100 prisoners: Difference between revisions
(Added Wren) |
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Optimal play success rate: 31.12990000000% |
Optimal play success rate: 31.12990000000% |
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Random play success rate: 0.00000000000%</pre> |
Random play success rate: 0.00000000000%</pre> |
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=={{header|Wren}}== |
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{{trans|Go}} |
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{{libheader|Wren-fmt}} |
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<lang ecmascript>import "random" for Random |
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import "/fmt" for Fmt |
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var rand = Random.new() |
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var doTrials = Fn.new{ |trials, np, strategy| |
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var pardoned = 0 |
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for (t in 0...trials) { |
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var drawers = List.filled(100, 0) |
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for (i in 0..99) drawers[i] = i |
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rand.shuffle(drawers) |
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var nextTrial = false |
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for (p in 0...np) { |
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var nextPrisoner = false |
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if (strategy == "optimal") { |
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var prev = p |
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for (d in 0..49) { |
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var curr = drawers[prev] |
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if (curr == p) { |
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nextPrisoner = true |
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break |
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} |
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prev = curr |
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} |
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} else { |
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var opened = List.filled(100, false) |
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for (d in 0..49) { |
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var n |
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while (true) { |
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n = rand.int(100) |
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if (!opened[n]) { |
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opened[n] = true |
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break |
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} |
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} |
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if (drawers[n] == p) { |
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nextPrisoner = true |
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break |
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} |
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} |
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} |
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if (!nextPrisoner) { |
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nextTrial = true |
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break |
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} |
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} |
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if (!nextTrial) pardoned = pardoned + 1 |
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} |
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var rf = pardoned/trials * 100 |
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Fmt.print(" strategy = $-7s pardoned = $,6d relative frequency = $5.2f\%\n", strategy, pardoned, rf) |
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} |
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var trials = 1e5 |
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for (np in [10, 100]) { |
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Fmt.print("Results from $,d trials with $d prisoners:\n", trials, np) |
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for (strategy in ["random", "optimal"]) doTrials.call(trials, np, strategy) |
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}</lang> |
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{{out}} |
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Sample run: |
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<pre> |
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Results from 100,000 trials with 10 prisoners: |
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strategy = random pardoned = 98 relative frequency = 0.10% |
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strategy = optimal pardoned = 31,212 relative frequency = 31.21% |
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Results from 100,000 trials with 100 prisoners: |
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strategy = random pardoned = 0 relative frequency = 0.00% |
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strategy = optimal pardoned = 31,139 relative frequency = 31.14% |
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</pre> |
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=={{header|zkl}}== |
=={{header|zkl}}== |
Revision as of 10:23, 28 July 2020
You are encouraged to solve this task according to the task description, using any language you may know.
- 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. If any don't then all sentences stand.
- The task
- Simulate several thousand instances of the game where the prisoners randomly open drawers
- 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).
- wp:100 prisoners problem
- 100 Prisoners Escape Puzzle DataGenetics.
- Random permutation statistics#One hundred prisoners on Wikipedia.
Ada
<lang Ada> package Prisoners is
type Win_Percentage is digits 2 range 0.0 .. 100.0; type Drawers is array (1 .. 100) of Positive;
function Play_Game (Repetitions : in Positive; Strategy : not null access function (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean) return Win_Percentage; -- Play the game with a specified number of repetitions, the chosen strategy -- is passed to this function
function Optimal_Strategy (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean;
function Random_Strategy (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean;
end Prisoners; </lang> <lang Ada> pragma Ada_2012; with Ada.Numerics.Discrete_Random; with Ada.Text_IO; use Ada.Text_IO;
package body Prisoners is
subtype Drawer_Range is Positive range 1 .. 100; package Random_Drawer is new Ada.Numerics.Discrete_Random (Drawer_Range); use Random_Drawer; -- Helper procedures to initialise and shuffle the drawers procedure Swap (A, B : Positive; Cupboard : in out Drawers) is Temp : Positive; begin Temp := Cupboard (B); Cupboard (B) := Cupboard (A); Cupboard (A) := Temp; end Swap;
procedure Shuffle (Cupboard : in out Drawers) is G : Generator; begin Reset (G); for I in Cupboard'Range loop Swap (I, Random (G), Cupboard); end loop; end Shuffle;
procedure Initialise_Drawers (Cupboard : in out Drawers) is begin for I in Cupboard'Range loop Cupboard (I) := I; end loop; Shuffle (Cupboard); end Initialise_Drawers; -- The two strategies for playing the game
function Optimal_Strategy (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean is Current_Card : Positive; begin Current_Card := Cupboard (Prisoner_Number); if Current_Card = Prisoner_Number then return True; else for I in Integer range 1 .. Max_Attempts loop Current_Card := Cupboard (Current_Card); if Current_Card = Prisoner_Number then return True; end if; end loop; end if; return False; end Optimal_Strategy;
function Random_Strategy (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean is Current_Card : Positive; G : Generator; begin Reset (G); Current_Card := Cupboard (Prisoner_Number); if Current_Card = Prisoner_Number then return True; else for I in Integer range 1 .. Max_Attempts loop Current_Card := Cupboard (Random (G)); if Current_Card = Prisoner_Number then return True; end if; end loop; end if; return False; end Random_Strategy;
function Prisoners_Attempts (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Strategy : not null access function (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean) return Boolean is begin for Prisoner_Number in Integer range 1 .. Max_Prisoners loop if not Strategy (Cupboard, Max_Prisoners, Max_Attempts, Prisoner_Number) then return False; end if; end loop; return True; end Prisoners_Attempts; -- The function to play the game itself
function Play_Game (Repetitions : in Positive; Strategy : not null access function (Cupboard : in Drawers; Max_Prisoners : Integer; Max_Attempts : Integer; Prisoner_Number : Integer) return Boolean) return Win_Percentage is Cupboard : Drawers; Win, Game_Count : Natural := 0; Number_Of_Prisoners : constant Integer := 100; Max_Attempts : constant Integer := 50; begin loop Initialise_Drawers (Cupboard); if Prisoners_Attempts (Cupboard => Cupboard, Max_Prisoners => Number_Of_Prisoners, Max_Attempts => Max_Attempts, Strategy => Strategy) then Win := Win + 1; end if; Game_Count := Game_Count + 1; exit when Game_Count = Repetitions; end loop; return Win_Percentage ((Float (Win) / Float (Repetitions)) * 100.0); end Play_Game;
end Prisoners; </lang> <lang Ada> with Prisoners; use Prisoners; with Ada.Text_IO; use Ada.Text_IO;
procedure Main is
Wins : Win_Percentage; package Win_Percentage_IO is new Float_IO (Win_Percentage);
begin
Wins := Play_Game (100_000, Optimal_Strategy'Access); Put ("Optimal Strategy = "); Win_Percentage_IO.Put (Wins, 2, 2, 0); Put ("%"); New_Line; Wins := Play_Game (100_000, Random_Strategy'Access); Put ("Random Strategy = "); Win_Percentage_IO.Put (Wins, 2, 2, 0); Put ("%");
end Main; </lang>
- Output:
Optimal Strategy = 31.80% Random Strategy = 0.00%
Applesoft BASIC
This is modified from the 100_prisoners#Commodore_BASIC listing. Here are some noted differences between the BASICs and platforms:
- UPPER CASE, for the 1970's Apple II and Apple II+
- GET in Applesoft waits for a keypress, so : IF K$ = "" THEN 1110 is not needed
- CLear Screen: PRINT CHR$ (147); on Commodore BASIC, HOME in Applesoft
- "{LEFT-CRSR}" is CHR$(8) on Apple II, but numbers printed in Applesoft don't have spaces appended to them
- but spaces need to be added in front and after numbers in Applesoft
- ; is optional for string concatenation
- Replace bare PRINT statement with M$ embedded in PRINT statements to visually compact the listing
And, minor speed tweaks:
- Remove REMs, adjust line numbers, move the two compacted methods to the beginning of the program
- Rename some two character variable names to single character names: 's/DR(/D(/' 's/IG(/J(/'
- Start at 0 and go up to 99, but don't regress into off by one bugs
- Inline the shuffle subroutine and hoist it out of the methods
- Embed the results in the loop because feedback can be helpful, otherwise it looks like the program froze
Actual test of 4000 trials for each method were run on the KEGSMAC emulator with MHz set to No Limit.
