100 prisoners: Difference between revisions
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There is some inefficiency entailed in repeatedly re-calculating the fixed sequence of drawers defined by index-chasing in the optimal strategy. Parts of the same paths from drawer to drawer are followed by several different prisoners. |
There is some inefficiency entailed in repeatedly re-calculating the fixed sequence of drawers defined by index-chasing in the optimal strategy. Parts of the same paths from drawer to drawer are followed by several different prisoners. |
||
We can avoid |
We can avoid redundant recalculation by first obtaining the full set of drawer-chasing cycles that are defined by the sequence of any given shuffle. |
||
We may also notice that the collective fate of the prisoners turns on whether any of the cyclical paths formed by a given shuffle are longer than 50 items. If a shuffle produces a single over-sized cycle, then not every prisoner will be able to reach their card in 50 moves. |
We may also notice that the collective fate of the prisoners turns on whether any of the cyclical paths formed by a given shuffle are longer than 50 items. If a shuffle produces a single over-sized cycle, then not every prisoner will be able to reach their card in 50 moves. |
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Revision as of 04:16, 7 November 2019
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at 100 prisoners problem. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
- 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-based numbering rather than 1-based numbering throughout. func doTrials(trials, np 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 < np; 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 = %5.2f%%\n\n", strategy, pardoned, rf)
}
func main() {
rand.Seed(time.Now().UnixNano())
const trials = 100_000
for _, np := range []int{10, 100} {
fmt.Printf("Results from %d trials with %d prisoners:\n\n", trials, np)
for _, strategy := range [2]string{"random", "optimal"} {
doTrials(trials, np, strategy)
}
}
}</lang>
- Output:
Results from 100000 trials with 10 prisoners: strategy = random pardoned = 99 relative frequency = 0.10% strategy = optimal pardoned = 31205 relative frequency = 31.20% Results from 100000 trials with 100 prisoners: strategy = random pardoned = 0 relative frequency = 0.00% strategy = optimal pardoned = 31154 relative frequency = 31.15%
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%
Pascal
<lang pascal>program Prisoners100;
const
rounds = 100000;
type
tValue = Uint32; tPrisNum = array of tValue;
var
drawers, PrisonersChoice : tPrisNum;
procedure shuffle(var N:tPrisNum); var
i,j,lmt : nativeInt; tmp: tValue;
Begin
lmt := High(N); For i := lmt downto 1 do begin //take on from index i..limit j := random(i+1); //exchange with i tmp := N[i];N[i]:= N[j];N[j]:= tmp; end;
end;
function PardonedRandom(maxTestNum: NativeInt):boolean; var
PrisNum,TestNum,Lmt : NativeUint; Pardoned : boolean;
Begin
IF maxTestNum <=0 then Begin PardonedRandom := false; EXIT; end; Lmt := High(drawers); IF (maxTestNum >= Lmt) then Begin PardonedRandom := true; EXIT; end;
shuffle(drawers);
PrisNum := 0;
repeat
//every prisoner uses his own list of drawers
shuffle(PrisonersChoice);
TestNum := 0;
repeat
Pardoned := drawers[PrisonersChoice[TestNum]] = PrisNum;
inc(TestNum);
until Pardoned OR (TestNum>=maxTestNum);
IF Not(Pardoned) then
BREAK;
inc(PrisNum);
until PrisNum>=Lmt;
PardonedRandom:= Pardoned;
end;
function PardonedOptimized(maxTestNum: NativeUint):boolean; var
PrisNum,TestNum,NextNum,Cnt,Lmt : NativeUint; Pardoned : boolean;
Begin
IF maxTestNum <=0 then Begin PardonedOptimized := false; EXIT; end; Lmt := High(drawers); IF (maxTestNum >= Lmt) then Begin PardonedOptimized := true; EXIT; end;
shuffle(drawers);
Lmt := High(drawers);
IF maxTestNum >= Lmt then
Begin
PardonedOptimized := true;
EXIT;
end;
PrisNum := 0;
repeat
Cnt := 0;
NextNum := PrisNum;
repeat
TestNum := NextNum;
NextNum := drawers[TestNum];
inc(cnt);
Pardoned := NextNum = PrisNum;
until Pardoned OR (cnt >=maxTestNum);
IF Not(Pardoned) then
BREAK;
inc(PrisNum);
until PrisNum>Lmt;
PardonedOptimized := Pardoned;
end;
procedure CheckRandom(testCount : NativeUint); var
i,cnt : NativeInt;
