       var killIndex = 0
       var people = self.lineUp(numberOfPeople: numberOfPeople)
       
       println("Prisoners executed in order:")
       while (people.count > save) {
           killIndex = (killIndex + spacing - 1) % people.count
           executeAndRemove(&people, killIndex: killIndex)
       }
       println()
       return people
   }

}

println("Josephus is number: \(Josephus.execute(numberOfPeople: 41, spacing: 3))") println() println("Survivors: \(Josephus.execucteAllButM(numberOfPeople: 41, spacing: 3, save: 3))")</lang>

Output:

Prisoners executed in order:
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 
Josephus is number: 30

Prisoners executed in order:
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 
Survivors: [15, 30, 34]

Tcl

<lang tcl>proc josephus {number step {survivors 1}} {

   for {set i 0} {$i<$number} {incr i} {lappend l $i}
   for {set i 1} {[llength $l]} {incr i} {

# If the element is to be killed, append to the kill sequence if {$i%$step == 0} { lappend killseq [lindex $l 0] set l [lrange $l 1 end] } else { # Roll the list set l [concat [lrange $l 1 end] [list [lindex $l 0]]] }

   }
   return [lrange $killseq end-[expr {$survivors-1}] end]

}</lang> Demonstrating: <lang tcl>puts "remaining: [josephus 41 3]" puts "remaining 4: [join [josephus 41 3 4] ,]"</lang>

Output:

remaining:   30
remaining 4: 3,34,15,30

VBScript

<lang vb> Function josephus(n,k,s) Set prisoner = CreateObject("System.Collections.ArrayList") For i = 0 To n - 1 prisoner.Add(i) Next index = -1 Do Until prisoner.Count = s step_count = 0 Do Until step_count = k If index+1 <= prisoner.Count-1 Then index = index+1 Else index = (index+1)-(prisoner.Count) End If step_count = step_count+1 Loop prisoner.RemoveAt(index) index = index-1 Loop For j = 0 To prisoner.Count-1 If j < prisoner.Count-1 Then josephus = josephus & prisoner(j) & "," Else josephus = josephus & prisoner(j) End If Next End Function

'testing the function WScript.StdOut.WriteLine josephus(5,2,1) WScript.StdOut.WriteLine josephus(41,3,1) WScript.StdOut.WriteLine josephus(41,3,3) </lang>

Output:

2
30
15,30,34

Vedit macro language

This macro first creates a list of prisoners in an edit buffer.
Then the prisoners are deleted in loop until specified number of survivors are left.
When the macro finishes, you can see the list of survivors in the edit buffer.

<lang vedit>#1 = 41 // number of prisoners

1. 2 = 3 // step size
2. 3 = 1 // number of survivors

Buf_Switch(Buf_Free) for (#5=0; #5<#1; #5++) {

   Ins_Text("prisoner ") Num_Ins(#5, LEFT)

}

BOF

1. 4=1

while (#1 > #3) {

   if (#4++ % #2 == 0) {

Del_Line(1)

       #1--
   } else {

Line(1)

   }
   if (At_EOF) { BOF }

}</lang>

Output:

prisoner 30

Output:

when the number of survivors is set to 3

prisoner 15
prisoner 30
prisoner 34

XPL0

<lang XPL0>include c:\cxpl\codes;

func Prisoner(N, K); \Return final surviving prisoner int N, K; \number of prisoners, number to skip int I, J; char A; [A:= Reserve(N); for I:= 0 to N-1 do A(I):= I; I:= 0; repeat I:= I+K-1; \skip to next prisoner

       I:= rem(I/N);                           \wrap to start if necessary
       IntOut(0, A(I)); ChOut(0, ^ );          \show killed prisoner
       for J:= I to N-2 do A(J):= A(J+1);      \shift survivors down
       N:= N-1;                                \one less prisoner

until N=1; return A(0); ];

[IntOut(0, Prisoner(5, 2)); CrLf(0);

IntOut(0, Prisoner(41, 3));  CrLf(0);

]</lang>

Output:

1 3 0 4 2
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 30

zkl

Translation of: Julia

<lang zkl>fcn j(n,k){

  reg p=[0..n-1].walk().copy(), i=0, seq=L();
  while(p){
     i=(i+k-1)%p.len();
     seq.append(p.pop(i));
  }
  "Prisoner killing order: %s.\nSurvivor: %d"
  .fmt(seq[0,-1].concat(","),seq[-1]);

}</lang>

Output:

j(41,3).println();
Prisoner killing order: 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.
Survivor: 30

<lang zkl>fcn j2(n,k,m){

  reg p=[0..n-1].walk().copy(), i=0, seq=L();
  while(p.len()>m){
     i=(i+k-1)%p.len();
     seq.append(p.pop(i));
  }
  "Prisoner killing order: %s.\nSurvivors: [%s]"
  .fmt(seq.concat(","),p.concat(","))

}</lang>

Output:

j2(41,3,3).println();
Prisoner killing order: 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.
Survivors: [15,30,34]