<lang gwbasic>0 GOTO 9
1 FOR X = 0 TO N:J(X) = X: NEXT: FOR I = 0 TO N:FOR X = 0 TO N:T = J(X):NP = INT ( RND (1) * H):J(X) = J(NP):J(NP) = T: NEXT :FOR G = 1 TO W:IF D(J(G)) = I THEN IP = IP + 1: NEXT I: RETURN 2 NEXT G:RETURN
3 FOR I = 0 TO N:NG = I: FOR G = 0 TO W:CD = D(NG):IF CD = I THEN IP = IP + 1: NEXT I: RETURN 4 NG = CD:IF CD = I THEN STOP 5 NEXT G: RETURN
9 H=100:N=H-1:DIM D(99),J(99):FOR I = 0 TO N:D(I) = I: NEXT:W=INT(H/2)-1:M$=CHR$(13):M$(1)="RANDOM GUESSING":M$(2)="CHAINED NUMBER PICKING"
1000 FOR Q = 0 TO 1 STEP 0 : HOME : PRINT "100 PRISONERS"M$: INPUT "HOW MANY TRIALS FOR EACH METHOD? "; TT 1010 VTAB 2:CALL-958:PRINT M$"RESULTS:"M$ 1020 FOR M = 1 TO 2: SU(M) = 0:FA(M) = 0 1030 FOR TN = 1 TO TT 1040 VTAB 4:PRINT M$ " OUT OF " TT " TRIALS, THE RESULTS ARE"M$" AS FOLLOWS..."; 1050 IP = 0: X = RND ( - TI): FOR I = 0 TO N:R = INT ( RND (1) * N):T = D(I):D(I) = D(R):D(R) = T: NEXT 1060 ON M GOSUB 1,3 : SU(M) = SU(M) + (IP = H):FA(M) = FA(M) + (IP < H) 1070 FOR Z = 1 TO 2 1071 PRINT M$M$Z". "M$(Z)":"M$ 1073 PRINT " "SU(Z)" SUCCESSES"TAB(21) 1074 PRINT " "FA(Z)" FAILURES"M$ 1075 PRINT " "(SU(Z) / TT) * 100"% SUCCESS RATE.";:CALL-868 1090 NEXT Z,TN,M
1100 PRINT M$M$"AGAIN?" 1110 GET K$ 1120 Q = K$ <> "Y" AND K$ <> CHR$(ASC("Y") + 32) : NEXT Q </lang>
- Output:
100 PRISONERS RESULTS: OUT OF 4000 TRIALS, THE RESULTS ARE AS FOLLOWS... 1. RANDOM GUESSING: 0 SUCCESSES 4000 FAILURES 0% SUCCESS RATE. 2. CHAINED NUMBER PICKING: 1278 SUCCESSES 2722 FAILURES 31.95% SUCCESS RATE.
C
<lang C>
- include<stdbool.h>
- include<stdlib.h>
- include<stdio.h>
- include<time.h>
- define LIBERTY false
- define DEATH true
typedef struct{ int cardNum; bool hasBeenOpened; }drawer;
drawer *drawerSet;
void initialize(int prisoners){ int i,j,card; bool unique;
drawerSet = ((drawer*)malloc(prisoners * sizeof(drawer))) -1;
card = rand()%prisoners + 1; drawerSet[1] = (drawer){.cardNum = card, .hasBeenOpened = false};
for(i=1 + 1;i<prisoners + 1;i++){ unique = false; while(unique==false){ for(j=0;j<i;j++){ if(drawerSet[j].cardNum == card){ card = rand()%prisoners + 1; break; } } if(j==i){ unique = true; } } drawerSet[i] = (drawer){.cardNum = card, .hasBeenOpened = false}; }
}
void closeAllDrawers(int prisoners){ int i; for(i=1;i<prisoners + 1;i++) drawerSet[i].hasBeenOpened = false; }
bool libertyOrDeathAtRandom(int prisoners,int chances){ int i,j,chosenDrawer;
for(i= 1;i<prisoners + 1;i++){ bool foundCard = false; for(j=0;j<chances;j++){ do{ chosenDrawer = rand()%prisoners + 1; }while(drawerSet[chosenDrawer].hasBeenOpened==true); if(drawerSet[chosenDrawer].cardNum == i){ foundCard = true; break; } drawerSet[chosenDrawer].hasBeenOpened = true; } closeAllDrawers(prisoners); if(foundCard == false) return DEATH; }
return LIBERTY; }
bool libertyOrDeathPlanned(int prisoners,int chances){ int i,j,chosenDrawer; for(i=1;i<prisoners + 1;i++){ chosenDrawer = i; bool foundCard = false; for(j=0;j<chances;j++){ drawerSet[chosenDrawer].hasBeenOpened = true;
if(drawerSet[chosenDrawer].cardNum == i){ foundCard = true; break; } if(chosenDrawer == drawerSet[chosenDrawer].cardNum){ do{
chosenDrawer = rand()%prisoners + 1;
}while(drawerSet[chosenDrawer].hasBeenOpened==true); } else{ chosenDrawer = drawerSet[chosenDrawer].cardNum; }
}
closeAllDrawers(prisoners); if(foundCard == false) return DEATH; }
return LIBERTY; }
int main(int argc,char** argv) { int prisoners, chances; unsigned long long int trials,i,count = 0;
char* end;
if(argc!=4) return printf("Usage : %s <Number of prisoners> <Number of chances> <Number of trials>",argv[0]);
prisoners = atoi(argv[1]); chances = atoi(argv[2]); trials = strtoull(argv[3],&end,10);
srand(time(NULL));
printf("Running random trials..."); for(i=0;i<trials;i+=1L){ initialize(prisoners);
count += libertyOrDeathAtRandom(prisoners,chances)==DEATH?0:1; }
printf("\n\nGames Played : %llu\nGames Won : %llu\nChances : %lf %% \n\n",trials,count,(100.0*count)/trials);
count = 0;
printf("Running strategic trials..."); for(i=0;i<trials;i+=1L){ initialize(prisoners);
count += libertyOrDeathPlanned(prisoners,chances)==DEATH?0:1; }
printf("\n\nGames Played : %llu\nGames Won : %llu\nChances : %lf %% \n\n",trials,count,(100.0*count)/trials); return 0; }
</lang>
$ gcc 100prisoners.c && ./a.out 100 50 10000 Running random trials... Games Played : 10000 Games Won : 0 Chances : 0.000000 % Running strategic trials... Games Played : 10000 Games Won : 3051 Chances : 30.510000 %
C#
<lang csharp>using System; using System.Linq;
namespace Prisoners {
class Program { static bool PlayOptimal() { var secrets = Enumerable.Range(0, 100).OrderBy(a => Guid.NewGuid()).ToList();
for (int p = 0; p < 100; p++) { bool success = false;
var choice = p; for (int i = 0; i < 50; i++) { if (secrets[choice] == p) { success = true; break; } choice = secrets[choice]; }
if (!success) { return false; } }
return true; }
static bool PlayRandom() { var secrets = Enumerable.Range(0, 100).OrderBy(a => Guid.NewGuid()).ToList();
for (int p = 0; p < 100; p++) { var choices = Enumerable.Range(0, 100).OrderBy(a => Guid.NewGuid()).ToList();
bool success = false; for (int i = 0; i < 50; i++) { if (choices[i] == p) { success = true; break; } }
if (!success) { return false; } }
return true; }
static double Exec(uint n, Func<bool> play) { uint success = 0; for (uint i = 0; i < n; i++) { if (play()) { success++; } } return 100.0 * success / n; }
static void Main() { const uint N = 1_000_000; Console.WriteLine("# of executions: {0}", N); Console.WriteLine("Optimal play success rate: {0:0.00000000000}%", Exec(N, PlayOptimal)); Console.WriteLine(" Random play success rate: {0:0.00000000000}%", Exec(N, PlayRandom)); } }
}</lang>
- Output:
# of executions: 1000000 Optimal play success rate: 31.21310000000% Random play success rate: 0.00000000000%
C++
<lang cpp>#include <iostream> //for output
- include <algorithm> //for shuffle
- include <stdlib.h> //for rand()
using namespace std;
int* setDrawers() { int drawers[100]; for (int i = 0; i < 100; i++) { drawers[i] = i; } random_shuffle(&drawers[0], &drawers[99]); return drawers; }
bool playRandom() { int* drawers = setDrawers(); bool openedDrawers[100] = { 0 }; for (int prisonerNum = 0; prisonerNum < 100; prisonerNum++) { //loops through prisoners numbered 0 through 99 bool prisonerSuccess = false; for (int i = 0; i < 50; i++) { //loops through 50 draws for each prisoner int drawerNum; while (true) { drawerNum = rand() % 100; if (!openedDrawers[drawerNum]) { openedDrawers[drawerNum] = true; cout << endl; break; } } if (*(drawers + drawerNum) == prisonerNum) { prisonerSuccess = true; break; } } if (!prisonerSuccess) return false; } return true; }
bool playOptimal() { int* drawers = setDrawers(); for (int prisonerNum = 0; prisonerNum < 100; prisonerNum++) { bool prisonerSuccess = false; int checkDrawerNum = prisonerNum; for (int i = 0; i < 50; i++) { if (*(drawers + checkDrawerNum) == prisonerNum) { prisonerSuccess = true; break; } else checkDrawerNum = *(drawers + checkDrawerNum); } if (!prisonerSuccess) return false; } return true; }
double simulate(string strategy) { int numberOfSuccesses = 0; for (int i = 0; i <= 10000; i++) { if ((strategy == "random" && playRandom()) || (strategy == "optimal" && playOptimal())) //will run playRandom or playOptimal but not both becuase of short-circuit evaluation numberOfSuccesses++; } return numberOfSuccesses / 100.0; }
int main() { cout << "Random Strategy: " << simulate("random") << "%" << endl; cout << "Optimal Strategy: " << simulate("optimal") << "%" << endl; system("PAUSE"); return 0; }</lang>
- Output:
Random Strategy: 0% Optimal Strategy: 31.51%
Clojure
<lang Clojure>(ns clojure-sandbox.prisoners)
(defn random-drawers []
"Returns a list of shuffled numbers" (-> 100 range shuffle))
(defn search-50-random-drawers [prisoner-number drawers]
"Select 50 random drawers and return true if the prisoner's number was found" (->> drawers shuffle ;; Put drawer contents in random order (take 50) ;; Select first 50, equivalent to selecting 50 random drawers (filter (fn [x] (= x prisoner-number))) ;; Filter to include only those that match prisoner number count (= 1))) ;; Returns true if the number of matching numbers is 1
(defn search-50-optimal-drawers [prisoner-number drawers]
