100 prisoners
You are encouraged to solve this task according to the task description, using any language you may know.
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- The Problem
- 100 prisoners are individually numbered 1 to 100
- A room having a cupboard of 100 opaque drawers numbered 1 to 100, that cannot be seen from outside.
- Cards numbered 1 to 100 are placed randomly, one to a drawer, and the drawers all closed; at the start.
- Prisoners start outside the room
- They can decide some strategy before any enter the room.
- Prisoners enter the room one by one, can open a drawer, inspect the card number in the drawer, then close the drawer.
- A prisoner can open no more than 50 drawers.
- A prisoner tries to find his own number.
- A prisoner finding his own number is then held apart from the others.
- If all 100 prisoners find their own numbers then they will all be pardoned.
- The task
- Simulate several thousand instances of the game where the prisoners randomly open draws
- Simulate several thousand instances of the game where the prisoners use the optimal strategy mentioned in the wikipedia article, of:
- First opening the drawer whose outside number is his prisoner number.
- If the card within has his number then he succeeds otherwise he opens the drawer with the same number as that of the revealed card. (until he opens his maximum).
Show and compare the computed probabilities of success for the two strategies, here, on this page.
- References
- The unbelievable solution to the 100 prisoner puzzle standupmaths (Video).
- 100 Prisoners Escape Puzzle DataGenetics.
EasyLang
<lang EasyLang>for i range 100
drawer[] &= i sampler[] &= i
. subr shuffle_drawer
for i = len drawer[] downto 2 r = random i swap drawer[r] drawer[i - 1] .
. subr play_random
call shuffle_drawer
found = 1
prisoner = 0
while prisoner < 100 and found = 1
found = 0
i = 0
while i < 50 and found = 0
r = random (100 - i)
card = drawer[sampler[r]]
swap sampler[r] sampler[100 - i - 1]
if card = prisoner
found = 1
.
i += 1
.
prisoner += 1
.
. subr play_optimal
call shuffle_drawer
found = 1
prisoner = 0
while prisoner < 100 and found = 1
reveal = prisoner
found = 0
i = 0
while i < 50 and found = 0
card = drawer[reveal]
if card = prisoner
found = 1
.
reveal = card
i += 1
.
prisoner += 1
.
. n = 10000 pardoned = 0 for round range n
call play_random pardoned += found
. print "random: " & 100.0 * pardoned / n & "%"
pardoned = 0 for round range n
call play_optimal pardoned += found
. print "optimal: " & 100.0 * pardoned / n & "%"</lang>
- Output:
random: 0.000% optimal: 30.800%
Factor
<lang factor>USING: arrays formatting fry io kernel math random sequences ;
- setup ( -- seq seq ) 100 <iota> dup >array randomize ;
- rand ( -- ? )
setup [ 50 sample member? not ] curry find nip >boolean not ;
- trail ( m seq -- n )
50 pick '[ [ nth ] keep over _ = ] replicate [ t = ] any? 2nip ;
- optimal ( -- ? ) setup [ trail ] curry [ and ] map-reduce ;
- simulate ( m quot -- x )
dupd replicate [ t = ] count swap /f 100 * ; inline
"Simulation count: 10,000" print 10,000 [ rand ] simulate "Random play success: " 10,000 [ optimal ] simulate "Optimal play success: " [ write "%.2f%%\n" printf ] 2bi@</lang>
- Output:
Simulation count: 10,000 Random play success: 0.00% Optimal play success: 31.11%
Go
<lang go>package main
import (
"fmt" "math/rand" "time"
)
// Uses 0-99 numbering rather than 1-100 numbering throughout. func doTrials(trials int, strategy string) {
pardoned := 0
trial:
for t := 0; t < trials; t++ {
var drawers [100]int
for i := 0; i < 100; i++ {
drawers[i] = i
}
rand.Shuffle(100, func(i, j int) {
drawers[i], drawers[j] = drawers[j], drawers[i]
})
prisoner:
for p := 0; p < 100; p++ {
if strategy == "optimal" {
prev := p
for d := 0; d < 50; d++ {
this := drawers[prev]
if this == p {
continue prisoner
}
prev = this
}
} else {
// Assumes a prisoner remembers previous drawers (s)he opened
// and chooses at random from the others.
