Josephus problem: Difference between revisions
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Revision as of 18:24, 26 May 2015
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.
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
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
C
<lang c>#include <stdio.h>
// m-th on the reversed kill list; m = 0 is final survivor int jos(int n, int k, int m) { int a; for (a = m + 1; a <= n; a++) m = (m + k) % a; return m; }
typedef unsigned long long xint;
// same as jos(), useful if n is large and k is not xint jos_large(xint n, xint k, xint m) { if (k <= 1) return n - m - 1;
xint a = m; while (a < n) { xint q = (a - m + k - 2) / (k - 1);
if (a + q > n) q = n - a; else if (!q) q = 1;
m = (m + q * k) % (a += q); }
return m; }
int main(void) { xint n, k, i;
n = 41; k = 3; printf("n = %llu, k = %llu, final survivor: %d\n", n, k, jos(n, k, 0));
n = 9876543210987654321ULL; k = 12031; printf("n = %llu, k = %llu, three survivors:", n, k);
for (i = 3; i--; ) printf(" %llu", jos_large(n, k, i)); putchar('\n');
return 0; }</lang>
- Output:
n = 41, k = 3, final survivor: 30 n = 9876543210987654321, k = 12031, three survivors: 6892710366467541051 1946357796579138992 3554846299321782413
C++
<lang cpp>
- 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,
Common Lisp
Using a loop: <lang lisp>(defun kill (n k &aux (m 0))
(loop for a from (1+ m) upto n do
(setf m (mod (+ m k) a)))
m)</lang>
Using a circular list. <lang lisp>(defun make-circular-list (n)
(let* ((list (loop for i below n
collect i))
(last (last list)))
(setf (cdr last) list)
list))
(defun kill (n d)
(let ((list (make-circular-list n)))
(flet ((one-element-clist-p (list)
(eq list (cdr list)))
(move-forward ()
(loop repeat (1- d)
until (eq list (cdr list))
do (setf list (cdr list))))
(kill-item ()
(setf (car list) (cadr list)
(cdr list) (cddr list))))
(loop until (one-element-clist-p list) do
(move-forward)
(kill-item))
(first list))))</lang>
- Example:
CL-USER > (kill 41 3) 30
D
<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
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, i: INTEGER prisoners: LINKED_LIST [INTEGER] do create prisoners.make from i := 0 until i = n loop prisoners.extend (i) i := i + 1 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 + " ") prisoners.go_i_th (killidx) prisoners.remove end io.new_line Result := prisoners.at (1) ensure Result_in_range: Result >= 0 and Result < n end
end </lang>
Prisoners are executed in the order: 3 7 11 4 9 2 10 6 5 8 1 Survivor is prisoner: 0
Erlang
<lang Erlang> -module( josephus_problem ).
-export( [general_solution/3, task/0] ).
general_solution( Prisoners, Kill, Survive ) -> general_solution( Prisoners, Kill, Survive, erlang:length(Prisoners), [] ).
task() -> general_solution( lists:seq(0, 40), 3, 1 ).
general_solution( Prisoners, _Kill, Survive, Survive, Kills ) ->
{Prisoners, lists:reverse(Kills)};
general_solution( Prisoners, Kill, Survive, Prisoners_length, Kills ) ->
{Skipped, [Killed | Rest]} = kill( Kill, Prisoners, Prisoners_length ),
general_solution( Rest ++ Skipped, Kill, Survive, Prisoners_length - 1, [Killed | Kills] ).
kill( Kill, Prisoners, Prisoners_length ) when Kill < Prisoners_length ->
lists:split( Kill - 1, Prisoners );
kill( Kill, Prisoners, Prisoners_length ) ->
kill_few( Kill rem Prisoners_length, Prisoners ).
kill_few( 0, Prisoners ) ->
[Last | Rest] = lists:reverse( Prisoners ),
{lists:reverse( Rest ), [Last]};
kill_few( Kill, Prisoners ) ->
lists:split( Kill - 1, Prisoners ).
</lang>
- Output:
11> josephus_problem:task().