"Open 50 drawers according to the agreed strategy, returning true if prisoner's number was found" (loop [next-drawer prisoner-number ;; The drawer index to start on is the prisoner's number drawers-opened 0] ;; To keep track of how many have been opened as 50 is the maximum (if (= drawers-opened 50) false ;; If 50 drawers have been opened, the prisoner's number has not been found (let [result (nth drawers next-drawer)] ;; Open the drawer given by next number (if (= result prisoner-number) ;; If prisoner number has been found true ;; No need to keep opening drawers - return true (recur result (inc drawers-opened))))))) ;; Restart the loop using the resulting number as the drawer number
(defn try-luck [drawers drawer-searching-function]
"Returns 1 if all prisoners find their number otherwise 0" (loop [prisoners (range 100)] ;; Start with 100 prisoners (if (empty? prisoners) ;; If they've all gone and found their number 1 ;; Return true- they'll all live (let [res (-> prisoners first (drawer-searching-function drawers))] ;; Otherwise, have the first prisoner open drawers according to the specified method (if (false? res) ;; If this prisoner didn't find their number 0 ;; no prisoners will be freed so we can return false and stop (recur (rest prisoners))))))) ;; Otherwise they've found the number, so we remove them from the queue and repeat with the others
(defn simulate-100-prisoners []
"Simulates all prisoners searching the same drawers by both strategies, returns map showing whether each was successful" (let [drawers (random-drawers)] ;; Create 100 drawers with randomly ordered prisoner numbers {:random (try-luck drawers search-50-random-drawers) ;; True if all prisoners found their number using random strategy :optimal (try-luck drawers search-50-optimal-drawers)})) ;; True if all prisoners found their number using optimal strategy
(defn simulate-n-runs [n]
"Simulate n runs of the 100 prisoner problem and returns a success count for each search method" (loop [random-successes 0 optimal-successes 0 run-count 0] (if (= n run-count) ;; If we've done the loop n times {:random-successes random-successes ;; return results :optimal-successes optimal-successes :run-count run-count} (let [next-result (simulate-100-prisoners)] ;; Otherwise, run for another batch of prisoners (recur (+ random-successes (:random next-result)) ;; Add result of run to the total successs count (+ optimal-successes (:optimal next-result)) (inc run-count)))))) ;; increment run count and run again
(defn -main [& args]
"For 5000 runs, print out the success frequency for both search methods" (let [{:keys [random-successes optimal-successes run-count]} (simulate-n-runs 5000)] (println (str "Probability of survival with random search: " (float (/ random-successes run-count)))) (println (str "Probability of survival with ordered search: " (float (/ optimal-successes run-count))))))</lang>
- Output:
Probability of survival with random search: 0.0 Probability of survival with ordered search: 0.3062
Commodore BASIC
It should be noted that this is a very time consuming process for a ~1 MHz 8-bit computer. Evaluating 1000 trials of each method with the algorithm below takes about 3.5 hours on the BASIC system clock (TIME$) of a stock NTSC Commodore 64, even with screen blanking. (Screen blanking seems to achieve only a 3% improvement in speed.) Actual test of 4000 trials for each method were run on the VICE emulator with warp speed engaged, otherwise the user would have had to wait a day and a half for results.
Another concern is when the prisoner's number is found. When this happens it becomes unnecessary to use whatever guesses are remaining; we should simply move on to the next prisoner. Furthermore, if any prisoner uses all 50 guesses with no luck, then everyone is out of luck and the trial is over, which means no other prisoner needs to make the attempt.
This potentially could cause problems on the stack with unfinished guessing (or prisoner) loops, especially where stack limits are extremely small however, a few things are happening to prevent this (See C64-Wiki "NEXT: Early Exits..." for reference.):
- The prisoner loop, and each prisoner's 50-guesses loop, are contained within a subroutine. The RETURN at the end of either subroutine terminates any unfinished loops and keeps the stack clean.
- When the NEXT belonging to loop 'i' is encountered, any inner loops ('g') are terminated.
- Similar to above, any new loop using an existing loop's variable terminates the old loop, and any nested loops within it.
The key here is avoiding the use of GOTO as a means of exiting a loop early.
<lang gwbasic> 10 rem 100 prisoners 20 rem set arrays 30 rem dr = drawers containing card values 40 rem ig = a list of numbers 1 through 100, shuffled to become the 41 rem guess sequence for each inmate - method 1 50 dim dr(100),ig(100) 55 rem initialize drawers with own card in each drawer 60 for i=1 to 100:dr(i)=i:next
1000 print chr$(147);"how many trials for each method";:input tt 1010 for m=1 to 2:su(m)=0:fa(m)=0 1015 for tn=1 to tt 1020 on m gosub 2000,3000 1025 rem ip = number of inmates who passed 1030 if ip=100 then su(m)=su(m)+1 1040 if ip<100 then fa(m)=fa(m)+1 1045 next tn 1055 next m
1060 print chr$(147);"Results:":print 1070 print "Out of";tt;"trials, the results are" 1071 print "as follows...":print 1072 print "1. Random Guessing:" 1073 print " ";su(1);"successes" 1074 print " ";fa(1);"failures" 1075 print " ";su(1)/tn;"{left-crsr}% success rate.":print 1077 print "2. Chained Number Picking:" 1078 print " ";su(2);"successes" 1079 print " ";fa(2);"failures" 1080 print " ";(su(2)/tn)*100;"{left-crsr}% success rate.":print 1100 print:print "Again?" 1110 get k$:if k$="" then 1110 1120 if k$="y" then 1000 1500 end
2000 rem random guessing method 2005 for x=1 to 100:ig(x)=x:next:ip=0:gosub 4000 2007 for i=1 to 100 2010 for x=1 to 100:t=ig(x):np=int(rnd(1)*100)+1:ig(x)=ig(np):ig(np)=t:next 2015 for g=1 to 50 2020 if dr(ig(g))=i then ip=ip+1:next i:return 2025 next g 2030 return
3000 rem chained method 3005 ip=0:gosub 4000 3007 rem iterate through each inmate 3010 fori=1to100 3015 ng=i:forg=1to50 3020 cd=dr(ng) 3025 ifcd=ithenip=ip+1:nexti:return 3030 ifcd<>ithenng=cd 3035 nextg:return
4000 rem shuffle the drawer cards randomly 4010 x=rnd(-ti) 4020 for i=1 to 100 4030 r=int(rnd(1)*100)+1:t=dr(i):dr(i)=dr(r):dr(r)=t:next 4040 return </lang>
- Output:
Results: Out of 4000 trials the percentage of success is as follows... 1. Random Guessing: 0 successes 4000 failures 0% success rate. 2. Chained Number Picking: 1274 successes 2726 failures 31.85% success rate.
Common Lisp
<lang lisp> (defparameter *samples* 10000) (defparameter *prisoners* 100) (defparameter *max-guesses* 50)
(defun range (n)
"Returns a list from 0 to N." (loop for i below n collect i))
(defun nshuffle (list)
"Returns a shuffled LIST." (loop for i from (length list) downto 2 do (rotatef (nth (random i) list) (nth (1- i) list))) list)
(defun build-drawers ()
"Returns a list of shuffled drawers." (nshuffle (range *prisoners*)))
(defun strategy-1 (drawers p)
"Returns T if P is found in DRAWERS under *MAX-GUESSES* using a random strategy." (loop for i below *max-guesses* thereis (= p (nth (random *prisoners*) drawers))))
(defun strategy-2 (drawers p)
"Returns T if P is found in DRAWERS under *MAX-GUESSES* using an optimal strategy." (loop for i below *max-guesses* for j = p then (nth j drawers) thereis (= p (nth j drawers))))
(defun 100-prisoners-problem (strategy &aux (drawers (build-drawers)))
"Returns T if all prisoners find their number using the given STRATEGY." (every (lambda (e) (eql T e)) (mapcar (lambda (p) (funcall strategy drawers p)) (range *prisoners*))))
(defun sampling (strategy)
(loop repeat *samples* for result = (100-prisoners-problem strategy) count result))
(defun compare-strategies ()
(format t "Using a random strategy in ~4,2F % of the cases the prisoners are free.~%" (* (/ (sampling #'strategy-1) *samples*) 100)) (format t "Using an optimal strategy in ~4,2F % of the cases the prisoners are free.~%" (* (/ (sampling #'strategy-2) *samples*) 100)))
</lang>
- Output:
CL-USER> (compare-strategies) Using a random strategy in 0.00 % of the cases the prisoners are free. Using an optimal strategy in 31.34 % of the cases the prisoners are free.
