# 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: ${\displaystyle n}$ prisoners are standing on a circle, sequentially numbered from ${\displaystyle 0}$ to ${\displaystyle n-1}$.

An executioner walks along the circle, starting from prisoner ${\displaystyle 0}$, removing every ${\displaystyle k}$-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 ${\displaystyle n=5}$ prisoners and ${\displaystyle k=2}$, 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.

Given any   ${\displaystyle n,k>0}$,   find out which prisoner will be the final survivor.

In one such incident, there were 41 prisoners and every 3rd prisoner was being killed   (${\displaystyle k=3}$).

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 ${\displaystyle m}$ survivors free,
and Josephus might just have   ${\displaystyle m-1}$   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 ${\displaystyle O(kn)}$. However, individually it takes no more than ${\displaystyle O(m)}$ to find out which prisoner is the ${\displaystyle m}$-th to die.
2. If it's more convenient, you can number prisoners from   ${\displaystyle 1}$ to ${\displaystyle n}$   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.

## 360 Assembly

Translation of: REXX

The program uses two ASSIST macros (XDECO,XPRNT) to keep the code as short as possible.

*      Josephus problem               10/02/2017
JOSEPH CSECT
USING JOSEPH,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) prolog
ST R13,4(R15) " <-
ST R15,8(R13) " ->
LA R7,1 m=1
DO WHILE=(C,R7,LE,=A(NPROB)) do m=1 to nprob
LR R1,R7 m
MH R1,=H'6' *6
LH R2,PROB-6(R1)
ST R2,N n=prob(m,1)
LH R2,PROB-4(R1)
ST R2,W w=prob(m,2)
LH R2,PROB-2(R1)
ST R2,S s=prob(m,3)
MVC PG,=CL80'josephus' init buffer
L R1,N n
XDECO R1,DEC edit
MVC PG+8(4),DEC+8 output
L R1,W w
XDECO R1,DEC edit
MVC PG+12(4),DEC+8 output
L R1,S s
XDECO R1,DEC edit
MVC PG+16(4),DEC+8 output
XPRNT PG,L'PG print buffer
L R11,N nx=n
L R8,=F'-1' p=-1
DO UNTIL=(C,R11,EQ,S) do until n=s
SR R9,R9 found=0
DO UNTIL=(C,R9,EQ,W) do until found=w
LA R8,1(R8) p=p+1
IF C,R8,EQ,N THEN if p=nn then
SR R8,R8 p=0
ENDIF , end if
IF CLI,0(R2),EQ,X'00' THEN if not dead(p+1) then
LA R9,1(R9) found=found+1
ENDIF , end if
ENDDO , end do
BCTR R11,0 nx=nx-1
ENDDO , end do
MVC PG,=CL80' ' clear buffer
LA R10,PG ipg=0
L R9,N nn
BCTR R9,0 nn-1
SR R6,R6 i=0
DO WHILE=(CR,R6,LE,R9) do i=0 to nn-1
IF CLI,0(R2),EQ,X'00' THEN if not dead(i+1) then
XDECO R6,DEC edit i
MVC 0(4,R10),DEC+8 output
LA R10,4(R10) ipg=ipg+4
ENDIF , end if
LA R6,1(R6) i=i+1
ENDDO , end do
XPRNT PG,L'PG print buffer
LA R7,1(R7) m=m+1
ENDDO , end do
L R13,4(0,R13) epilog
LM R14,R12,12(R13) " restore
XR R15,R15 " rc=0
BR R14 exit
PROB DC H'41',H'3',H'1' round 1
DC H'41',H'3',H'3' round 2
NPROB EQU (*-PROB)/6 number of rounds
N DS F n number of prisoners
W DS F w killing count
S DS F s number of prisoners to survive
PG DS CL80 buffer
DEC DS CL12 temp for xdeco
YREGS
END JOSEPH
Output:
josephus  41   3   1
30
josephus  41   3   3
15  30  34

## 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.)

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
STA \$D2
CMP \$D0  ; have we killed all but 1?
BPL RETURN
LDY #\$00
JMP FIND
RETURN: TXA  ; a <- index of survivor
RTS

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.

procedure Josephus is

function Arg(Idx, Default: Positive) return Positive is -- read Argument(Idx)

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
("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
end if;
for Execution in reverse 1 .. Prisoners loop
if Execution = Survivors then
Print := True;
end if;
if Print then
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;
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