{[30],
[2,5,8,11,14,17,20,23,26,29,32,35,38,0,4,9,13,18,22,27,31,
36,40,6,12,19,25|...]}
The general solution can handle other items than numbers.
12> josephus_problem:general_solution( [joe, jack, william, averell, ratata], 2, 1 ).
{[william],[jack,averell,joe,ratata]}
ERRE
<lang ERRE> PROGRAM JOSEPHUS
! ! for rosettacode.org !
!$INTEGER
DIM DEAD[100]
PROCEDURE MAIN(N,K,S->ERRORS) ! n - number of prisoners ! k - kill every k'th prisoner ! s - number of survivors
LOCAL KILLED$,SURVIVED$,FOUND,P,NN,I
ERRORS=0
FOR I=0 TO 100 DO
DEAD[I]=0
END FOR ! prepare array
PRINT("N=";N,"K=";K,"S=";S) ! show arguments
IF S>N THEN PRINT("S>N";) ERRORS+=1 END IF
IF K<=0 THEN PRINT("K<=0";) ERRORS+=1 END IF
IF ERRORS>0 THEN EXIT PROCEDURE END IF
NN=N ! wrap around boundary
P=-1 ! start here
WHILE N<>S DO ! until survivor count is met
FOUND=0 ! start looking
WHILE FOUND<>K DO ! until we have the k-th prisoner
P+=1
IF P=NN THEN P=0 END IF ! wrap around
IF DEAD[P]<>1 THEN
FOUND+=1
END IF ! if prisoner is alive increment found
END WHILE
DEAD[P]=1 ! kill the unlucky one
KILLED$=KILLED$+STR$(P) ! build killed list
N-=1 ! reduce size of circle
END WHILE
FOR I=0 TO NN-1 DO
IF DEAD[I]<>1 THEN
SURVIVED$=SURVIVED$+STR$(I) ! build survivor list
END IF
END FOR
PRINT("Killed:";KILLED$)
PRINT("Survived:";SURVIVED$)
END PROCEDURE
BEGIN
ERRORS=0 MAIN(5,2,1->ERRORS) MAIN(41,3,1->ERRORS) MAIN(41,3,3->ERRORS)
END PROGRAM </lang> Note: Adapted from AWK version! Output is the same.
Factor
<lang factor>USING: kernel locals math math.ranges sequences ; IN: josephus
- josephus ( k n -- m )
n [1,b] 0 [ [ k + ] dip mod ] reduce ;</lang>
IN: scratchpad 3 41 josephus . 30
Forth
<lang forth>: josephus 0 1 begin dup 41 <= while swap 3 + over mod swap 1+ repeat drop ;</lang>
josephus . 30
Fortran
Naive approach: prisonners are put in a "linked buffer" (implemented as an array giving number of "next living prisonner"). Then we iterate, killing one after each loop, until there is only one left. <lang fortran>program josephus
implicit none
integer :: n, i, k, p
integer, allocatable :: next(:)
read *, n, k
allocate(next(0:n - 1))
do i = 0, n - 2
next(i) = i + 1
end do
next(n - 1) = 0
p = 0
do while(next(p) /= p)
do i = 1, k - 2
p = next(p)
end do
print *, "Kill", next(p)
next(p) = next(next(p))
p = next(p)
end do
print *, "Alive", p
deallocate(next)
end program</lang>
friendly interactive shell
<lang fishshell>function execute
# If the list is empty, don't do anything.
test (count $argv) -ge 2; or return
# If the list has only one element, return it
if test (count $argv) -eq 2
echo $argv[2]
return
end
# Rotate prisoners
for i in (seq 2 $argv[1])
set argv $argv[1 3..-1 2]
end
# Mention killed prisoner
echo $argv[2]
# Kill rest recursively
execute $argv[1 3..-1]
end
echo Prisoner (execute 3 (seq 0 40))[-1] survived.</lang>
- Output:
Prisoner 30 survived.
It's also possible to calculate more than one survivor. <lang fishshell>echo Prisoners (execute 3 (seq 0 40))[-3..-1] survived.</lang>
- Output:
Prisoners 34 15 30 survived.