Crystal
Based on the Ruby implementation
<lang crystal>prisoners = (1..100).to_a N = 100_000 generate_rooms = ->{ (1..100).to_a.shuffle }
res = N.times.count do
rooms = generate_rooms.call prisoners.all? { |pr| rooms[1, 100].sample(50).includes?(pr) }
end puts "Random strategy : %11.4f %%" % (res.fdiv(N) * 100)
res = N.times.count do
rooms = generate_rooms.call prisoners.all? do |pr| cur_room = pr 50.times.any? do cur_room = rooms[cur_room - 1] found = (cur_room == pr) found end end
end puts "Optimal strategy: %11.4f %%" % (res.fdiv(N) * 100)</lang>
- Output:
Random strategy : 0.0000 % Optimal strategy: 31.3190 %
D
<lang d>import std.array; import std.random; import std.range; import std.stdio; import std.traits;
bool playOptimal() {
auto secrets = iota(100).array.randomShuffle();
prisoner: foreach (p; 0..100) { auto choice = p; foreach (_; 0..50) { if (secrets[choice] == p) continue prisoner; choice = secrets[choice]; } return false; }
return true;
}
bool playRandom() {
auto secrets = iota(100).array.randomShuffle();
prisoner: foreach (p; 0..100) { auto choices = iota(100).array.randomShuffle(); foreach (i; 0..50) { if (choices[i] == p) continue prisoner; } return false; }
return true;
}
double exec(const size_t n, bool function() play) {
size_t success = 0; for (int i = n; i > 0; i--) { if (play()) { success++; } } return 100.0 * success / n;
}
void main() {
enum N = 1_000_000; writeln("# of executions: ", N); writefln("Optimal play success rate: %11.8f%%", exec(N, &playOptimal)); writefln(" Random play success rate: %11.8f%%", exec(N, &playRandom));
}</lang>
- Output:
# of executions: 1000000 Optimal play success rate: 31.16100000% Random play success rate: 0.00000000%
EasyLang
<lang EasyLang>intvars 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%
FreeBASIC
<lang freebasic>#include once "knuthshuf.bas" 'use the routines in https://rosettacode.org/wiki/Knuth_shuffle#FreeBASIC
function gus( i as long, strat as boolean ) as long
if strat then return i return 1+int(rnd*100)
end function
sub trials( byref c_success as long, byref c_fail as long, byval strat as boolean )
dim as long i, j, k, guess, drawer(1 to 100) for i = 1 to 100 drawer(i) = i next i for j = 1 to 1000000 'one million trials of prisoners knuth_up( drawer() ) 'shuffles the cards in the drawers for i = 1 to 100 'prisoner number guess = gus(i, strat) for k = 1 to 50 'each prisoner gets 50 tries if drawer(guess) = i then goto next_prisoner guess = gus(drawer(guess), strat) next k c_fail += 1 goto next_trial next_prisoner: next i c_success += 1 next_trial: next j
end sub
randomize timer dim as long c_fail=0, c_success=0
trials( c_success, c_fail, false )
print using "For prisoners guessing randomly we had ####### successes and ####### failures.";c_success;c_fail
c_success = 0 c_fail = 0
trials( c_success, c_fail, true )
print using "For prisoners using the strategy we had ####### successes and ####### failures.";c_success;c_fail</lang>
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%
Groovy
<lang groovy>import java.util.function.Function import java.util.stream.Collectors import java.util.stream.IntStream
class Prisoners {
private static boolean playOptimal(int n) { List<Integer> secretList = IntStream.range(0, n).boxed().collect(Collectors.toList()) Collections.shuffle(secretList)
prisoner: for (int i = 0; i < secretList.size(); ++i) { int prev = i for (int j = 0; j < secretList.size() / 2; ++j) { if (secretList.get(prev) == i) { continue prisoner } prev = secretList.get(prev) } return false } return true }
private static boolean playRandom(int n) { List<Integer> secretList = IntStream.range(0, n).boxed().collect(Collectors.toList()) Collections.shuffle(secretList)
prisoner: for (Integer i : secretList) { List<Integer> trialList = IntStream.range(0, n).boxed().collect(Collectors.toList()) Collections.shuffle(trialList)
for (int j = 0; j < trialList.size() / 2; ++j) { if (Objects.equals(trialList.get(j), i)) { continue prisoner } }
return false } return true }
private static double exec(int n, int p, Function<Integer, Boolean> play) { int succ = 0 for (int i = 0; i < n; ++i) { if (play.apply(p)) { succ++ } } return (succ * 100.0) / n }
static void main(String[] args) { final int n = 100_000 final int p = 100 System.out.printf("# of executions: %d\n", n) System.out.printf("Optimal play success rate: %f%%\n", exec(n, p, Prisoners.&playOptimal)) System.out.printf("Random play success rate: %f%%\n", exec(n, p, Prisoners.&playRandom)) }
}</lang>
- Output:
# of executions: 100000 Optimal play success rate: 31.215000% Random play success rate: 0.000000%
Haskell
<lang haskell>import System.Random import Control.Monad.State
numRuns = 10000 numPrisoners = 100 numDrawerTries = 50 type Drawers = [Int] type Prisoner = Int type Prisoners = [Int]
main = do
gen <- getStdGen putStrLn $ "Chance of winning when choosing randomly: " ++ (show $ evalState runRandomly gen) putStrLn $ "Chance of winning when choosing optimally: " ++ (show $ evalState runOptimally gen)
runRandomly :: State StdGen Double
runRandomly =
let runResults = replicateM numRuns $ do drawers <- state $ shuffle [1..numPrisoners] allM (\prisoner -> openDrawersRandomly drawers prisoner numDrawerTries) [1..numPrisoners] in ((/ fromIntegral numRuns) . fromIntegral . sum . map fromEnum) `liftM` runResults
openDrawersRandomly :: Drawers -> Prisoner -> Int -> State StdGen Bool openDrawersRandomly drawers prisoner triesLeft = go triesLeft []
where go 0 _ = return False go triesLeft seenDrawers = do try <- state $ randomR (1, numPrisoners) case try of x | x == prisoner -> return True | x `elem` seenDrawers -> go triesLeft seenDrawers | otherwise -> go (triesLeft - 1) (x:seenDrawers)
runOptimally :: State StdGen Double runOptimally =
let runResults = replicateM numRuns $ do drawers <- state $ shuffle [1..numPrisoners] return $ all (\prisoner -> openDrawersOptimally drawers prisoner numDrawerTries) [1..numPrisoners] in ((/ fromIntegral numRuns) . fromIntegral . sum . map fromEnum) `liftM` runResults
openDrawersOptimally :: Drawers -> Prisoner -> Int -> Bool openDrawersOptimally drawers prisoner triesLeft = go triesLeft prisoner
where go 0 _ = False go triesLeft drawerToTry = let thisDrawer = drawers !! (drawerToTry - 1) in if thisDrawer == prisoner then True else go (triesLeft - 1) thisDrawer
-- Haskel stdlib is lacking big time, so here some necessary 'library' functions
-- make a list of 'len' random values in range 'range' from 'gen' randomLR :: Integral a => Random b => a -> (b, b) -> StdGen -> ([b], StdGen) randomLR 0 range gen = ([], gen) randomLR len range gen =
let (x, newGen) = randomR range gen (xs, lastGen) = randomLR (len - 1) range newGen in (x : xs, lastGen)
-- shuffle a list by a generator
shuffle :: [a] -> StdGen -> ([a], StdGen)
shuffle list gen = (shuffleByNumbers numbers list, finalGen)
where n = length list (numbers, finalGen) = randomLR n (0, n-1) gen shuffleByNumbers :: [Int] -> [a] -> [a] shuffleByNumbers [] _ = [] shuffleByNumbers _ [] = [] shuffleByNumbers (i:is) xs = let (start, x:rest) = splitAt (i `mod` length xs) xs in x : shuffleByNumbers is (start ++ rest)
-- short-circuit monadic all allM :: Monad m => (a -> m Bool) -> [a] -> m Bool allM func [] = return True allM func (x:xs) = func x >>= \res -> if res then allM func xs else return False </lang>
- Output:
Chance of winning when choosing randomly: 0.0 Chance of winning when choosing optimally: 0.3188
J
<lang J> NB. game is solvable by optimal strategy when the length (#) of the NB. longest (>./) cycle (C.) is at most 50. opt=: 50 >: [: >./ [: > [: #&.> C.