var opened [100]bool
for d := 0; d < 50; d++ {
var n int
for {
n = rand.Intn(100)
if !opened[n] {
opened[n] = true
break
}
}
if drawers[n] == p {
continue prisoner
}
}
}
continue trial
}
pardoned++
}
rf := float64(pardoned) / float64(trials) * 100
fmt.Printf("strategy = %-7s pardoned = %-6d relative frequency = %4.1f%%\n\n", strategy, pardoned, rf)
}
func main() {
rand.Seed(time.Now().UnixNano())
const trials = 100_000
fmt.Printf("Results from %d trials:\n\n", trials)
for _, strategy := range [2]string{"random", "optimal"} {
doTrials(trials, strategy)
}
}</lang>
- Output:
Results from 100000 trials: strategy = random pardoned = 0 relative frequency = 0.0% strategy = optimal pardoned = 31060 relative frequency = 31.1%
Kotlin
<lang Kotlin>val playOptimal: () -> Boolean = {
val secrets = (0..99).toMutableList()
var ret = true
secrets.shuffle()
prisoner@ for(i in 0 until 100){
var prev = i
draw@ for(j in 0 until 50){
if (secrets[prev] == i) continue@prisoner
prev = secrets[prev]
}
ret = false
break@prisoner
}
ret
}
val playRandom: ()->Boolean = {
var ret = true
val secrets = (0..99).toMutableList()
secrets.shuffle()
prisoner@ for(i in 0 until 100){
val opened = mutableListOf<Int>()
val genNum : () ->Int = {
var r = (0..99).random()
while (opened.contains(r)) {
r = (0..99).random()
}
r
}
for(j in 0 until 50){
val draw = genNum()
if ( secrets[draw] == i) continue@prisoner
opened.add(draw)
}
ret = false
break@prisoner
}
ret
}
fun exec(n:Int, play:()->Boolean):Double{
var succ = 0
for (i in IntRange(0, n-1)){
succ += if(play()) 1 else 0
}
return (succ*100.0)/n
}
fun main() {
val N = 100_000
println("# of executions: $N")
println("Optimal play success rate: ${exec(N, playOptimal)}%")
println("Random play success rate: ${exec(N, playRandom)}%")
} </lang>
- Output:
# of executions: 100000 Optimal play success rate: 31.451% Random play success rate: 0.0%
Perl 6
<lang perl6>my @prisoners = ^100; my $half = floor +@prisoners / 2;
sub random ($n) {
^$n .race.map( {
my @drawers = @prisoners.pick: *;
@prisoners.map( -> $prisoner {
my $found = 0;
for @drawers.pick($half) -> $card {
$found = 1 and last if $card == $prisoner
}
last unless $found;
$found
}
).sum == @prisoners
}
).grep( *.so ).elems / $n * 100
}
sub optimal ($n) {
^$n .race.map( {
my @drawers = @prisoners.pick: *;
@prisoners.map( -> $prisoner {
my $found = 0;
my $card = @drawers[$prisoner];
if $card == $prisoner {
$found = 1
} else {
for ^($half - 1) {
$card = @drawers[$card];
$found = 1 and last if $card == $prisoner
}
}
last unless $found;
$found
}
).sum == @prisoners
}
).grep( *.so ).elems / $n * 100
}
my $n = 10_000; say " Simulation count: $n\n" ~ sprintf(" Random play wins: %4.1f%% of simulations\n", random $n) ~ sprintf("Optimal play wins: %4.1f%% of simulations\n", optimal $n);</lang>
- Output:
Simulation count: 10000 Random play wins: 0.0% of simulations Optimal play wins: 31.9% of simulations
Python
<lang python>import random
def play_random(n):
# using 0-99 instead of ranges 1-100
pardoned = 0
in_drawer = list(range(100))
sampler = list(range(100))
for _round in range(n):
random.shuffle(in_drawer)
found = False
for prisoner in range(100):
found = False
for reveal in random.sample(sampler, 50):
card = in_drawer[reveal]
if card == prisoner:
found = True
break
if not found:
break
if found:
pardoned += 1
return pardoned / n * 100 # %
def play_optimal(n):
# using 0-99 instead of ranges 1-100
pardoned = 0
in_drawer = list(range(100))
for _round in range(n):
random.shuffle(in_drawer)
for prisoner in range(100):
reveal = prisoner
found = False
for go in range(50):
card = in_drawer[reveal]
if card == prisoner:
found = True
break
reveal = card
if not found:
break
if found:
pardoned += 1
return pardoned / n * 100 # %
if __name__ == '__main__':
n = 100_000
print(" Simulation count:", n)
print(f" Random play wins: {play_random(n):4.1f}% of simulations")
print(f"Optimal play wins: {play_optimal(n):4.1f}% of simulations")</lang>
- Output:
Simulation count: 100000 Random play wins: 0.0% of simulations Optimal play wins: 31.1% of simulations
REXX
<lang rexx>/*REXX program to simulate the problem of 100 prisoners: random, and optimal strategy.*/ parse arg men trials seed . /*obtain optional arguments from the CL*/ if men== | men=="," then men= 100 /*number of prisoners for this run.*/ if trials== | trials=="," then trials= 100000 /* " " simulations " " " */ if datatype(seed, 'W') then call random ,,seed /*seed for the random number generator.*/ $.1= ' a simple '; $.2= "an optimal" /*literals used for the SAY instruction*/ try= men % 2 /*number tries for searching for a card*/