Prisoners don't have to be numbers. <lang fishshell>echo Prisoner (execute 2 Joe Jack William Averell Rantanplan)[-1] survived.</lang>
- Output:
Prisoner William survived.
Groovy
<lang groovy>int[] Josephus (int size, int kill, int survivors) {
// init user pool
def users = new int[size];
// give initial values such that [0] = 1 (first person) [1] = 2 (second person) etc
users.eachWithIndex() {obj, i -> users[i] = i + 1};
// keep track of which person we are on (ranging from 1 to kill)
def person = 1;
// keep going until we have the desired number of survivors
while (users.size() > survivors)
{
// for each person, if they are the kill'th person, set them to -1 to show eliminated
users.eachWithIndex() {obj, i ->
if (person++ % kill == 0) {
users[i] = -1;
}
// if person overflowed kill then reset back to 1
if (person > kill) {person = 1;}
}
// clear out all eliminated persons
users = users.findAll{w -> w >= 0};
}
// resulting set is the safe positions
return users;
}
// Run some test cases
println "Final survivor for n = 10201 and k = 17: " + Josephus(10201,17,1)[0];
println "4 safe spots for n = 10201 and k = 17: " + Josephus(10201,17,4); </lang>
- Output:
Final survivor for n = 10201 and k = 17: 7450 4 safe spots for n = 10201 and k = 17: [3413, 7244, 7450, 7605]
Go
<lang go>package main
import "fmt"
// basic task function func finalSurvivor(n, k int) int {
// argument validation omitted
circle := make([]int, n)
for i := range circle {
circle[i] = i
}
k--
exPos := 0
for len(circle) > 1 {
exPos = (exPos + k) % len(circle)
circle = append(circle[:exPos], circle[exPos+1:]...)
}
return circle[0]
}
// extra func position(n, k, pos int) int {
// argument validation omitted
circle := make([]int, n)
for i := range circle {
circle[i] = i
}
k--
exPos := 0
for len(circle) > 1 {
exPos = (exPos + k) % len(circle)
if pos == 0 {
return circle[exPos]
}
pos--
circle = append(circle[:exPos], circle[exPos+1:]...)
}
return circle[0]
}
func main() {
// show basic task function on given test case
fmt.Println(finalSurvivor(41, 3))
// show extra function on all positions of given test case
fmt.Println("Position Prisoner")
for i := 0; i < 41; i++ {
fmt.Printf("%5d%10d\n", i, position(41, 3, i))
}
}</lang>
- Output:
30
Position Prisoner
0 2
1 5
2 8
3 11
4 14
5 17
6 20
7 23
8 26
9 29
10 32
11 35
12 38
13 0
14 4
15 9
16 13
17 18
18 22
19 27
20 31
21 36
22 40
23 6
24 12
25 19
26 25
27 33
28 39
29 7
30 16
31 28
32 37
33 10
34 24
35 1
36 21
37 3
38 34
39 15
40 30
Haskell
Shows only the surviving prisoners. Change "print $ snd" to just "print" to show the killed prisoners, too. The arguments to the "main" function are: n = number of prisoners, k = kill every kth prisoner, m = show at most m survivors <lang Haskell>import Data.List ((\\)) import System.Environment (getArgs)
prisoners :: Int -> [Int] prisoners n = [0 .. n - 1]
counter :: Int -> [Int] counter k = cycle [k, k-1 .. 1]
killList :: [Int] -> [Int] -> ([Int], [Int], [Int]) killList xs cs = (killed, survivors, newCs)
where
(killed, newCs) = kill xs cs []
survivors = xs \\ killed
kill [] cs rs = (rs, cs)
kill (x:xs) (c:cs) rs
| c == 1 =
let ts = rs ++ [x]
in kill xs cs ts
| otherwise =
kill xs cs rs
killRecursive :: [Int] -> [Int] -> Int -> ([Int], [Int]) killRecursive xs cs m = killR ([], xs, cs)
where
killR (killed, remaining, counter)
| length remaining <= m = (killed, remaining)
| otherwise =
let (newKilled, newRemaining, newCounter) =
killList remaining counter
allKilled = killed ++ newKilled
in killR (allKilled, newRemaining, newCounter)
main :: IO () main = do
args <- getArgs
case args of
[n, k, m] -> print $ snd $ killRecursive (prisoners (read n))
(counter (read k)) (read m)
_ -> print $ snd $ killRecursive (prisoners 41) (counter 3) 1
</lang>
Using modulo and list split, indices are 1-based. This is much faster than cycled list for larger numbers: <lang Haskell>jseq n k = f n [1 .. n] where
f 0 _ = []
f m s = x:f (m-1) (right ++ left) where
(left,x:right) = splitAt ((k-1) `mod` m) s
-- the final survivor is ((k + ...((k + ((k + 0)`mod` 1)) `mod` 2) ... ) `mod` n) jos n k = 1 + foldl (\x->((k+x)`mod`)) 0 [2..n]
main = do
print $ jseq 41 3 print $ jos 10000 100</lang>
Icon and Unicon
The following works in both languages.