NB. for each prisoner randomly open 50 boxes ((50?100){y) and see if NB. the right card is there (p&e.). if not return 0. rand=: monad define for_p. i.100 do. if. -.p e.(50?100){y do. 0 return. end. end. 1 )
NB. use both strategies on the same shuffles y times. simulate=: monad define 'o r'=. y %~ 100 * +/ ((rand,opt)@?~)"0 y # 100 ('strategy';'win rate'),('random';(":o),'%'),:'optimal';(":r),'%' )</lang>
- Output:
simulate 10000000 ┌────────┬────────┐ │strategy│win rate│ ├────────┼────────┤ │random │0% │ ├────────┼────────┤ │optimal │31.1816%│ └────────┴────────┘
Java
<lang java>import java.util.Collections; import java.util.List; import java.util.Objects; import java.util.function.Function; import java.util.function.Supplier; import java.util.stream.Collectors; import java.util.stream.IntStream;
public class Main {
private static boolean playOptimal(int n) { List<Integer> secretList = IntStream.range(0, n).boxed().collect(Collectors.toList()); Collections.shuffle(secretList);
prisoner: for (int i = 0; i < secretList.size(); ++i) { int prev = i; for (int j = 0; j < secretList.size() / 2; ++j) { if (secretList.get(prev) == i) { continue prisoner; } prev = secretList.get(prev); } return false; } return true; }
private static boolean playRandom(int n) { List<Integer> secretList = IntStream.range(0, n).boxed().collect(Collectors.toList()); Collections.shuffle(secretList);
prisoner: for (Integer i : secretList) { List<Integer> trialList = IntStream.range(0, n).boxed().collect(Collectors.toList()); Collections.shuffle(trialList);
for (int j = 0; j < trialList.size() / 2; ++j) { if (Objects.equals(trialList.get(j), i)) { continue prisoner; } }
return false; } return true; }
private static double exec(int n, int p, Function<Integer, Boolean> play) { int succ = 0; for (int i = 0; i < n; ++i) { if (play.apply(p)) { succ++; } } return (succ * 100.0) / n; }
public static void main(String[] args) { final int n = 100_000; final int p = 100; System.out.printf("# of executions: %d\n", n); System.out.printf("Optimal play success rate: %f%%\n", exec(n, p, Main::playOptimal)); System.out.printf("Random play success rate: %f%%\n", exec(n, p, Main::playRandom)); }
}</lang>
- Output:
# of executions: 100000 Optimal play success rate: 31.343000% Random play success rate: 0.000000%
JavaScript
<lang javascript> const _ = require('lodash');
const numPlays = 100000;
const setupSecrets = () => { // setup the drawers with random cards let secrets = [];
for (let i = 0; i < 100; i++) { secrets.push(i); }
return _.shuffle(secrets); }
const playOptimal = () => {
let secrets = setupSecrets();
// Iterate once per prisoner
loop1:
for (let p = 0; p < 100; p++) {
// whether the prisoner succeedss let success = false;
// the drawer number the prisoner chose let choice = p;
// The prisoner can choose up to 50 cards
loop2:
for (let i = 0; i < 50; i++) {
// if the card in the drawer that the prisoner chose is his card if (secrets[choice] === p){ success = true; break loop2; }
// the next drawer the prisoner chooses will be the number of the card he has. choice = secrets[choice];
} // each prisoner gets 50 chances
if (!success) return false;
} // iterate for each prisoner
return true; }
const playRandom = () => {
let secrets = setupSecrets();
// iterate for each prisoner for (let p = 0; p < 100; p++) {
let choices = setupSecrets();
let success = false;
for (let i = 0; i < 50; i++) {
if (choices[i] === p) { success = true; break; } }
if (!success) return false; }
return true; }
const execOptimal = () => {
let success = 0;
for (let i = 0; i < numPlays; i++) {
if (playOptimal()) success++;
}
return 100.0 * success / 100000; }
const execRandom = () => {
let success = 0;
for (let i = 0; i < numPlays; i++) {
if (playRandom()) success++;
}
return 100.0 * success / 100000; }
console.log("# of executions: " + numPlays); console.log("Optimal Play Success Rate: " + execOptimal()); console.log("Random Play Success Rate: " + execRandom()); </lang>
Julia
<lang julia>using Random, Formatting
function randomplay(n, numprisoners=100)
pardoned, indrawer, found = 0, collect(1:numprisoners), false for i in 1:n shuffle!(indrawer) for prisoner in 1:numprisoners found = false for reveal in randperm(numprisoners)[1:div(numprisoners, 2)] indrawer[reveal] == prisoner && (found = true) && break end !found && break end found && (pardoned += 1) end return 100.0 * pardoned / n
end
function optimalplay(n, numprisoners=100)
pardoned, indrawer, found = 0, collect(1:numprisoners), false for i in 1:n shuffle!(indrawer) for prisoner in 1:numprisoners reveal = prisoner found = false for j in 1:div(numprisoners, 2) card = indrawer[reveal] card == prisoner && (found = true) && break reveal = card end !found && break end found && (pardoned += 1) end return 100.0 * pardoned / n end
const N = 100_000 println("Simulation count: $N") println("Random play wins: ", format(randomplay(N), precision=8), "% of simulations.") println("Optimal play wins: ", format(optimalplay(N), precision=8), "% of simulations.")
</lang>
- Output:
Simulation count: 100000 Random play wins: 0.00000000% of simulations. Optimal play wins: 31.18100000% of simulations.
Kotlin
<lang scala>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%
Lua
<lang lua>function shuffle(tbl)
for i = #tbl, 2, -1 do local j = math.random(i) tbl[i], tbl[j] = tbl[j], tbl[i] end return tbl
end
function playOptimal()
local secrets = {} for i=1,100 do secrets[i] = i end shuffle(secrets)
for p=1,100 do local success = false
local choice = p for i=1,50 do if secrets[choice] == p then success = true break end choice = secrets[choice] end
if not success then return false end end
return true
end
function playRandom()
local secrets = {} for i=1,100 do secrets[i] = i end shuffle(secrets)
for p=1,100 do local choices = {} for i=1,100 do choices[i] = i end shuffle(choices)
local success = false for i=1,50 do if choices[i] == p then success = true break end end
if not success then return false end end
return true
end
function exec(n,play)
local success = 0 for i=1,n do if play() then success = success + 1 end end return 100.0 * success / n
end
function main()
local N = 1000000 print("# of executions: "..N) print(string.format("Optimal play success rate: %f", exec(N, playOptimal))) print(string.format("Random play success rate: %f", exec(N, playRandom)))
end
main()</lang>
- Output:
# of executions: 1000000 Optimal play success rate: 31.237500 Random play success rate: 0.000000
MATLAB
<lang MATLAB>function [randSuccess,idealSuccess]=prisoners(numP,numG,numT)
%numP is the number of prisoners %numG is the number of guesses %numT is the number of trials randSuccess=0; %Random for trial=1:numT drawers=randperm(numP); won=1; for i=1:numP correct=0; notopened=drawers; for j=1:numG ind=randi(numel(notopened)); m=notopened(ind); if m==i correct=1; break; end notopened(ind)=[]; end if correct==0 won=0; break; end end randSuccess=randSuccess*(trial-1)/trial+won/trial; end
%Ideal idealSuccess=0;
for trial=1:numT drawers=randperm(numP); won=1; for i=1:numP correct=0; guess=i; for j=1:numG m=drawers(guess); if m==i correct=1; break; end guess=m; end if correct==0 won=0; break; end end idealSuccess=idealSuccess*(trial-1)/trial+won/trial; end disp(['Probability of success with random strategy: ' num2str(randSuccess*100) '%']); disp(['Probability of success with ideal strategy: ' num2str(idealSuccess*100) '%']);
end</lang>
- Output:
>> [randSuccess,idealSuccess]=prisoners(100,50,10000); Probability of success with random strategy: 0% Probability of success with ideal strategy: 31.93%
MiniScript
<lang MiniScript>playRandom = function(n)
// using 0-99 instead of 1-100 pardoned = 0 numInDrawer = range(99) choiceOrder = range(99) for round in range(1, n) numInDrawer.shuffle choiceOrder.shuffle for prisoner in range(99) found = false for card in choiceOrder[:50] if card == prisoner then found = true break end if end for if not found then break end for if found then pardoned = pardoned + 1 end for return pardoned / n * 100
end function
playOptimal = function(n)
// using 0-99 instead of 1-100 pardoned = 0 numInDrawer = range(99) for round in range(1, n) numInDrawer.shuffle for prisoner in range(99) found = false
drawer = prisoner
for i in range(1,50) card = numInDrawer[drawer] if card == prisoner then found = true break end if drawer = card end for if not found then break end for if found then pardoned = pardoned + 1 end for return pardoned / n * 100
end function
print "Random: " + playRandom(10000) + "%" print "Optimal: " + playOptimal(10000) + "%"</lang>
- Output:
Random: 0% Optimal: 31.06%
Nim
Imperative style. <lang Nim>import random, sequtils, strutils
type
Sample = tuple succ: int fail: int
const
numPrisoners = 100 numDrawsEachPrisoner = numPrisoners div 2 numDrawings: Positive = 1_000_000 div 1
proc `$`(s: Sample): string =
"Succs: $#\tFails: $#\tTotal: $#\tSuccess Rate: $#%." % [$s.succ, $s.fail, $(s.succ + s.fail), $(s.succ.float / (s.succ + s.fail).float * 100.0)]
proc prisonersWillBeReleasedSmart(): bool =
result = true var drawers = toSeq(0..<numPrisoners) drawers.shuffle for prisoner in 0..<numPrisoners: var drawer = prisoner block inner: for _ in 0..<numDrawsEachPrisoner: if drawers[drawer] == prisoner: break inner drawer = drawers[drawer] return false
proc prisonersWillBeReleasedRandom(): bool =
result = true var drawers = toSeq(0..<numPrisoners) drawers.shuffle for prisoner in 0..<numPrisoners: var selectDrawer = toSeq(0..<numPrisoners) selectDrawer.shuffle block inner: for i in 0..<numDrawsEachPrisoner: if drawers[selectDrawer[i]] == prisoner: break inner return false
proc massDrawings(prisonersWillBeReleased: proc(): bool): Sample =
var success = 0 for i in 1..numDrawings: if prisonersWillBeReleased(): inc(success) return (success, numDrawings - success)
randomize() echo $massDrawings(prisonersWillBeReleasedSmart) echo $massDrawings(prisonersWillBeReleasedRandom)</lang>
- Output:
Succs: 312225 Fails: 687775 Total: 1000000 Success Rate: 31.2225%. Succs: 0 Fails: 1000000 Total: 1000000 Success Rate: 0.0%.