do strategy=1 for 2; pardons= 0 /*perform the two types of strategies. */
do trials; call gCards /*do trials for a strategy; gen cards.*/
do p=1 for men until failure /*have each prisoner go through trial. */
if strategy==1 then failure= simple() /*Is 1st strategy? Use simple strategy*/
else failure= pick() /* " 2nd " " optimal " */
end /*p*/ /*FAILURE ≡ 1? Then a prisoner failed.*/
if #==men then pardons= pardons + 1 /*was there a pardon of all prisoners? */
end /*trials*/ /*if 1 prisoner fails, then they all do*/
say commas(men) 'prisoners in' commas(trials) "trials, complete pardons using",
$.strategy "strategy: " left(, pardons==0)format(pardons/trials*100, , 1)"%"
end /*strategy*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do c=length(_)-3 to 1 by -3; _= insert(',', _, c); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ gCards: #=0; cards=; m=men; do j=1 for m; cards= cards j /*define seq. of cards*/
end /*j*/ /*same as seq. of men.*/
do r=1 for men-1; x= random(1,m); @.r= word(cards,x) /*pick a random card. */
cards= delword(cards, x, 1); m= m - 1 /*del a card; new cnt.*/
end /*r*/ /*only one card left. */
@.men= strip(cards); return /*it has extra blank. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ simple: do try; ?= random(1, men); if @.?==p then do; #= #+1; return 0; end
end /*try*/; return 1 /* [↑] has the prisoner found his card?*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ pick: ?=p; do try; if @.?==p then do; #= #+1; return 0; end /*Found his own card? */
?= @.? /*choose next drawer from previous card*/
end /*try*/; return 1 /*only choose 1/2 of the num of drawers*/</lang>
- output when using the default inputs:
100 prisoners in 100,000 trials, complete pardons using a simple strategy: 0.0% 100 prisoners in 100,000 trials, complete pardons using an optimal strategy: 31.2%
Rust
Fairly naive implementation. Could probably be made more idiomatic. Depends on extern rand crate.
Cargo.toml <lang toml>[dependencies] rand = '0.7.2'</lang>
src/main.rs <lang rust>extern crate rand;
use rand::prelude::*;
// Do a full run of checking boxes in a random order for a single prisoner fn check_random_boxes(prisoner: u8, boxes: &[u8]) -> bool {
let checks = {
let mut b: Vec<u8> = (1u8..=100u8).collect();
b.shuffle(&mut rand::thread_rng());
b
};
checks.into_iter().take(50).any(|check| boxes[check as usize - 1] == prisoner)
}
// Do a full run of checking boxes in the optimized order for a single prisoner fn check_ordered_boxes(prisoner: u8, boxes: &[u8]) -> bool {
let mut next_check = prisoner;
(0..50).any(|_| {
next_check = boxes[next_check as usize - 1];
next_check == prisoner
})
}
fn main() {
let mut boxes: Vec<u8> = (1u8..=100u8).collect();
let trials = 100000;
let ordered_successes = (0..trials).filter(|_| {
boxes.shuffle(&mut rand::thread_rng());
(1u8..=100u8).all(|prisoner| check_ordered_boxes(prisoner, &boxes))
}).count();
let random_successes = (0..trials).filter(|_| {
boxes.shuffle(&mut rand::thread_rng());
(1u8..=100u8).all(|prisoner| check_random_boxes(prisoner, &boxes))
}).count();
println!("{} / {} ({:.02}%) successes in ordered", ordered_successes, trials, ordered_successes as f64 * 100.0 / trials as f64);
println!("{} / {} ({:.02}%) successes in random", random_successes, trials, random_successes as f64 * 100.0 / trials as f64);
}</lang>
- Output:
31106 / 100000 (31.11%) successes in ordered 0 / 100000 (0.00%) successes in random
zkl
<lang zkl>const P=100, TRIES=50, N=5_000; fcn oneHundredJDI{ // just do it strategy
cupboard,picks := [0..P-1].walk().shuffle(), cupboard.copy();
// if this prisoner can't find their number in TRIES, all fail
foreach p in (P){ if(picks.shuffle().find(p)>=TRIES) return(False); }
True
} fcn oneHundredO{ // Optimal strategy
cupboard := [0..P-1].walk().shuffle();
foreach p in (P){
n:=p;
do(TRIES){ if((n=cupboard[n]) == p) continue(2) }
return(False); // this prisoner failed to find their number, all fail
}
True // all found their number
}</lang> <lang zkl>s:=N.pump(Ref(0).incN,oneHundredJDI).value.toFloat()/N*100; println("Just do it strategy (%,d simulatations): %.2f%%".fmt(N,s));
s:=N.pump(Ref(0).incN,oneHundredO).value.toFloat()/N*100; println("Optimal strategy (%,d simulatations): %.2f%%".fmt(N,s));</lang>
- Output:
Just do it strategy (10,000 simulatations): 0.00% Optimal strategy (10,000 simulatations): 31.16%