Step by Step solution about How to resolve the algorithm 100 prisoners step by step in the Mathematica/Wolfram Language programming language

This Wolfram code simulates two strategies for playing a game where you choose a number from 1 to 100, and the goal is to avoid having your number drawn from a sample of 50 random numbers. The first strategy (PlayRandom) chooses a random number and keeps drawing random numbers until your number is drawn, while the second strategy (PlayOptimal) chooses a random number and then sequentially checks the numbers in the sample until your number is found.

Here's a detailed breakdown of the code:

1. ClearAll[PlayRandom, PlayOptimal]: This line clears any previous definitions of the PlayRandom and PlayOptimal functions.

2. PlayRandom[n_] :=: This line defines the PlayRandom function, which takes one argument, n, representing the number of times to play the game.

3. Inside the PlayRandom function:

   • Module[{pardoned = 0, sampler, indrawer, found, reveal}, ...]: This line creates a local scope for the variables pardoned, sampler, indrawer, found, and reveal`.
   • sampler = indrawer = Range[100];: This line initializes both sampler and indrawer to a list of numbers from 1 to 100. sampler will be used to randomly select numbers, while indrawer will hold the remaining undrawn numbers.
   • Do[ ..., {n}]: This line starts a loop that runs n times, simulating n games.
   • Inside the loop:
     • indrawer //= RandomSample;: This line randomly shuffles the indrawer list.
     • found = 0;: This line initializes the found variable to 0, indicating that the player's number has not yet been found.
     • Do[ ..., {p, 100}]: This line starts a nested loop that iterates through all numbers from 1 to 100.
     • Inside the nested loop:
       • reveal = RandomSample[sampler, 50];: This line randomly selects 50 numbers from the sampler list and assigns them to the reveal variable.
       • If[MemberQ[indrawer[[reveal]], p], ...]: This conditional statement checks if the player's number p is among the 50 numbers in reveal. If it is, the found variable is incremented by 1.
     • If[found == 100, pardoned++];: This conditional statement checks if the player's number was found in all 100 iterations of the nested loop. If it was, the pardoned variable is incremented by 1, indicating that the player successfully avoided having their number drawn.
   • Finally, the function returns the fraction of games where the player's number was not drawn (N[pardoned/n]).

4. PlayOptimal[n_] :=: This line defines the PlayOptimal function, which also takes one argument, n, representing the number of times to play the game.

5. Inside the PlayOptimal function:

   • Module[{pardoned = 0, indrawer, reveal, found, card}, ...]: This line creates a local scope for the variables pardoned, indrawer, reveal, found, and card.
   • indrawer = Range[100];: This line initializes indrawer to a list of numbers from 1 to 100.
   • Do[ ..., {n}]: This line starts a loop that runs n times, simulating n games.
   • Inside the loop:
     • indrawer //= RandomSample;: This line randomly shuffles the indrawer list.
     • Do[ ..., {p, 100}]: This line starts a nested loop that iterates through all numbers from 1 to 100.
     • Inside the nested loop:
       • reveal = p;: This line sets reveal to the current number p, which represents the player's number.
       • found = False;: This line initializes found to False, indicating that the player's number has not yet been found.
       • Do[ ..., {g, 50}]: This line starts a loop that iterates 50 times.
       • Inside the innermost loop:
         • card = indrawer[[reveal]];: This line assigns the value at the reveal index of indrawer to the card variable.
         • If[card == p, ...]: This conditional statement checks if the card value is equal to the player's number p. If they are equal, it means the player's number has been found, so found is set to True, and the Break[] statement is used to exit the innermost loop.
         • reveal = card;: If the card value is not equal to the player's number, reveal is updated to the value of card.
       • After the innermost loop completes, if found is still False, it means the player's number was not found in the 50 iterations, and the outer loop is exited using the Break[] statement.
     • If[found, pardoned++];: This conditional statement checks if the player's number was found. If it was not found, pardoned is incremented by 1, indicating that the player successfully avoided having their number drawn.
   • Finally, the function returns the fraction of games where the player's number was not drawn (N[pardoned/n]).

6. PlayRandom[1000]: This line calls the PlayRandom function 1000 times and prints the fraction of games where the player's number was not drawn.

7. PlayOptimal[10000]: This line calls the PlayOptimal function 10000 times and prints the fraction of games where the player's number was not drawn.

ClearAll[PlayRandom, PlayOptimal]
PlayRandom[n_] := 
 Module[{pardoned = 0, sampler, indrawer, found, reveal},
  sampler = indrawer = Range[100];
  Do[
   indrawer //= RandomSample;
   found = 0;
   Do[
    reveal = RandomSample[sampler, 50];
    If[MemberQ[indrawer[[reveal]], p],
     found++;
     ]
    ,
    {p, 100}
    ];
   If[found == 100, pardoned++];
   ,
   {n}
   ];
  N[pardoned/n]
  ]
PlayOptimal[n_] := 
 Module[{pardoned = 0, indrawer, reveal, found, card},
  indrawer = Range[100];
  Do[
   indrawer //= RandomSample;
   Do[
    reveal = p;
    found = False;
    Do[
     card = indrawer[[reveal]];
     If[card == p,
      found = True;
      Break[];
      ];
     reveal = card;
     ,
     {g, 50}
     ];
    If[! found, Break[]];
    ,
    {p, 100}
    ];
   If[found, pardoned++];
   ,
   {n}
   ];
  N[pardoned/n]
  ];
PlayRandom[1000]
PlayOptimal[10000]