Pascal
searching the longest cycle length as stated on talk page and increment an counter for that cycle length. <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
Alternative for optimized
<lang pascal>program Prisoners100; {$IFDEF FPC}
{$MODE DELPHI}{$OPTIMIZATION ON,ALL}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF} type
tValue = NativeUint; tpValue = pNativeUint; tPrisNum = array of tValue;
const
rounds = 1000000; cAlreadySeen = High(tValue);
var
drawers, Visited, CntToPardoned : tPrisNum; PrisCount : NativeInt;
procedure shuffle(var N:tPrisNum;lmt : nativeInt = 0); var
pN : tpValue; i,j : nativeInt; tmp: tValue;
Begin
pN := @N[0]; if lmt = 0 then lmt := High(N); For i := lmt downto 1 do begin //take one from index [0..i] j := random(i+1); //exchange with i tmp := pN[i];pN[i]:= pN[j];pN[j]:= tmp; end;
end;
procedure CopyDrawers2Visited; //drawers and Visited are of same size, so only moving values Begin
Move(drawers[0],Visited[0],SizeOf(tValue)*PrisCount);
end;
function GetMaxCycleLen:NativeUint; var
pVisited : tpValue; cycleLen,MaxCycLen,Num,NumBefore : NativeUInt;
Begin
CopyDrawers2Visited; pVisited := @Visited[0]; MaxCycLen := 0; cycleLen := MaxCycLen; Num := MaxCycLen; repeat NumBefore := Num; Num := pVisited[Num]; pVisited[NumBefore] := cAlreadySeen; inc(cycleLen); IF (Num= NumBefore) or (Num = cAlreadySeen) then begin IF Num = cAlreadySeen then dec(CycleLen); IF MaxCycLen < cycleLen then MaxCycLen := cycleLen; Num := 0; while (Num< PrisCount) AND (pVisited[Num] = cAlreadySeen) do inc(Num); //all cycles found IF Num >= PrisCount then BREAK; cycleLen :=0; end; until false; GetMaxCycleLen := MaxCycLen-1;
end;
procedure CheckOptimized(testCount : NativeUint); var
factor: extended; i,sum,digit,delta : NativeInt;
Begin
For i := 1 to rounds do begin shuffle(drawers); inc(CntToPardoned[GetMaxCycleLen]); end;
digit := 0; sum := rounds; while sum > 100 do Begin inc(digit); sum := sum DIV 10; end; factor := 100.0/rounds;
delta :=0; sum := 0; For i := 0 to High(drawers) do Begin inc(sum,CntToPardoned[i]); dec(delta); IF delta <= 0 then Begin writeln(sum*factor:Digit+5:Digit,'% get pardoned checking max ',i+1); delta := delta+Length(drawers) DIV 10; end; end;
end;
procedure OneCompareRun(PrisCnt:NativeInt); var
i,lmt :nativeInt;
begin
PrisCount := PrisCnt; setlength(drawers,PrisCnt); For i := 0 to PrisCnt-1 do drawers[i] := i; setlength(Visited,PrisCnt); setlength(CntToPardoned,PrisCnt); //test writeln('Checking ',PrisCnt,' prisoners for ',rounds,' rounds'); lmt := PrisCnt; CheckOptimized(lmt); writeln;
setlength(CntToPardoned,0); setlength(Visited,0); setlength(drawers,0);
end;
Begin
randomize; OneCompareRun(10); OneCompareRun(100); OneCompareRun(1000);
end.</lang>
- Output:
Checking 10 prisoners for 1000000 rounds 0.0000% get pardoned checking max 1 0.2584% get pardoned checking max 2 4.7431% get pardoned checking max 3 17.4409% get pardoned checking max 4 35.4983% get pardoned checking max 5 52.1617% get pardoned checking max 6 66.4807% get pardoned checking max 7 78.9761% get pardoned checking max 8 90.0488% get pardoned checking max 9 100.0000% get pardoned checking max 10 Checking 100 prisoners for 1000000 rounds 0.0000% get pardoned checking max 1 0.0000% get pardoned checking max 10 0.0459% get pardoned checking max 20 2.5996% get pardoned checking max 30 13.5071% get pardoned checking max 40 31.2258% get pardoned checking max 50 49.3071% get pardoned checking max 60 64.6128% get pardoned checking max 70 77.8715% get pardoned checking max 80 89.5385% get pardoned checking max 90 100.0000% get pardoned checking max 100 Checking 1000 prisoners for 1000000 rounds 0.0000% get pardoned checking max 1 0.0000% get pardoned checking max 100 0.0374% get pardoned checking max 200 2.3842% get pardoned checking max 300 13.1310% get pardoned checking max 400 30.7952% get pardoned checking max 500 48.9710% get pardoned checking max 600 64.3555% get pardoned checking max 700 77.6950% get pardoned checking max 800 89.4515% get pardoned checking max 900 100.0000% get pardoned checking max 1000 real 0m9,975s
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
Phix
<lang Phix>function play(integer prisoners, iterations, bool optimal)
sequence drawers = shuffle(tagset(prisoners)) integer pardoned = 0 bool found = false for i=1 to iterations do drawers = shuffle(drawers) for prisoner=1 to prisoners do found = false integer drawer = iff(optimal?prisoner:rand(prisoners)) for j=1 to prisoners/2 do drawer = drawers[drawer] if drawer==prisoner then found = true exit end if if not optimal then drawer = rand(prisoners) end if end for if not found then exit end if end for pardoned += found end for return 100*pardoned/iterations
end function
constant iterations = 100_000 printf(1,"Simulation count: %d\n",iterations) for prisoners=10 to 100 by 90 do
atom random = play(prisoners,iterations,false), optimal = play(prisoners,iterations,true) printf(1,"Prisoners:%d, random:%g, optimal:%g\n",{prisoners,random,optimal})
end for</lang>
- Output:
Simulation count: 100000 Prisoners:10, random:0.006, optimal:35.168 Prisoners:100, random:0, optimal:31.098
PowerShell
<lang PowerShell>
- Clear Screen from old Output
Clear-Host
Function RandomOpening ()
{ $Prisoners = 1..100 | Sort-Object {Get-Random} $Cupboard = 1..100 | Sort-Object {Get-Random} ## Loop for the Prisoners $Survived = $true for ($I=1;$I -le 100;$i++) { $OpeningListe = 1..100 | Sort-Object {Get-Random} $Gefunden = $false ## Loop for the trys of every prisoner for ($X=1;$X -le 50;$X++) { $OpenNumber = $OpeningListe[$X] IF ($Cupboard[$OpenNumber] -eq $Prisoners[$I]) { $Gefunden = $true } ## Cancel loop if prisoner found his number (yeah i know, dirty way ^^ ) IF ($Gefunden) { $X = 55 } } IF ($Gefunden -eq $false) { $I = 120 $Survived = $false } } Return $Survived }
Function StrategyOpening () { $Prisoners = 1..100 | Sort-Object {Get-Random} $Cupboard = 1..100 | Sort-Object {Get-Random} $Survived = $true for ($I=1;$I -le 100;$i++) { $Gefunden = $false $OpeningNumber = $Prisoners[$I-1] for ($X=1;$X -le 50;$X++) { IF ($Cupboard[$OpeningNumber-1] -eq $Prisoners[$I-1]) { $Gefunden = $true } else { $OpeningNumber = $Cupboard[$OpeningNumber-1] } IF ($Gefunden) { $X = 55 } } IF ($Gefunden -eq $false) { $I = 120 $Survived = $false } } Return $Survived }
$MaxRounds = 10000
Function TestRandom
{ $WinnerRandom = 0 for ($Round = 1; $Round -le $MaxRounds;$Round++) { IF (($Round%1000) -eq 0) { $Time = Get-Date Write-Host "Currently we are at rount $Round at $Time" } $Rueckgabewert = RandomOpening IF ($Rueckgabewert) { $WinnerRandom++ } } $Prozent = (100/$MaxRounds)*$WinnerRandom Write-Host "There are $WinnerRandom survivors whit random opening. This is $Prozent percent" }
Function TestStrategy
{ $WinnersStrategy = 0 for ($Round = 1; $Round -le $MaxRounds;$Round++) { IF (($Round%1000) -eq 0) { $Time = Get-Date Write-Host "Currently we are at $Round at $Time" } $Rueckgabewert = StrategyOpening IF ($Rueckgabewert) { $WinnersStrategy++ } } $Prozent = (100/$MaxRounds)*$WinnersStrategy Write-Host "There are $WinnersStrategy survivors whit strategic opening. This is $Prozent percent" }
Function Main ()
{ Clear-Host TestRandom TestStrategy }
Main </lang>
- Output:
# of executions: 10000 There are 0 survivors whit random opening. This is 0 percent There are 3104 survivors whit strategic opening. This is 31,04 percent"
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
Or, an alternative procedural approach:
<lang python># http://rosettacode.org/wiki/100_prisoners
import random
def main():
NUM_DRAWERS = 10 NUM_REPETITIONS = int(1E5)
print('{:15}: {:5} ({})'.format('approach', 'wins', 'ratio')) for approach in PrisionersGame.approaches: num_victories = 0 for _ in range(NUM_REPETITIONS): game = PrisionersGame(NUM_DRAWERS) num_victories += PrisionersGame.victory(game.play(approach))
print('{:15}: {:5} ({:.2%})'.format( approach.__name__, num_victories, num_victories / NUM_REPETITIONS))
class PrisionersGame:
"""docstring for PrisionersGame""" def __init__(self, num_drawers): assert num_drawers % 2 == 0 self.num_drawers = num_drawers self.max_attempts = int(self.num_drawers / 2) self.drawer_ids = list(range(1, num_drawers + 1)) shuffled = self.drawer_ids[:] random.shuffle(shuffled) self.drawers = dict(zip(self.drawer_ids, shuffled))
def play_naive(self, player_number): """ Randomly open drawers """ for attempt in range(self.max_attempts): if self.drawers[random.choice(self.drawer_ids)] == player_number: return True
return False
def play_naive_mem(self, player_number): """ Randomly open drawers but avoiding repetitions """ not_attemped = self.drawer_ids[:] for attempt in range(self.max_attempts): guess = random.choice(not_attemped) not_attemped.remove(guess)
if self.drawers[guess] == player_number: return True
return False
def play_optimum(self, player_number): """ Open the drawer that matches the player number and then open the drawer with the revealed number. """ prev_attempt = player_number for attempt in range(self.max_attempts): if self.drawers[prev_attempt] == player_number: return True else: prev_attempt = self.drawers[prev_attempt]
return False
@classmethod def victory(csl, results): """Defines a victory of a game: all players won""" return all(results)
approaches = [play_naive, play_naive_mem, play_optimum]
def play(self, approach): """Plays this game and returns a list of booleans with True if a player one, False otherwise""" return [approach(self, player) for player in self.drawer_ids]
if __name__ == '__main__':
main()</lang>
- Output:
With 10 drawers (100k runs) approach : wins (ratio) play_naive : 14 (0.01%) play_naive_mem : 74 (0.07%) play_optimum : 35410 (35.41%) With 100 drawers (10k runs) approach : wins (ratio) play_naive : 0 (0.00%) play_naive_mem : 0 (0.00%) play_optimum : 3084 (30.84%)
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 none of the index-chasing cycles in a shuffled sample of [1..n] cards are longer than half the sample size. 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%
Racket
<lang racket>#lang racket (require srfi/1)
(define current-samples (make-parameter 10000)) (define *prisoners* 100) (define *max-guesses* 50)
(define (evaluate-strategy instance-solved? strategy (s (current-samples)))
(/ (for/sum ((_ s) #:when (instance-solved? strategy)) 1) s))
(define (build-drawers)
(list->vector (shuffle (range *prisoners*))))
(define (100-prisoners-problem strategy)
(every (strategy (build-drawers)) (range *prisoners*)))
(define ((strategy-1 drawers) p)
(any (λ (_) (= p (vector-ref drawers (random *prisoners*)))) (range *max-guesses*)))
(define ((strategy-2 drawers) p)
(define-values (_ found?) (for/fold ((d p) (found? #f)) ((_ *max-guesses*)) #:break found? (let ((card (vector-ref drawers d))) (values card (= card p))))) found?)