<lang unicon>procedure main(A)
m := integer(A[1]) | 41
c := integer(A[2]) | 3
write("With ",m," men, counting to ",c," last position is: ", j(m,c))
end
procedure j(m,c)
return if m==1 then 0 else (j(m-1,c)+c)%m
end</lang>
- Output:
->josephus With 41 men, counting to 3 last position is: 30 ->
Extra 'credit' version:
This is done awkwardly, but I've had this laying around since the late 1980's...
<lang unicon>procedure main(args)
n := total := integer(args[1]) | 41 # Number of people
k := count := integer(args[2]) | 3 # Count
s := integer(args[3])-1 | 0 # Number to save
write("With ",n," people, counting by ",k,", the ",s+1," safe places are:")
every write("\t",j(n,k,(n-s) to n))
end
procedure j(n,k,s)
a := k*(n-s) + 1
q := k/(k-1.0)
nk := n*k
olda := a
while a <= nk do {
olda := a
a := ceil(a,q)
}
t := nk - olda
return t
end
procedure ceil(a,q)
n := a*q
if n = integer(n) then return integer(n)
n ?:= integer(tab(upto('.'))) + 1
return n
end</lang>
Sample run:
->josephus2 41 3 4
With 41 people, counting by 3, the 4 safe places are:
3
34
15
30
->
J
Using the executioner's algorithm.
Tacit version
<lang J> 3 ([ (1 }. <:@[ |. ])^:(1 < #@])^:_ i.@]) 41 30</lang> Structured derivation of the fixed tacit code <lang J> DropNext=. 1 }. <:@[ |. ]
MoreThanOne=. 1 < #@] WhileMoreThanOne=. (^:MoreThanOne f.) (^:_) prisoners=. i.@] [ DropNext WhileMoreThanOne prisoners f.
[ (1 }. <:@[ |. ])^:(1 < #@])^:_ i.@]</lang>
Explicit version
<lang J>Josephus =: dyad define NB. explicit form, assume executioner starts at position 0
NB. use: SKIP josephus NUMBER_OF_PRISONERS N =: y K =: N | x EXECUTIONER =: 0 PRISONERS =: i. N kill =: ] #~ (~: ([: i. #)) while. 1 (< #) PRISONERS do. EXECUTIONER =: (# PRISONERS) | <: K + EXECUTIONER PRISONERS =: EXECUTIONER kill PRISONERS end.
)
3 Josephus 41
30</lang>
Explicit version 2
<lang J> Josephus2 =: 4 : '(|x&+)/i.->:y' NB. this is a direct translation of the algo from C code above.