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

## ANSI Standard BASIC

Translated from ALGOL 68

100 FUNCTION josephus (n, k, m)
110 ! Return m-th on the reversed kill list; m=0 is final survivor.
120 LET lm = m  ! Local copy OF m
130 FOR a = m+1 TO n
140 LET lm = MOD(lm+k, a)
150 NEXT a
160 LET josephus = lm
170 END FUNCTION
180 LET n = 41
190 LET k=3
200 PRINT "n =";n, "k =";k,"final survivor =";josephus(n, k, 0)
210 END

## AutoHotkey

; 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
Output:
31

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

### Using Objects

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

## AWK

# syntax: GAWK -f JOSEPHUS_PROBLEM.AWK
# converted from PL/I
BEGIN {
main(5,2,1)
main(41,3,1)
main(41,3,3)
exit(0)
}
# n - number of prisoners
# k - kill every k'th prisoner
# 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++) {
survived = survived i " " # build survivor list
}
}
printf("killed: %s\n",killed)
printf("survived: %s\n",survived)
return(1)
}

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

## BASIC

Unstructured implementation: see solutions listed under specific BASIC dialects for structured versions.

10 N=41
20 K=3
30 M=0
40 FOR I=M+1 TO N
50 M=INT(I*((M+K)/I-INT((M+K)/I))+0.5)
60 NEXT I
70 PRINT "Survivor is number";M
Output:
Survivor is number 30

### Applesoft BASIC

Translated from the BASIC implementation above and the ANSI Standard BASIC.

10 DEF FN MOD(X) = X - INT (X / A) * A
20 LM = 0: INPUT "GIVE N AND K (N,K): ";N,K
30 IF N < 1 or K < 1 THEN GOTO 20
40 FOR A = 1 TO N: LM = FN MOD(LM + K): NEXT A
50 PRINT "N = ";N;", K = ";K;", SURVIVOR: ";LM

Output:
GIVE N AND K (N,K): 41,3
N = 41, K = 3, SURVIVOR: 30

### IS-BASIC

100 PROGRAM "Josephus.bas"
110 INPUT PROMPT "Number of prisoners: ":NP
120 INPUT PROMPT "Execution step: ":EX
130 INPUT PROMPT "How many survivors: ":SU
140 PRINT "Survivors:";
150 FOR S=0 TO SU-1
160 PRINT JOSEPHUS(NP,EX,S);
170 NEXT
180 DEF JOSEPHUS(N,K,M)
190 FOR I=M+1 TO N
200 LET M=MOD((M+K),I)
210 NEXT
220 LET JOSEPHUS=M
230 END DEF

## Batch File

Uses C's jos() function.

Translation of: C
@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
Output:
30
34 15 30
Press any key to continue . . .

## BBC BASIC

REM >josephus
PRINT "Survivor is number "; FNjosephus(41, 3, 0)
END
:
DEF FNjosephus(n%, k%, m%)
LOCAL i%
FOR i% = m% + 1 TO n%
m% = (m% + k%) MOD i%
NEXT
= m%
Output:
Survivor is number 30

## Befunge

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

>0" :srenosirP">:#,_&>>00p>>v
v0p01<&_,#!>#:<"Step size: "<
>1+:20p00g`!#v_0"  :rovivru"v
^g02%g02+g01<<@.\$_,#!>#:<"S"<
Output:
Prisoners: 41
Step size: 3
Survivor:  30

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

## C#

namespace Josephus
{
using System;
using System.Collections;
using System.Collections.Generic;

public class Program
{
public static int[] JosephusProblem(int n, int m)
{
var circle = new List<int>();
var order = new int[n];

for (var i = 0; i < n; ++i)
{
}

var l = 0;
var j = 0;
var k = 0;

while (circle.Count != 0)
{
j++;
if (j == m)
{
order[k] = circle[l];
circle.RemoveAt(l);

k++;
l--;
j = 0;
}

if (k == n - 1)
{
order[k] = circle[0];
circle.RemoveAt(0);
}

if (l == circle.Count - 1)
{
l = 0;
}
else
{
l++;
}
}

return order;
}

static void Main(string[] args)
{
try
{
var n = 7;
var m = 2;

var result = JosephusProblem(n, m);

for (var i = 0; i < result.Length; i++)
{
Console.WriteLine(result[i]);//1 3 5 0 4 2 6
}
}
catch (Exception e)
{
Console.WriteLine(e);
}
finally
{
}
}

}
}

## C++

#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,

## 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.")))
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:

(defun kill (n k &aux (m 0))
(loop for a from (1+ m) upto n do
(setf m (mod (+ m k) a)))
m)

Using a circular list.