(define (print-sample-percentage caption f (s (current-samples)))
(printf "~a: ~a%~%" caption (real->decimal-string (* 100 f) (- (order-of-magnitude s) 2))))
(module+ main
(print-sample-percentage "random" (evaluate-strategy 100-prisoners-problem strategy-1)) (print-sample-percentage "optimal" (evaluate-strategy 100-prisoners-problem strategy-2)))</lang>
- Output:
random: 0.00% optimal: 31.18%
Raku
(formerly 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
Red
<lang Rebol> Red []
K_runs: 100000 repeat n 100 [append rand_arr: [] n] ;; define array/series with numbers 1..100
- -------------------------------
strat_optimal: function [pris ][
- -------------------------------
locker: pris ;; start with locker equal to prisoner number loop 50 [ if Board/:locker = pris [ return true ] ;; locker with prisoner number found locker: Board/:locker ] false ;; number not found - fail
]
- -------------------------------
strat_rand: function [pris ][
- -------------------------------
random rand_arr ;; define set of random lockers repeat n 50 [ if Board/(rand_arr/:n) = pris [ return true ] ] ;; try first 50, found ? then return success false
]
- ------------------------------
check_board: function [ strat][
- ------------------------------
repeat pris 100 [ ;; for each prisoner
either strat = 'optimal [ unless strat_optimal pris [return false ] ] [ unless strat_rand pris [return false ] ]
]
true ;; all 100 prisoners passed test
]
saved: saved_rand: 0 ;; count all saved runs per strategy loop K_runs [
Board: random copy rand_arr ;; new board for every run if check_board 'optimal [saved: saved + 1] ;; optimal stategy if check_board 'rand [saved_rand: saved_rand + 1] ;; random strategy
]
print ["runs" k_runs newline "Percent saved opt.strategy:" saved * 100.0 / k_runs ] print ["Percent saved random strategy:" saved_rand * 100.0 / k_runs ] </lang>
- Output:
runs 100000 Percent saved opt.strategy: 31.165 Percent saved random strategy: 0.0
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.*/ try= men % 2; swaps= men * 3 /*number tries for searching for a card*/ $.1= ' a simple '; $.2= "an optimal" /*literals used for the SAY instruction*/ say center(' running' commas(trials) "trials with" commas(men) 'prisoners ', 70, "═") say
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 process*/ if strategy==1 then failure= simple() /*Is 1st strategy? Use simple strategy*/ else failure= picker() /* " 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*/
pc= format( pardons/trials*100, , 3); _= left(, pc<10) say right('Using', 9) $.strategy "strategy yields pardons " _||pc"% of the time." 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; do j=1 for men; @.j= j /*define seq. of cards*/
end /*j*/ /*same as seq. of men.*/ do swaps; a= random(1, men) /*get 1st rand number.*/ do until b\==a; b= random(1, men) /* " 2nd " " */ end /*until*/ /* [↑] ensure A ¬== B */ parse value @.a @.b with @.b @.a /*swap 2 random cards.*/ end /*swaps*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/ simple: !.= 0; do try; do until !.?==0; ?= random(1, men) /*get random card ··· */
end /*until*/ /*··· not used before.*/ if @.?==p then do; #= #+1; return 0; end /*found his own card? */ !.?= 1 /*flag as being used. */ end /*try*/; return 1 /*didn't find his card*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ picker: ?= p; do try; if @.?==p then do; #= #+1; return 0 /*Found his own card? */
end /* [↑] indicate success for prisoner. */ ?= @.? /*choose next drawer from current card.*/ end /*try*/; return 1 /*choose half of the number of drawers.*/</lang>
- output when using the default inputs:
══════════════ running 100,000 trials with 100 prisoners ══════════════ Using a simple strategy yields pardons 0.000% of the time. Using an optimal strategy yields pardons 31.186% of the time.
- output when using the input of: 10
══════════════ running 100,000 trials with 10 prisoners ══════════════ Using a simple strategy yields pardons 0.086% of the time. Using an optimal strategy yields pardons 31.204% of the time.