Begin
cnt := 0;
For i := 1 to rounds do
IF PardonedRandom(TestCount) then
inc(cnt);
writeln('Randomly ',cnt/rounds*100:7:2,'% get pardoned out of ',rounds,' checking max ',TestCount);
end;
procedure CheckOptimized(testCount : NativeUint); var
i,cnt : NativeInt;
Begin
cnt := 0;
For i := 1 to rounds do
IF PardonedOptimized(TestCount) then
inc(cnt);
writeln('Optimized ',cnt/rounds*100:7:2,'% get pardoned out of ',rounds,' checking max ',TestCount);
end;
procedure OneCompareRun(PrisCnt:NativeInt); var
i,lmt :nativeInt;
begin
setlength(drawers,PrisCnt); For i := 0 to PrisCnt-1 do drawers[i] := i; PrisonersChoice := copy(drawers);
//test
writeln('Checking ',PrisCnt,' prisoners');
lmt := PrisCnt; repeat CheckOptimized(lmt); dec(lmt,PrisCnt DIV 10); until lmt < 0; writeln;
lmt := PrisCnt; repeat CheckRandom(lmt); dec(lmt,PrisCnt DIV 10); until lmt < 0; writeln; writeln;
end;
Begin
//init randomize; OneCompareRun(20); OneCompareRun(100);
end.</lang>
- Output:
Checking 20 prisoners Optimized 100.00% get pardoned out of 100000 checking max 20 Optimized 89.82% get pardoned out of 100000 checking max 18 Optimized 78.25% get pardoned out of 100000 checking max 16 Optimized 65.31% get pardoned out of 100000 checking max 14 Optimized 50.59% get pardoned out of 100000 checking max 12 Optimized 33.20% get pardoned out of 100000 checking max 10 Optimized 15.28% get pardoned out of 100000 checking max 8 Optimized 3.53% get pardoned out of 100000 checking max 6 Optimized 0.10% get pardoned out of 100000 checking max 4 Optimized 0.00% get pardoned out of 100000 checking max 2 Optimized 0.00% get pardoned out of 100000 checking max 0 Randomly 100.00% get pardoned out of 100000 checking max 20 Randomly 13.55% get pardoned out of 100000 checking max 18 Randomly 1.38% get pardoned out of 100000 checking max 16 Randomly 0.12% get pardoned out of 100000 checking max 14 Randomly 0.00% get pardoned out of 100000 checking max 12 Randomly 0.00% get pardoned out of 100000 checking max 10 Randomly 0.00% get pardoned out of 100000 checking max 8 Randomly 0.00% get pardoned out of 100000 checking max 6 Randomly 0.00% get pardoned out of 100000 checking max 4 Randomly 0.00% get pardoned out of 100000 checking max 2 Randomly 0.00% get pardoned out of 100000 checking max 0 Checking 100 prisoners Optimized 100.00% get pardoned out of 100000 checking max 100 Optimized 89.48% get pardoned out of 100000 checking max 90 Optimized 77.94% get pardoned out of 100000 checking max 80 Optimized 64.48% get pardoned out of 100000 checking max 70 Optimized 49.35% get pardoned out of 100000 checking max 60 Optimized 31.10% get pardoned out of 100000 checking max 50 Optimized 13.38% get pardoned out of 100000 checking max 40 Optimized 2.50% get pardoned out of 100000 checking max 30 Optimized 0.05% get pardoned out of 100000 checking max 20 Optimized 0.00% get pardoned out of 100000 checking max 10 Optimized 0.00% get pardoned out of 100000 checking max 0 Randomly 100.00% get pardoned out of 100000 checking max 100 Randomly 0.01% get pardoned out of 100000 checking max 90 Randomly 0.00% get pardoned out of 100000 checking max 80 Randomly 0.00% get pardoned out of 100000 checking max 70 Randomly 0.00% get pardoned out of 100000 checking max 60 Randomly 0.00% get pardoned out of 100000 checking max 50 Randomly 0.00% get pardoned out of 100000 checking max 40 Randomly 0.00% get pardoned out of 100000 checking max 30 Randomly 0.00% get pardoned out of 100000 checking max 20 Randomly 0.00% get pardoned out of 100000 checking max 10 Randomly 0.00% get pardoned out of 100000 checking max 0
Perl
<lang perl>use strict; use warnings; use feature 'say'; use List::Util 'shuffle';
sub simulation {
my($population,$trials,$strategy) = @_; my $optimal = $strategy =~ /^o/i ? 1 : 0; my @prisoners = 0..$population-1; my $half = int $population / 2; my $pardoned = 0;
for (1..$trials) {
my @drawers = shuffle @prisoners;
my $total = 0;
for my $prisoner (@prisoners) {
my $found = 0;
if ($optimal) {
my $card = $drawers[$prisoner];
if ($card == $prisoner) {
$found = 1;
} else {
for (1..$half-1) {
$card = $drawers[$card];
($found = 1, last) if $card == $prisoner
}
}
} else {
for my $card ( (shuffle @drawers)[0..$half]) {
($found = 1, last) if $card == $prisoner