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]
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
Julia
Translated from python <lang julia> function j(n,k)
p, i, seq=[0:n-1], 0, Int[]
while !isempty(p)
i=(i+k-1)%length(p)
push!(seq,splice!(p,i+1))
end
@sprintf("Prisoner killing order: %s.\nSurvivor: %i",replace(chomp(string(seq[1:end-1])),"\n",", "),seq[end])
end</lang>
- Output:
<lang julia>julia> print(j(41,3)) Prisoner killing order: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 0, 4, 9, 13, 18, 22, 27, 31, 36, 40, 6, 12, 19, 25, 33, 39, 7, 16, 28, 37, 10, 24, 1, 21, 3, 34, 15. Survivor: 30</lang> Getting the remaining m survivors <lang julia> function j2(n,k,m)
p, i, seq=[0:n-1], 0, Int[]
while length(p)>m
i=(i+k-1)%length(p)
push!(seq,splice!(p,i+1))
end
prt_array(x)=replace(chomp(string(x)),"\n",", ")
@sprintf("Prisoner killing order: %s.\nSurvivors: %s",prt_array(seq),"["*prt_array(p)*"]")
end </lang>
- Output:
<lang julia> julia> print(j2(41,3,3)) Prisoner killing order: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 0, 4, 9, 13, 18, 22, 27, 31, 36, 40, 6, 12, 19, 25, 33, 39, 7, 16, 28, 37, 10, 24, 1, 21, 3. Survivors: [15, 30, 34] </lang>
jq
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]
Lua
Lua indexes tables starting at 1. Positions are stored from 0,n-1. <lang lua>function josephus(n, k, m)
local positions={}
for i=1,n do
table.insert(positions, i-1)
end
local i,j=1,1
local s='Execution order: '
while #positions>m do
if j==k then
s=s .. positions[i] .. ', '
table.remove(positions, i)
i=i-1
end
i=i+1
j=j+1
if i>#positions then i=1 end
if j>k then j=1 end
end
print(s:sub(1,#s-2) .. '.')
local s='Survivors: '
for _,v in pairs(positions) do s=s .. v .. ', ' end
print(s:sub(1,#s-2) .. '.')
end josephus(41,3, 1) </lang>
- Output:
Execution order: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 0, 4, 9, 13, 18, 22, 27, 31, 36, 40, 6, 12, 19, 25, 33, 39, 7, 16, 28, 37, 10, 24, 1, 21, 3, 34, 15. Survivors: 30.
Mathematica
<lang mathematica>survivor[n_, k_] := Nest[Most[RotateLeft[#, k]] &, Range[0, n - 1], n - 1] survivor[41, 3]</lang>
- Output:
{30}
NetRexx
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>func: josephus(n, k) { | prisoners killed i |
ListBuffer newSize(n) dup addAll(n seq) ->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 << cr System.Out "Survivor : " << 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 is rw, $k) {
until @prisoner == 1 {
@prisoner.=rotate($k - 1); @prisoner.shift;
}
}
my @prisoner = ^41; Execute @prisoner, 3; say "Prisoner {@prisoner} survived.";</lang>
- Output:
Prisoner 30 survived.
We don't have to use numbers. Any list will do: <lang Perl6>my @dalton = <Joe Jack William Averell Rantanplan>; Execute @dalton, 2; say "{@dalton} survived.";</lang>
- Output:
William survived.
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
Python
<lang python>>>> def j(n, k): p, i, seq = list(range(n)), 0, [] while p: i = (i+k-1) % len(p) seq.append(p.pop(i)) return 'Prisoner killing order: %s.\nSurvivor: %i' % (', '.join(str(i) for i in seq[:-1]), seq[-1])
>>> print(j(5, 2)) Prisoner killing order: 1, 3, 0, 4. Survivor: 2 >>> print(j(41, 3)) Prisoner killing order: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 0, 4, 9, 13, 18, 22, 27, 31, 36, 40, 6, 12, 19, 25, 33, 39, 7, 16, 28, 37, 10, 24, 1, 21, 3, 34, 15. Survivor: 30 >>> </lang>
Faster way to solve in python, it does not show the killing order. <lang python>>>>def josephus(n, k):
r = 0
for i in xrange(1, n+1):
r = (r+k)%i
return 'Survivor: %d' %r
>>> print(josephus(5, 2)) Survivor: 2 >>> print(josephus(41, 3)) Survivor: 30 >>> </lang>
Alternate solution with a circular linked list
The function returns the killing order. The last in the list stays alive. Notice that the result is a permutation of [0, 1, ... n - 1]. In the program, a[p] is the index of the next living prisoner after 'p'. The program stops when p = a[p], that is, when there remains only one living prisoner.