Ruby
<lang ruby>prisoners = [*1..100] N = 10_000 generate_rooms = ->{ [nil]+[*1..100].shuffle }
res = N.times.count do
rooms = generate_rooms[] prisoners.all? {|pr| rooms[1,100].sample(50).include?(pr)}
end puts "Random strategy : %11.4f %%" % (res.fdiv(N) * 100)
res = N.times.count do
rooms = generate_rooms[] prisoners.all? do |pr| cur_room = pr 50.times.any? do found = (rooms[cur_room] == pr) cur_room = rooms[cur_room] found end end
end puts "Optimal strategy: %11.4f %%" % (res.fdiv(N) * 100) </lang>
- Output:
Random strategy : 0.0000 % Optimal strategy: 30.7400 %
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
Scala
<lang scala>import scala.util.Random import scala.util.control.Breaks._
object Main {
def playOptimal(n: Int): Boolean = { val secretList = Random.shuffle((0 until n).toBuffer)
for (i <- secretList.indices) { var prev = i breakable { for (_ <- 0 until secretList.size / 2) { if (secretList(prev) == i) { break() } prev = secretList(prev) } return false } }
true }
def playRandom(n: Int): Boolean = { val secretList = Random.shuffle((0 until n).toBuffer)
for (i <- secretList.indices) { val trialList = Random.shuffle((0 until n).toBuffer)
breakable { for (j <- 0 until trialList.size / 2) { if (trialList(j) == i) { break() } } return false } }
true }
def exec(n: Int, p: Int, play: Int => Boolean): Double = { var succ = 0.0 for (_ <- 0 until n) { if (play(p)) { succ += 1 } } (succ * 100.0) / n }
def main(args: Array[String]): Unit = { val n = 100000 val p = 100 printf("# of executions: %,d\n", n) printf("Optimal play success rate: %f%%\n", exec(n, p, playOptimal)) printf("Random play success rate: %f%%\n", exec(n, p, playRandom)) }
}</lang>
- Output:
# of executions: 100,000 Optimal play success rate: 31.201000% Random play success rate: 0.000000%
Swift
<lang swift>import Foundation
struct PrisonersGame {
let strategy: Strategy let numPrisoners: Int let drawers: [Int]
init(numPrisoners: Int, strategy: Strategy) { self.numPrisoners = numPrisoners self.strategy = strategy self.drawers = (1...numPrisoners).shuffled() }
@discardableResult func play() -> Bool { for num in 1...numPrisoners { guard findNumber(num) else { return false } }
return true }
private func findNumber(_ num: Int) -> Bool { var tries = 0 var nextDrawer = num - 1
while tries < 50 { tries += 1
switch strategy { case .random where drawers.randomElement()! == num: return true case .optimum where drawers[nextDrawer] == num: return true case .optimum: nextDrawer = drawers[nextDrawer] - 1 case _: continue } }
return false }
enum Strategy { case random, optimum }
}
let numGames = 100_000 let lock = DispatchSemaphore(value: 1) var done = 0
print("Running \(numGames) games for each strategy")
DispatchQueue.concurrentPerform(iterations: 2) {i in
let strat = i == 0 ? PrisonersGame.Strategy.random : .optimum var numPardoned = 0
for _ in 0..<numGames { let game = PrisonersGame(numPrisoners: 100, strategy: strat)
if game.play() { numPardoned += 1 } }
print("Probability of pardon with \(strat) strategy: \(Double(numPardoned) / Double(numGames))")
lock.wait() done += 1 lock.signal()
if done == 2 { exit(0) }
}
dispatchMain()</lang>
- Output:
Running 100000 games for each strategy Probability of pardon with optimum strategy: 0.31099 Probability of pardon with random strategy: 0.0
VBA
<lang VBA>Sub HundredPrisoners()
NumberOfPrisoners = Int(InputBox("Number of Prisoners", "Prisoners", 100)) Tries = Int(InputBox("Numer of Tries", "Tries", 1000)) Selections = Int(InputBox("Number of Selections", "Selections", NumberOfPrisoners / 2))
StartTime = Timer
AllFoundOptimal = 0 AllFoundRandom = 0 AllFoundRandomMem = 0
For i = 1 To Tries
OptimalCount = HundredPrisoners_Optimal(NumberOfPrisoners, Selections) RandomCount = HundredPrisoners_Random(NumberOfPrisoners, Selections) RandomMemCount = HundredPrisoners_Random_Mem(NumberOfPrisoners, Selections) If OptimalCount = NumberOfPrisoners Then AllFoundOptimal = AllFoundOptimal + 1 End If If RandomCount = NumberOfPrisoners Then AllFoundRandom = AllFoundRandom + 1 End If If RandomMemCount = NumberOfPrisoners Then AllFoundRandomMem = AllFoundRandomMem + 1 End If
Next i
ResultString = "Optimal: " & AllFoundOptimal & " of " & Tries & ": " & AllFoundOptimal / Tries * 100 & "%"
ResultString = ResultString & Chr(13) & "Random: " & AllFoundRandom & " of " & Tries & ": " & AllFoundRandom / Tries * 100 & "%"
ResultString = ResultString & Chr(13) & "RandomMem: " & AllFoundRandomMem & " of " & Tries & ": " & AllFoundRandomMem / Tries * 100 & "%"
EndTime = Timer
ResultString = ResultString & Chr(13) & "Elapsed Time: " & Round(EndTime - StartTime, 2) & " s" ResultString = ResultString & Chr(13) & "Trials/sec: " & Tries / Round(EndTime - StartTime, 2)
MsgBox ResultString, vbOKOnly, "Results"
End Sub
Function HundredPrisoners_Optimal(ByVal NrPrisoners, ByVal NrSelections) As Long
Dim DrawerArray() As Long ReDim DrawerArray(NrPrisoners - 1) For Counter = LBound(DrawerArray) To UBound(DrawerArray) DrawerArray(Counter) = Counter + 1 Next Counter
FisherYates DrawerArray For i = 1 To NrPrisoners NumberFromDrawer = DrawerArray(i - 1) For j = 1 To NrSelections - 1 If NumberFromDrawer = i Then FoundOwnNumber = FoundOwnNumber + 1 GoTo Finish End If NumberFromDrawer = DrawerArray(NumberFromDrawer - 1) Next j
Finish:
Next i
HundredPrisoners_Optimal = FoundOwnNumber End Function
Function HundredPrisoners_Random(ByVal NrPrisoners, ByVal NrSelections) As Long
Dim DrawerArray() As Long ReDim DrawerArray(NrPrisoners - 1) FoundOwnNumber = 0 For Counter = LBound(DrawerArray) To UBound(DrawerArray) DrawerArray(Counter) = Counter + 1 Next Counter
FisherYates DrawerArray For i = 1 To NrPrisoners For j = 1 To NrSelections RandomDrawer = Int(NrPrisoners * Rnd) NumberFromDrawer = DrawerArray(RandomDrawer) If NumberFromDrawer = i Then FoundOwnNumber = FoundOwnNumber + 1 GoTo Finish End If Next j
Finish:
Next i
HundredPrisoners_Random = FoundOwnNumber End Function
Function HundredPrisoners_Random_Mem(ByVal NrPrisoners, ByVal NrSelections) As Long
Dim DrawerArray() As Long Dim SelectionArray() As Long ReDim DrawerArray(NrPrisoners - 1) ReDim SelectionArray(NrPrisoners - 1) HundredPrisoners_Random_Mem = 0 FoundOwnNumberMem = 0 For Counter = LBound(DrawerArray) To UBound(DrawerArray) DrawerArray(Counter) = Counter + 1 Next Counter For Counter = LBound(SelectionArray) To UBound(SelectionArray) SelectionArray(Counter) = Counter + 1 Next Counter
FisherYates DrawerArray For i = 1 To NrPrisoners FisherYates SelectionArray For j = 1 To NrSelections NumberFromDrawer = DrawerArray(SelectionArray(j - 1) - 1) If NumberFromDrawer = i Then FoundOwnNumberMem = FoundOwnNumberMem + 1 GoTo Finish2 End If Next j
Finish2:
Next i
HundredPrisoners_Random_Mem = FoundOwnNumberMem End Function
Sub FisherYates(ByRef InputArray() As Long)
Dim Temp As Long Dim PosRandom As Long Dim Counter As Long Dim Upper As Long Dim Lower As Long Lower = LBound(InputArray) Upper = UBound(InputArray) Randomize For Counter = Upper To (Lower + 1) Step -1 PosRandom = CLng(Int((Counter - Lower + 1) * Rnd + Lower)) Temp = InputArray(Counter) InputArray(Counter) = InputArray(PosRandom) InputArray(PosRandom) = Temp Next Counter
End Sub</lang>
- Output:
Optimal: 29090 of 100000: 29.09% Random: 0 of 100000: 0% RandomMem: 0 of 100000: 0% Elapsed Time: 388.41 s
Visual Basic .NET
<lang vbnet>Module Module1
Function PlayOptimal() As Boolean Dim secrets = Enumerable.Range(0, 100).OrderBy(Function(a) Guid.NewGuid).ToList
For p = 1 To 100 Dim success = False
Dim choice = p - 1 For i = 1 To 50 If secrets(choice) = p - 1 Then success = True Exit For End If choice = secrets(choice) Next
If Not success Then Return False End If Next
Return True End Function
Function PlayRandom() As Boolean Dim secrets = Enumerable.Range(0, 100).OrderBy(Function(a) Guid.NewGuid).ToList
For p = 1 To 100 Dim choices = Enumerable.Range(0, 100).OrderBy(Function(a) Guid.NewGuid).ToList
Dim success = False For i = 1 To 50 If choices(i - 1) = p Then success = True Exit For End If Next
If Not success Then Return False End If Next
Return True End Function
Function Exec(n As UInteger, play As Func(Of Boolean)) Dim success As UInteger = 0 For i As UInteger = 1 To n If play() Then success += 1 End If Next Return 100.0 * success / n End Function
Sub Main() Dim N = 1_000_000 Console.WriteLine("# of executions: {0}", N) Console.WriteLine("Optimal play success rate: {0:0.00000000000}%", Exec(N, AddressOf PlayOptimal)) Console.WriteLine(" Random play success rate: {0:0.00000000000}%", Exec(N, AddressOf PlayRandom)) End Sub
End Module</lang>
- Output:
# of executions: 1000000 Optimal play success rate: 31.12990000000% Random play success rate: 0.00000000000%
Wren
<lang ecmascript>import "random" for Random import "/fmt" for Fmt
var rand = Random.new()
var doTrials = Fn.new{ |trials, np, strategy|
var pardoned = 0 for (t in 0...trials) { var drawers = List.filled(100, 0) for (i in 0..99) drawers[i] = i rand.shuffle(drawers) var nextTrial = false for (p in 0...np) { var nextPrisoner = false if (strategy == "optimal") { var prev = p for (d in 0..49) { var curr = drawers[prev] if (curr == p) { nextPrisoner = true break } prev = curr } } else { var opened = List.filled(100, false) for (d in 0..49) { var n while (true) { n = rand.int(100) if (!opened[n]) { opened[n] = true break } } if (drawers[n] == p) { nextPrisoner = true break } } } if (!nextPrisoner) { nextTrial = true break } } if (!nextTrial) pardoned = pardoned + 1 } var rf = pardoned/trials * 100 Fmt.print(" strategy = $-7s pardoned = $,6d relative frequency = $5.2f\%\n", strategy, pardoned, rf)
}
var trials = 1e5 for (np in [10, 100]) {
Fmt.print("Results from $,d trials with $d prisoners:\n", trials, np) for (strategy in ["random", "optimal"]) doTrials.call(trials, np, strategy)
}</lang>
- Output:
Sample run:
Results from 100,000 trials with 10 prisoners: strategy = random pardoned = 98 relative frequency = 0.10% strategy = optimal pardoned = 31,212 relative frequency = 31.21% Results from 100,000 trials with 100 prisoners: strategy = random pardoned = 0 relative frequency = 0.00% strategy = optimal pardoned = 31,139 relative frequency = 31.14%
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 Raku 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%
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