}
}
last unless $found;
$total++;
}
$pardoned++ if $total == $population;
}
$pardoned / $trials * 100
}
my $population = 100; my $trials = 10000; say " Simulation count: $trials\n" . (sprintf " Random strategy pardons: %6.3f%% of simulations\n", simulation $population, $trials, 'random' ) . (sprintf "Optimal strategy pardons: %6.3f%% of simulations\n", simulation $population, $trials, 'optimal');
$population = 10; $trials = 100000; say " Simulation count: $trials\n" . (sprintf " Random strategy pardons: %6.3f%% of simulations\n", simulation $population, $trials, 'random' ) . (sprintf "Optimal strategy pardons: %6.3f%% of simulations\n", simulation $population, $trials, 'optimal');</lang>
- Output:
Simulation count: 10000 Random strategy pardons: 0.000% of simulations Optimal strategy pardons: 31.510% of simulations Simulation count: 1000000 Random strategy pardons: 0.099% of simulations Optimal strategy pardons: 35.420% of simulations
Perl 6
Accepts command line parameters to modify the number of prisoners and the number of simulations to run.
Also test with 10 prisoners to verify that the logic is correct for random selection. Random selection should succeed with 10 prisoners at a probability of (1/2)**10, so in 100_000 simulations, should get pardons about .0977 percent of the time.
<lang perl6>unit sub MAIN (:$prisoners = 100, :$simulations = 10000); my @prisoners = ^$prisoners; 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
}
say "Testing $simulations simulations with $prisoners prisoners."; printf " Random play wins: %.3f%% of simulations\n", random $simulations; printf "Optimal play wins: %.3f%% of simulations\n", optimal $simulations;</lang>
- Output:
With defaults
Testing 10000 simulations with 100 prisoners. Random play wins: 0.000% of simulations Optimal play wins: 30.510% of simulations
With passed parameters: --prisoners=10, --simulations=100000
Testing 100000 simulations with 10 prisoners. Random play wins: 0.099% of simulations Optimal play wins: 35.461% of simulations
Python
Procedural
<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
Functional
There is some inefficiency entailed in repeatedly re-calculating the fixed sequence of drawers defined by index-chasing in the optimal strategy. Parts of the same paths from drawer to drawer are followed by several different prisoners.
We can avoid redundant recalculation by first obtaining the full set of drawer-chasing cycles that are defined by the sequence of any given shuffle.
We may also notice that the collective fate of the prisoners turns on whether any of the cyclical paths formed by a given shuffle are longer than 50 items. If a shuffle produces a single over-sized cycle, then not every prisoner will be able to reach their card in 50 moves.
The computation below returns a survival failure as soon as a cycle of more than 50 items is found for any given shuffle:
<lang python>100 Prisoners
from random import randint, sample
- allChainedPathsAreShort :: Int -> IO (0|1)
def allChainedPathsAreShort(n):
1 if all of the index-chasing cycles in a shuffled
sample of [1..n] cards are shorter than half
the sample size (ensuring general survival).
Otherwise 0.
limit = n // 2
xs = range(1, 1 + n)
shuffled = sample(xs, k=n)
# A cycle of boxes, drawn from a shuffled
# sample, which includes the given target.
def cycleIncluding(target):
boxChain = [target]
v = shuffled[target - 1]
while v != target:
boxChain.append(v)
v = shuffled[v - 1]
return boxChain
# Nothing if the target list is empty, or if the cycle which contains the
# first target is larger than half the sample size.
# Otherwise, just a cycle of enchained boxes containing the first target
# in the list, tupled with the residue of any remaining targets which
# fall outside that cycle.
def boxCycle(targets):
if targets:
boxChain = cycleIncluding(targets[0])
return Just((
difference(targets[1:])(boxChain),
boxChain
)) if limit >= len(boxChain) else Nothing()
else:
return Nothing()
# No cycles longer than half of total box count ? return int(n == sum(map(len, unfoldr(boxCycle)(xs))))
- randomTrialResult :: RandomIO (0|1) -> Int -> (0|1)
def randomTrialResult(coin):
1 if every one of the prisoners finds their ticket
in an arbitrary half of the sample. Otherwise 0.