(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 ()
(cdr list) (cddr list))))
(loop until (one-element-clist-p list) do
(move-forward)
(kill-item))
(first list))))
Example:
CL-USER > (kill 41 3)
30

## D

Translation of: Python
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;
}
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
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);
}}
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.

;; 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

Output:

;; 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 … ∞)

## 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
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

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

## 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)

Output:
30

## Emacs Lisp

(defun jo(n k)
(if (= 1 n) 1 (1+ (% (+ (1- k)
(jo (1- n) k)) n ) ) ))
(princ-list (jo 50 2) "\n" (jo 60 3))
Output:
37
41

## Erlang

-module( josephus_problem ).

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 ).

Output:
{[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

PROGRAM JOSEPHUS

!
! for rosettacode.org
!

!\$INTEGER

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
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
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
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

Note: Adapted from AWK version! Output is the same.

## Factor

USING: kernel locals math math.ranges sequences ;
IN: josephus

:: josephus ( k n -- m )
n [1,b] 0 [ [ k + ] dip mod ] reduce ;
IN: scratchpad 3 41 josephus .
30

## Forth

: josephus  0 1 begin dup 41 <= while  swap 3 + over mod swap  1+ repeat drop ;
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.

program josephus
implicit none
integer :: n, i, k, p
integer, allocatable :: next(:)
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

## friendly interactive shell

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.
Output:
Prisoner 30 survived.

It's also possible to calculate more than one survivor.

echo Prisoners (execute 3 (seq 0 40))[-3..-1] survived.
Output:
Prisoners 34 15 30 survived.

Prisoners don't have to be numbers.

echo Prisoner (execute 2 Joe Jack William Averell Rantanplan)[-1] survived.
Output:
Prisoner William survived.

## 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);

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

## Go

package main

import "fmt"

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))
}
}
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

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

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))
_ -> print \$ snd \$ killRecursive (prisoners 41) (counter 3) 1

Using modulo and list split, indices are 1-based. This is much faster than cycled list for larger numbers:

jseq :: Int -> Int -> [Int]
jseq n k = f n [1 .. n]
where
f 0 _ = []
f m s = x : f (m - 1) (right ++ left)
where
(left, x:right) = splitAt (mod (k - 1) m) s

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

main :: IO ()
main = do
print \$ jseq 41 3
print \$ jos 10000 100

## Icon and Unicon

The following works in both languages.

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
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...

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

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

3 ([ (1 }. <:@[ |. ])^:(1 < #@])^:_ [email protected]]) 41
30

Structured derivation of the fixed tacit code

DropNext=. 1 }. <:@[ |. ]
MoreThanOne=. 1 < #@]
WhileMoreThanOne=. (^:MoreThanOne f.) (^:_)
prisoners=. [email protected]]

[ DropNext WhileMoreThanOne prisoners f.
[ (1 }. <:@[ |. ])^:(1 < #@])^:_ [email protected]]

### Explicit version

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

### Explicit version 2

NB. this is a direct translation of the algo from C code above.
Josephus2 =: 4 : '(| x&+)/i. - 1+y'

3 Josephus2 41
30

## Java

Works with: Java version 1.5+
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++){
}
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++){
}
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));
}
}
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
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;
}
}
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:

var Josephus = {
init: function(n) {
for (var i = 0; i < n-1; i++) {
current.label = i+1;
current.next = {prev: current};
current = current.next;
}
current.label = n;
return this;
},
kill: function(spacing) {
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;
}
}
Output:
> Josephus.init(30).kill(2)
29

With Array methods:

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)+',...' , '<br>')
return [ps, ks]
}
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

Works with: Julia version 0.6

Recursive (with Memoize):

using Memoize
@memoize josephus(n::Integer, k::Integer, m::Integer=1) = n == m ? collect(0:m .- 1) : mod.(josephus(n - 1, k, m) + k, n)

@show josephus(41, 3)
@show josephus(41, 3, 5)
Output:
josephus(41, 3) = [30]
josephus(41, 3, 5) = [3, 15, 21, 30, 34]

Iterative:

function josephus(n::Integer, k::Integer, m::Integer=1)
p, i, seq = collect(0:n-1), 0, Vector{typeof(n)}(0)
while length(p) > m
i = (i + k - 1) % length(p)
push!(seq, splice!(p, i + 1))
end
return seq, p
end

seq, surv = josephus(41, 3)
println("Prisoner killing in order: \$seq\nSurvivor: \$surv")

seq, surv = josephus(41, 3, 3)
println("Prisoner killing in order: \$seq\nSurvivor: \$surv")
Output:
Prisoner killing 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]
Prisoner killing 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]
Survivor: [15, 30, 34]