<lang python>def josephus(n, k):
a = list(range(1, n + 1))
a[n - 1] = 0
p = 0
v = []
while a[p] != p:
for i in range(k - 2):
p = a[p]
v.append(a[p])
a[p] = a[a[p]]
p = a[p]
v.append(p)
return v
josephus(10, 2) [1, 3, 5, 7, 9, 2, 6, 0, 8, 4]
josephus(41, 3)[-1] 30</lang>
R
<lang R> jose <-function(s, r,n){ y <- 0:(r-1)
for (i in (r+1):n) y <- (y + s) %% i return(y)
} > jose(3,1,41) # r is the number of remained prisoner. [1] 30 </lang>
Racket
<lang Racket>#lang racket (define (josephus n k (m 0))
(for/fold ((m (add1 m))) ((a (in-range (add1 m) (add1 n)))) (remainder (+ m k) a)))
(josephus 41 3) ; ->30</lang>
REBOL
Works in Rebol 2 or 3 <lang REBOL>Rebol []
execute: func [death-list [block!] kill [integer!]] [
assert [not empty? death-list]
until [
loop kill - 1 [append death-list take death-list]
(1 == length? remove death-list)
]
]
prisoner: [] for n 0 40 1 [append prisoner n] execute prisoner 3 print ["Prisoner" prisoner "survived"]</lang>
- Output:
Prisoner 30 survived
And any kind of list will do: <lang REBOL>for-the-chop: [Joe Jack William Averell Rantanplan] execute for-the-chop 2 print [for-the-chop "survived"]</lang>
- Output:
William survived
REXX
version 1
<lang rexx>/* REXX **************************************************************
- 15.11.2012 Walter Pachl - my own solution
- 16.11.2012 Walter Pachl generalized n prisoners + w killing distance
- and s=number of survivors
- 09.05.2013 Walter Pachl accept arguments n w s and fix output
- thanks for the review/test
- I see no need for specifying a start count (actually a start number)
- This program should work on EVERY REXX.
- Pls report if this is not the case and let us know what's a problem.
- /
Parse Arg n w s . If n='?' Then Do
Say 'Invoke the program with the following arguments:' Say 'n number of prisoners (default 41)' Say 'w killing count (default 3)' Say 's number of prisoners to survive (default 1)' Exit End
If n= Then n=41 /* number of alive prisoners */ If w= Then w=3 /* killing count */ If s= Then s=1 /* nuber of survivors */ dead.=0 /* nobody's dead yet */ nn=n /* wrap around boundary */ p=-1 /* start here */ killed= /* output of killings */ Do until n=s /* until one alive prisoner */
found=0 /* start looking */
Do Until found=w /* until we have the third */
p=p+1 /* next position */
If p=nn Then p=0 /* wrap around */
If dead.p=0 Then /* a prisoner who is alive */
found=found+1 /* increment found count */
End
dead.p=1
/*
Say 'killing' p 'now'
*/
n=n-1 /* shoot the one on this pos. */
killed=killed p /* add to output */
End /* End of main loop */
Say 'killed:'killed /* output killing sequence */ s= Do i=0 To nn-1 /* look for the surviving p's */
If dead.i=0 Then s=s i /* found one */ End
Say 'Survivor(s):'s /* show */</lang>
- Output:
killed: 2 5 8 11 14 17 20 23 26 29 32 35 38 0 4 9 13 18 22 27 31 36 40 6 12 19 25 33 39 7 16 28 37 10 24 1 21 3 34 15 Survivor(s): 30
version 2
This version allows the user to specify:
- 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 pgm, Josephus problem: N men standing in a circle, every Kth kilt*/ parse arg N K Z R . /*get optional arguments. */ if N==',' | N== then N = 41 /*no #prisoners? Then use default*/ if K==',' | K== then K = 3 /*no kill count? " " " */ if Z==',' | Z== then Z = 0 /*no initial # ? " " " */ if R==',' | R== then R = 1 /*no remaining#? " " " */ $=; x=; do pop=Z for N; $=$ pop; end /*populate the prisoner's circle.*/ c=0 /*initial prisoner count-off num.*/
do remove=0; p=words($) /*keep removing until R are left.*/
