Josephus problem: Difference between revisions
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===version 2=== |
===version 2=== |
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This version allows the user to specify: |
This REXX version allows the user to specify: |
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::* the number of prisoners |
::* the number of prisoners |
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::* the count-off [every K<sup>th</sup> prisoner] |
::* the count-off [every K<sup>th</sup> prisoner] |
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::* the number of survivors |
::* the number of survivors |
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::* the solving of the extra credit task requirement of multiple survivors |
::* the solving of the extra credit task requirement of multiple survivors |
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The output echoes the choices specified and was made "English" readable. |
The output echoes the choices specified and was made "English" readable. |
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Revision as of 01:08, 25 January 2020
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: prisoners are standing on a circle, sequentially numbered from to .
An executioner walks along the circle, starting from prisoner , removing every -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 prisoners and , 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 , find out which prisoner will be the final survivor.
In one such incident, there were 41 prisoners and every 3rd prisoner was being killed ().
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 survivors free,
and Josephus might just have friends to save.
Provide a way to calculate which prisoner is at any given position on the killing sequence.
- Notes
- 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 . However, individually it takes no more than to find out which prisoner is the -th to die.
- If it's more convenient, you can number prisoners from to instead. If you choose to do so, please state it clearly.
- 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
The program uses two ASSIST macros (XDECO,XPRNT) to keep the code as short as possible. <lang 360asm>* 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) " ->
LR R13,R15 " addressability
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
MVI DEAD,X'00' dead(1)='0'B;
MVC DEAD+1(255),DEAD dead(*)='0'B;
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
LA R2,DEAD(R8) @dead(p+1)
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
LA R2,DEAD(R8) @dead(p+1)
MVI 0(R2),X'01' dead(p+1)='1'B
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
LA R2,DEAD(R6) @dead(i+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 DEAD DS 256X n max
YREGS
END JOSEPH</lang>
- 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.) <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
ANSI Standard BASIC
Translated from ALGOL 68 <lang ANSI Standard BASIC>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 </lang>
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>
- 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)
} function main(n,k,s, dead,errors,found,i,killed,nn,p,survived) {
- 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++) {
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
BASIC
Unstructured implementation: see solutions listed under specific BASIC dialects for structured versions. <lang basic>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</lang>
- Output:
Survivor is number 30
Applesoft BASIC
Translated from the BASIC implementation above and the ANSI Standard BASIC. <lang Applesoft 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
</lang>
- Output:
GIVE N AND K (N,K): 41,3 N = 41, K = 3, SURVIVOR: 30
IS-BASIC
<lang 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</lang>
Batch File
Uses C's jos() function.
<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 . . .
BBC BASIC
<lang bbcbasic>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%</lang>
- Output:
Survivor is number 30
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 csharp> 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)
{
circle.Add(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
{
Console.ReadLine();
}
}
}
} </lang>
C++
<lang cpp>
- 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;
} //-------------------------------------------------------------------------------------------------- </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
Crystal
<lang ruby>n = ARGV.fetch(0, 41).to_i # n default is 41 or ARGV[0] k = ARGV.fetch(1, 3).to_i # k default is 3 or ARGV[1]
prisoners = (0...n).to_a while prisoners.size > 1; prisoners.rotate!(k-1).shift end puts "From #{n} prisoners, eliminating each prisoner #{k} leaves prisoner #{prisoners.first}." </lang>
- Output:
$ crystal josephus.cr From 41 prisoners, eliminating each prisoner 3 leaves prisoner 30. $ crystal josephus.cr 123 From 123 prisoners, eliminating each prisoner 3 leaves prisoner 54. $ crystal josephus.cr 123 47 From 123 prisoners, eliminating each prisoner 47 leaves prisoner 101.
D
<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
<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>
- Output:
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
Emacs Lisp
<lang 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))</lang>
- Output:
37 41
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
Fōrmulæ
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
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>
FreeBASIC
<lang freebasic> Function Josephus (n As Integer, k As Integer, m As Integer) As Integer
Dim As Integer lm = m
For i As Integer = m + 1 To n
lm = (lm + k) Mod i
Next i
Josephus = lm
End Function
Dim As Integer n = 41 'prisioneros Dim As Integer k = 3 'orden de ejecución
Print "n ="; n, "k ="; k, "superviviente = "; Josephus(n, k, 0) </lang>
- Output:
n = 41 k = 3 superviviente = 30
Frink
<lang frink> killingCycle[prisonerCount,killStep = 2] := {
i = 0
killed = new array
prisoners = array[0 to prisonerCount - 1]
while length[prisoners] > 1
{
i = (i + killStep - 1) mod length[prisoners]
killed.push[prisoners.remove[i]] // Remove the killed prisoner from the prisoners array and add it to the killed array.