## 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.

# A control structure, for convenience:
# 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;

# n is the initial number; every k-th prisoner is removed until m remain.
# 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] );

Examples:

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

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]

## Kotlin

// version 1.1.3

fun josephus(n: Int, k: Int, m: Int): Pair<List<Int>, List<Int>> {
require(k > 0 && m > 0 && n > k && n > m)
val killed = mutableListOf<Int>()
val survived = MutableList(n) { it }
var start = k - 1
outer@ while (true) {
val end = survived.size - 1
var i = start
var deleted = 0
while (i <= end) {
if (survived.size == m) break@outer
deleted++
i += k
}
start = i - end - 1
}
return Pair(survived, killed)
}

fun main(args: Array<String>) {
val triples = listOf(Triple(5, 2, 1), Triple(41, 3, 1), Triple(41, 3, 3))
for (triple in triples) {
val(n, k, m) = triple
println("Prisoners = \$n, Step = \$m, Survivors = \$m")
val (survived, killed) = josephus(n, k, m)
println("Survived  : \$survived")
println("Kill order : \$killed")
println()
}
}
Output:
Prisoners = 5, Step = 1, Survivors = 1
Survived   : [2]
Kill order : [1, 3, 0, 4]

Prisoners = 41, Step = 1, Survivors = 1
Survived   : [30]
Kill 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]

Prisoners = 41, Step = 3, Survivors = 3
Survived   : [15, 30, 34]
Kill 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]

## Lua

Lua indexes tables starting at 1. Positions are stored from 0,n-1.

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)

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.

## MATLAB

function [indAlive] = josephus(numPeople,count)
% Josephus: Given a circle of numPeople individuals, with a count of count,
% find the index (starting at 1) of the survivor [see Josephus Problem]

%% Definitions:
% 1 = alive position
% index = # of person

%% Setting up
arrPeople = ones(1, numPeople);
currInd = 0;

%% Counting
while (length(arrPeople(arrPeople == 1)) > 1) % While more than 1 person is alive
counter = 0;
while counter ~= count % Counting until we hit the count
currInd = currInd + 1; % Move to the next person

if currInd > numPeople % If overflow, wraparound
currInd = currInd - numPeople;
end

if arrPeople(currInd) % If the current person is alive
counter = counter + 1; % Add 1 person to the count
%fprintf("Index: %d \t| Counter: %d\n", currInd, counter)  % Uncomment to display index and counter location
end

end

arrPeople(currInd) = 0; % Kill the person we reached
%fprintf("Killed person %d \n", currInd)  % Uncomment to display order of killing
%disp(arrPeople)  % Uncomment to display current status of people
end

indAlive = find(arrPeople);

end

## Mathematica

survivor[n_, k_] := Nest[Most[RotateLeft[#, k]] &, Range[0, n - 1], n - 1]
survivor[41, 3]
Output:
{30}

## Modula-2

MODULE Josephus;
FROM FormatString IMPORT FormatString;

PROCEDURE Josephus(n,k : INTEGER) : INTEGER;
VAR a,m : INTEGER;
BEGIN
m := 0;
FOR a:=1 TO n DO
m := (m + k) MOD a;
END;
RETURN m
END Josephus;

VAR
buf : ARRAY[0..63] OF CHAR;
n,k,i : INTEGER;
nl,kl,il : LONGCARD;
BEGIN
n := 41;
k := 3;
FormatString("n = %i, k = %i, final survivor: %i\n", buf, n, k, Josephus(n, k));
WriteString(buf);

END Josephus.

## NetRexx

Translation of: REXX

Hardly any changes at all...

/* 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
**********************************************************************/

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
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
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
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
system.delete(p, i)

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

echo j(5,2)
echo j(41,3)
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

class Josephus {
function : Execute(n : Int, k : Int) ~ Int {
killIdx := 0;
prisoners := Collection.IntVector->New();
for(i := 0;i < n;i+=1;){
};

"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 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();
};
};
}
}

## Oforth

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

: 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
] drop

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

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

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

## Perl

Translation of: Perl6
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"
Output:
Prisoner 30 survived.