c=c+K /*bump prisoner count-off by K.*/
if c>p then do /* [↓] remove some prisoner(s).*/
do j=1 for words(x); $=delword($,word(x,j)+1-j,1)
if words($)==R then leave remove /*slaying done yet?*/
end /*j*/
c=(c//p)//words($); x= /*adjust prisoner count-off &list*/
end
if c\==0 then x=x c /*list of prisoners to be removed*/
end /*remove*/ /*remove 'til R prisoners left.*/
say 'removing every ' th(K) " prisoner out of " N ' (starting at' Z") with ",
R ' survivor's(R)"," ; say 'leaving prisoner's(R)':' $
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────subroutines──────────────────────────*/ s: if arg(1)==1 then return arg(3); return word(arg(2) 's',1) th: arg y; return y||word('th st nd rd', 1+y//10*(y//100%10\==1)*(y//10<4))</lang>
- Output:
when using the default input
removing every 3rd prisoner out of 41 (starting at 0) with 1 survivor, leaving prisoner: 30
- Output:
when using the input of
removing every 3rd prisoner out of 41 (starting at 1) with 1 survivor, leaving prisoner: 31
- Output:
when using the input of
removing every 3rd prisoner out of 41 (starting at 1) with 2 survivors, leaving prisoners: 16 31
- Output:
when using the input of
removing every 2nd prisoner out of 5 (starting at 0) with 1 survivor, leaving prisoner: 2
Ruby
<lang ruby>def main
n = (ARGV[0] || 41).to_i k = (ARGV[1] || 3).to_i puts josephus(n,k)
end
def josephus(n, k)
prisoners = (0...n).to_a prisoners.rotate!(k-1).shift while prisoners.length > 1 return prisoners.first
end
main</lang>
Scala
Executioner's Solution, not Josephus'
(Prisoners labeled 0 to n-1) <lang scala>def executed( prisonerCount:Int, step:Int ) = {
val prisoners = ((0 until prisonerCount) map (_.toString)).toList
def behead( dead:Seq[String], alive:Seq[String] )(countOff:Int) : (Seq[String], Seq[String]) = {
val group = if( alive.size < countOff ) countOff - alive.size else countOff
(dead ++ alive.take(group).drop(group-1), alive.drop(group) ++ alive.take(group-1)) }
def beheadN( dead:Seq[String], alive:Seq[String] ) : (Seq[String], Seq[String]) = behead(dead,alive)(step)
def execute( t:(Seq[String], Seq[String]) ) : (Seq[String], Seq[String]) = t._2 match {
case x :: Nil => (t._1, Seq(x))
case x :: xs => execute(beheadN(t._1,t._2))
}
execute((List(),prisoners))
}
val (dead,alive) = executed(41,3)
println( "Prisoners executed in order:" ) print( dead.mkString(" ") )
println( "\n\nJosephus is prisoner " + alive(0) )</lang>
- Output:
Prisoners executed in order: 2 5 8 11 14 17 20 23 26 29 32 35 38 0 4 9 13 18 22 27 31 36 40 6 12 19 25 33 39 7 16 28 37 10 24 1 21 3 34 15 Josephus is prisoner 30
Seed7
The main task (find one survivor) is a special case of the extra task (find m survivors). The function executeAllButM solves the extra task and is called with m=1 to solve the main task. The function str converts an array of integer elements to a string. The function enable_output uses str to define everything necessary to write an array of integers. This way the main program can write the survivor array. <lang seed7>$ include "seed7_05.s7i";
const func array integer: executeAllButM (in integer: n, in integer: k, in integer: m) is func
result
var array integer: prisoners is [0 .. -1] times 0;
local
var integer: killIdx is 0;
var integer: prisonerNum is 0;
begin
for prisonerNum range 0 to pred(n) do
prisoners &:= prisonerNum;
end for;
writeln("Prisoners executed in order:");
while length(prisoners) > m do
killIdx := (killIdx + k - 1) rem length(prisoners);
write(prisoners[killIdx] <& " ");