}
killedResult = "Killed:"
for kill = killed // Loop through the killed array to format it nicely.
{
killedResult = killedResult + " " + kill
}
aliveResult = "Alive: " + prisoners@0 // Get the only item left in the array
return """$killedResult
$aliveResult""" }
println[killingCycle[41,3]] // Enter in total number of prisoners and the number to skip each cycle </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 Alive: 30
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 :: 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</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
<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]
<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 (with Memoize): <lang julia>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)</lang>
- Output:
josephus(41, 3) = [30] josephus(41, 3, 5) = [3, 15, 21, 30, 34]
Iterative: <lang julia>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")</lang>
- 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
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:
- 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] );</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]
Kotlin
<lang scala>// 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) {
killed.add(survived.removeAt(i - deleted))
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()
}
}</lang>
- 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. <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.
MATLAB
<lang 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: % 0 = dead position % 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 </lang>
Mathematica
<lang mathematica>survivor[n_, k_] := Nest[Most[RotateLeft[#, k]] &, Range[0, n - 1], n - 1] survivor[41, 3]</lang>
- Output:
{30}
Modula-2
<lang modula2>MODULE Josephus; FROM FormatString IMPORT FormatString; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
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);
ReadChar
END Josephus.</lang>
Nanoquery
<lang Nanoquery>def j(n, k)
p = list(range(0, n-1))
i = 0
seq = {}
while len(p) > 0
i = (i+k-1) % len(p)
seq.append(p[i])
p.remove(i)
end
sur = seq[len(seq) - 1]; seq.remove(len(seq) - 1)
return format("Prisoner killing order: %s\nSurvivor: %d", seq, sur)
end
println j(5,2) println println j(41,3)</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
NetRexx
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
<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
<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
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.";
- We don't have to use numbers. Any list will do:
my @dalton = <Joe Jack William Averell Rantanplan>; Execute @dalton, 2; say "{@dalton} survived.";</lang>
- 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. <lang Phix>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)</lang>
- Output:
{3}
{31}
{16,31,35}
{"William"}
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|PicoLisp}}==
The counting starts from one instead of zero. The last remaining person is returned.
<lang PicoLisp>
#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))
</lang>
{{out}}
<pre>
3
31
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>
- 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
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
}
</lang> <lang PowerShell>
- 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] </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
Processing
Translation of Java example. <lang processing>void setup() {
println("Survivor: " + execute(41, 3));
println("Survivors: " + executeAllButM(41, 3, 3));
}
int execute(int n, int k) {
int killIdx = 0;
IntList prisoners = new IntList(n);
for (int i = 0; i < n; i++) {
prisoners.append(i);
}
println("Prisoners executed in order:");
while (prisoners.size() > 1) {
killIdx = (killIdx + k - 1) % prisoners.size();
print(prisoners.get(killIdx) + " ");
prisoners.remove(killIdx);
}
println();
return prisoners.get(0);
}
IntList executeAllButM(int n, int k, int m) {
int killIdx = 0;
IntList prisoners = new IntList(n);
for (int i = 0; i < n; i++) {
prisoners.append(i);
}
println("Prisoners executed in order:");
while (prisoners.size() > m) {
killIdx = (killIdx + k - 1) % prisoners.size();
print(prisoners.get(killIdx) + " ");
prisoners.remove(killIdx);
}
println();
return prisoners;
}</lang>
PureBasic
<lang purebasic>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>
learning iter in python
<lang 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] </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 task states: n prisoners are standing on a circle,
- sequentially numbered from 0 to n-1. The 1st prisoner is 0.
- 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 REXX 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
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.*/ 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))</lang>
- 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
- output when using the input of: 5 2
removing every 2nd prisoner out of 5 (starting at 0) with 1 survivor, leaving prisoner: 2
Ring
<lang 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 </lang> Output:
n =41 k = 3 final survivor = 30
Ruby
<lang Ruby>n = (ARGV[0]{
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
- 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 }
}</lang>
- Output:
prisoner 30
- Output:
when the number of survivors is set to 3
prisoner 15 prisoner 30 prisoner 34
Visual Basic .NET
<lang vbnet>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</lang>
- 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
<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
<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]
ZX Spectrum Basic
<lang zxbasic>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 </lang>
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