## Perl 6

Works with: rakudo version 2015-11-12

Straightforward implementation of the executioner's algorithm:

sub Execute(@prisoner, \$k) {
until @prisoner == 1 {
@prisoner.=rotate(\$k - 1);
@prisoner.shift;
}
}

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

# We don't have to use numbers. Any list will do:

my @dalton = <Joe Jack William Averell Rantanplan>;
Execute @dalton, 2;
say "{@dalton} survived.";
Output:
Prisoner 30 survived.
William survived.

## Phix

Note indexes and results are 1-based. Prisoners do not have to be numbers. Based on AWK, but replacing killed prisoners in-situ.

function Josephus(sequence prisoners, integer step, survivors)
integer n = length(prisoners), nn = n
integer p = 0
while n>survivors do
integer found = 0
while found!=step do
p = iff(p=nn?1:p+1)
found += prisoners[p]!=-1
end while
-- (if you want a kill list, build it here!)
prisoners[p] = -1
n -= 1
end while
return remove_all(-1,prisoners)
end function

?Josephus(tagset(5),2,1)
?Josephus(tagset(41),3,1)
?Josephus(tagset(41),3,3)
?Josephus({"Joe","Jack","William","John","James"},2,1)
Output:
{3}
{31}
{16,31,35}
{"William"}

## 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
while((array_sum(array_count_values(\$prisoners))<\$n)){//while sum of count of unique values dead times < n (they start as all false)
\$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)))));
}else{\$duckpool++;}
}
}
}
return \$prisoners;
}
echo '<pre>'.print_r(Jotapata(41,3,5),true).'<pre>';

## PicoLisp

The counting starts from one instead of zero. The last remaining person is returned.

#general solution
(de jo (N K)
(if (=1 N)
1
(inc
(%
(+ (dec K) (jo (dec N) K))
N ) ) ) )

#special case when K is 2; much faster than general version.
(de jo2(N)
(let P 1
(while (<= P N)
(setq P (* 2 P))
(+ (- (* 2 N) P) 1) ) ) )

# find the survivor using an optimal solution
(de survivor (N K)
(if (=0 (% N 2))
(jo2 N)
(jo N K) ) )
(print (survivor 5 2))
(print (survivor 41 3))

Output:
3
31

## PL/I

*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 (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): ');
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;
Output:
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>
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
}

# Get the prisoner order for a circle of 41 prisoners, selecting every third
\$Prisoners = Get-JosephusPrisoners -N 41 -K 3

# Display the prisoner order
\$Prisoners -join " "

# Display the last remaining prisoner
"Last prisoner remmaining: " + \$Prisoners[-1]

# Display the last three remaining prisoners
\$S = 3
"Last \$S remaining: " + \$Prisoners[-\$S..-1]

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

NewList prisoners.i()

Procedure f2l(List p.i())
FirstElement(p())  : tmp.i=p()
DeleteElement(p(),1) : LastElement(p())
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
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

>>> 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
>>>

Faster way to solve in python, it does not show the killing order.

>>>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
>>>

### 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.

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

### learning iter in python

from itertools import compress, cycle
def josephus(prisoner, kill, surviver):
p = range(prisoner)
k = [0] * kill
k[kill-1] = 1
s = [1] * kill
s[kill -1] = 0
queue = p

queue = compress(queue, cycle(s))
try:
while True:
p.append(queue.next())
except StopIteration:
pass

kil=[]
killed = compress(p, cycle(k))
try:
while True:
kil.append(killed.next())
except StopIteration:
pass

print 'The surviver is: ', kil[-surviver:]
print 'The kill sequence is ', kil[:prisoner-surviver]

josephus(41,3,2)
The surviver is: [15, 30]
The kill sequence is [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]
josephus(5,2,1)
The surviver is: [2]
The kill sequence is [1, 3, 0, 4]

## 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

## Racket

#lang racket
(define (josephus n k (m 0))
(remainder (+ m k) a)))

(josephus 41 3) ; ->30

## REBOL

Works in Rebol 2 or 3

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"]
Output:
Prisoner 30 survived

And any kind of list will do:

for-the-chop: [Joe Jack William Averell Rantanplan]
execute for-the-chop 2
print [for-the-chop "survived"]
Output:
William survived

## REXX

### version 1

/* 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 */
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
/*
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 */
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 Kth prisoner]
•   the start count   [zero or one]
•   the number of survivors
•   the solving of the extra credit task requirement of multiple survivors

This solution is an   executor's   solution.