ignore(remove(prisoners, killIdx));
end while;
writeln;
end func;
const func string: str (in array integer: intArr) is func
result
var string: stri is "";
local
var integer: index is 0;
begin
for key index range intArr do
if index <> minIdx(intArr) then
stri &:= ", ";
end if;
stri &:= str(intArr[index]);
end for;
end func;
enable_output(array integer);
const proc: main is func
begin
writeln("Survivor: " <& executeAllButM(41, 3, 1));
writeln("Survivors: " <& executeAllButM(41, 3, 3));
end func;</lang>
- Output:
Prisoners executed in order: 2 5 8 11 14 17 20 23 26 29 32 35 38 0 4 9 13 18 22 27 31 36 40 6 12 19 25 33 39 7 16 28 37 10 24 1 21 3 34 15 Survivor: 30 Prisoners executed in order: 2 5 8 11 14 17 20 23 26 29 32 35 38 0 4 9 13 18 22 27 31 36 40 6 12 19 25 33 39 7 16 28 37 10 24 1 21 3 Survivors: 15, 30, 34
Sidef
Iterative: <lang ruby>func josephus(n, k) {
var prisoners = (0 .. n-1);
while (prisoners.len > 1) {
prisoners.rotate!(k - 1).shift;
};
return prisoners[0];
};</lang>
Recursive: <lang ruby>func josephus(n, k) {
n == 1 ? 0 : ((__FUNC__(n-1, k) + k) % n)
};</lang>
Calling the function: <lang ruby>var survivor = josephus(41, 3); say "Prisoner #{survivor} survived.";</lang>
- Output:
Prisoner 30 survived.
Swift
<lang Swift>class Josephus {
class func lineUp(#numberOfPeople:Int) -> [Int] {
var people = [Int]()
for (var i = 0; i < numberOfPeople; i++) {
people.append(i)
}
return people
}
class func execute(#numberOfPeople:Int, spacing:Int) -> Int {
var killIndex = 0
var people = self.lineUp(numberOfPeople: numberOfPeople)
println("Prisoners executed in order:")
while (people.count > 1) {
killIndex = (killIndex + spacing - 1) % people.count
executeAndRemove(&people, killIndex: killIndex)
}
println()
return people[0]
}
class func executeAndRemove(inout people:[Int], killIndex:Int) {
print("\(people[killIndex]) ")
people.removeAtIndex(killIndex)
}
class func execucteAllButM(#numberOfPeople:Int, spacing:Int, save:Int) -> [Int] {
var killIndex = 0
var people = self.lineUp(numberOfPeople: numberOfPeople)
println("Prisoners executed in order:")
while (people.count > save) {
killIndex = (killIndex + spacing - 1) % people.count
executeAndRemove(&people, killIndex: killIndex)
}
println()
return people
}
}
println("Josephus is number: \(Josephus.execute(numberOfPeople: 41, spacing: 3))") println() println("Survivors: \(Josephus.execucteAllButM(numberOfPeople: 41, spacing: 3, save: 3))")</lang>
- Output:
Prisoners executed in order: 2 5 8 11 14 17 20 23 26 29 32 35 38 0 4 9 13 18 22 27 31 36 40 6 12 19 25 33 39 7 16 28 37 10 24 1 21 3 34 15 Josephus is number: 30 Prisoners executed in order: 2 5 8 11 14 17 20 23 26 29 32 35 38 0 4 9 13 18 22 27 31 36 40 6 12 19 25 33 39 7 16 28 37 10 24 1 21 3 Survivors: [15, 30, 34]
Tcl
<lang tcl>proc josephus {number step {survivors 1}} {
for {set i 0} {$i<$number} {incr i} {lappend l $i}
for {set i 1} {[llength $l]} {incr i} {
# If the element is to be killed, append to the kill sequence if {$i%$step == 0} { lappend killseq [lindex $l 0] set l [lrange $l 1 end] } else { # Roll the list set l [concat [lrange $l 1 end] [list [lindex $l 0]]] }
}
return [lrange $killseq end-[expr {$survivors-1}] end]
}</lang> Demonstrating: <lang tcl>puts "remaining: [josephus 41 3]" puts "remaining 4: [join [josephus 41 3 4] ,]"</lang>
- Output:
remaining: 30 remaining 4: 3,34,15,30
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
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]
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