return lambda n: int(all(
coin(x) for x in range(1, 1 + n)
))
- TEST ----------------------------------------------------
- main :: IO ()
def main():
Two sampling techniques constrasted with 100 drawers
and 100 prisoners, over 100,000 trial runs.
halfOfDrawers = randomRInt(0)(1)
def optimalDrawerSampling(x):
return allChainedPathsAreShort(x)
def randomDrawerSampling(x):
return randomTrialResult(halfOfDrawers)(x)
# kSamplesWithNBoxes :: Int -> Int -> String
def kSamplesWithNBoxes(k):
tests = range(1, 1 + k)
return lambda n: '\n\n' + fTable(
str(k) + ' tests of optimal vs random drawer-sampling ' +
'with ' + str(n) + ' boxes: \n'
)(fName)(lambda r: '{:.2%}'.format(r))(
lambda f: sum(f(n) for x in tests) / k
)([
optimalDrawerSampling,
randomDrawerSampling,
])
print(kSamplesWithNBoxes(10000)(10))
print(kSamplesWithNBoxes(10000)(100))
print(kSamplesWithNBoxes(100000)(100))
- ------------------------DISPLAY--------------------------
- fTable :: String -> (a -> String) ->
- (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
def go(xShow, fxShow, f, xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
return s + '\n' + '\n'.join(map(
lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
xs, ys
))
return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
- fname :: (a -> b) -> String
def fName(f):
Name bound to the given function. return f.__name__
- ------------------------GENERIC -------------------------
- Just :: a -> Maybe a
def Just(x):
Constructor for an inhabited Maybe (option type) value.
Wrapper containing the result of a computation.
return {'type': 'Maybe', 'Nothing': False, 'Just': x}
- Nothing :: Maybe a
def Nothing():
Constructor for an empty Maybe (option type) value.
Empty wrapper returned where a computation is not possible.
return {'type': 'Maybe', 'Nothing': True}
- difference :: Eq a => [a] -> [a] -> [a]
def difference(xs):
All elements of xs, except any also found in ys. return lambda ys: list(set(xs) - set(ys))
- randomRInt :: Int -> Int -> IO () -> Int
def randomRInt(m):
The return value of randomRInt is itself
a function. The returned function, whenever
called, yields a a new pseudo-random integer
in the range [m..n].
return lambda n: lambda _: randint(m, n)
- unfoldr(lambda x: Just((x, x - 1)) if 0 != x else Nothing())(10)
- -> [10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
- unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
def unfoldr(f):
Dual to reduce or foldr.
Where catamorphism reduces a list to a summary value,
the anamorphic unfoldr builds a list from a seed value.
As long as f returns Just(a, b), a is prepended to the list,
and the residual b is used as the argument for the next
application of f.
When f returns Nothing, the completed list is returned.
def go(v):
xr = v, v
xs = []
while True:
mb = f(xr[0])
if mb.get('Nothing'):
return xs
else:
xr = mb.get('Just')
xs.append(xr[1])
return xs
return lambda x: go(x)
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
10000 tests of optimal vs random drawer-sampling with 10 boxes: optimalDrawerSampling -> 35.47% randomDrawerSampling -> 0.09% 10000 tests of optimal vs random drawer-sampling with 100 boxes: optimalDrawerSampling -> 30.40% randomDrawerSampling -> 0.00% 100000 tests of optimal vs random drawer-sampling with 100 boxes: optimalDrawerSampling -> 31.17% randomDrawerSampling -> 0.00%
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, , 3)"%"
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: 0.000% 100 prisoners in 100,000 trials, complete pardons using an optimal strategy: 31.184%
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 SLOTS=100, PRISONERS=100, TRIES=50, N=10_000; fcn oneHundredJDI{ // just do it strategy
cupboard,picks := [0..SLOTS-1].walk().shuffle(), cupboard.copy();
// if this prisoner can't find their number in TRIES, all fail
foreach p in (PRISONERS){ if(picks.shuffle().find(p)>=TRIES) return(False); }
True // all found their number
} fcn oneHundredO{ // Optimal strategy
cupboard := [0..SLOTS-1].walk().shuffle();
foreach p in (PRISONERS){
d:=p;
do(TRIES){ if((d=cupboard[d]) == p) continue(2) } // found my number
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%
And a sanity check (from the Perl6 entry): <lang zkl>const SLOTS=100, PRISONERS=10, TRIES=50, N=100_000;</lang>
- Output:
Just do it strategy (100,000 simulatations): 0.09% Optimal strategy (100,000 simulatations): 31.13%