/*REXX program solves  Josephus problem:   N  men standing in a circle,  every Kth kilt.*/
parse arg N K Z R . /*obtain optional arguments from the CL*/
if N=='' | N=="," then N= 41 /* men not specified? Use default.*/
if K=='' | K=="," then K= 3 /* kilt " " " " */
if Z=='' | Z=="," then Z= 0 /* start " " " " */
if R=='' | R=="," then R= 1 /*remaining " " " " */
\$=; do i=Z for N; \$=\$ i; end /*i*/ /*populate prisoner's circle (with a #)*/
x= /*the list of prisoners to be removed. */
do c=k by k; p=words(\$) /*keep removing until R are remaining*/
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 c /*The slaying finished? (R people left)*/
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) /*plurals*/
th: y=arg(1); return y || word('th st nd rd', 1+ y // 10 * (y//100%10\==1) * (y//10<4))
output   when using the default inputs:
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

{{out|output|text=  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

{{out|output|text=  when using the input of:   5   2

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

## Ring

n = 41
k=3
see "n =" + n + " k = " + k + " final survivor = " + josephus(n, k, 0) + nl

func josephus (n, k, m)
lm = m
for a = m+1 to n
lm = (lm+k) % a
next
josephus = lm
return josephus

Output:

n =41 k = 3 final survivor = 30

## Ruby

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

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

## Scala

Executioner's Solution, not Josephus'

(Prisoners labeled 0 to n-1)

def executed( prisonerCount:Int, step:Int ) = {

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

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 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))
}

println( "Prisoners executed in order:" )

println( "\n\nJosephus is prisoner " + alive(0) )
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.

\$ 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;
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

## SequenceL

Translation of: Python
main := josephus(41, 3);

josephus(n, k) := josephusHelper(n, k, 1, 0);

josephusHelper(n, k, i, r) :=
r when i > n
else
josephusHelper(n, k, i + 1, (r + k) mod i);
Output:
30

## Sidef

Iterative:

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

Recursive:

func josephus(n, k) {
n == 1 ? 0 : ((__FUNC__(n-1, k) + k) % n)
};

Calling the function:

var survivor = josephus(41, 3);
say "Prisoner #{survivor} survived.";
Output:
Prisoner 30 survived.

## 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))")
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

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]
}

Demonstrating:

puts "remaining:   [josephus 41 3]"
puts "remaining 4: [join [josephus 41 3 4] ,]"
Output:
remaining:   30
remaining 4: 3,34,15,30

## VBScript

Function josephus(n,k,s)
Set prisoner = CreateObject("System.Collections.ArrayList")
For i = 0 To n - 1
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)

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.

#1 = 41		// number of prisoners
#2 = 3 // step size
#3 = 1 // number of survivors

Buf_Switch(Buf_Free)
for (#5=0; #5<#1; #5++) {
Ins_Text("prisoner ") Num_Ins(#5, LEFT)
}

BOF
#4=1
while (#1 > #3) {
if (#4++ % #2 == 0) {
Del_Line(1)
#1--
} else {
Line(1)
}
if (At_EOF) { BOF }
}
Output:
prisoner 30
Output:
when the number of survivors is set to 3:
prisoner 15
prisoner 30
prisoner 34

## Visual Basic .NET

Translation of: D
Module Module1

'Determines the killing order numbering prisoners 1 to n
Sub Josephus(n As Integer, k As Integer, m As Integer)
Dim p = Enumerable.Range(1, n).ToList()
Dim i = 0

Console.Write("Prisoner killing order:")
While p.Count > 1
i = (i + k - 1) Mod p.Count
Console.Write(" {0}", p(i))
p.RemoveAt(i)
End While
Console.WriteLine()

Console.WriteLine("Survivor: {0}", p(0))
End Sub

Sub Main()
Josephus(5, 2, 1)
Console.WriteLine()
Josephus(41, 3, 1)
End Sub

End Module
Output:
Prisoner killing order: 2 4 1 5
Survivor: 3

Prisoner killing order: 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

## 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;
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);
]
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
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]);
}
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
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(","))
}
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]

## ZX Spectrum Basic

Translation of: ANSI Standard BASIC
10 LET n=41: LET k=3: LET m=0
20 GO SUB 100
30 PRINT "n= ";n;TAB (7);"k= ";k;TAB (13);"final survivor= ";lm
40 STOP
100 REM Josephus
110 REM Return m-th on the reversed kill list; m=0 is final survivor.
120 LET lm=m: REM Local copy of m
130 FOR a=m+1 TO n
140 LET lm=FN m(lm+k,a)
150 NEXT a
160 RETURN
200 DEF FN m(x,y)=x-INT (x/y)*y: REM MOD function