Step by Step solution about How to resolve the algorithm 100 prisoners step by step in the Julia programming language

Overview:

The provided Julia code simulates a game played by prisoners and studies the effectiveness of two strategies: random play and optimal play.

Code Details:

1. Importing Libraries:

using Random, Formatting

• Imports the Random library for generating random numbers.
• Imports the Formatting library for formatting output values.

2. randomplay Function:

function randomplay(n, numprisoners=100)
   ...
end

This function simulates the game with a random play strategy.

Input Parameters:

• n: Number of simulations to run.
• numprisoners: Number of prisoners (default: 100).

Function Body:

• Initializes pardoned (number of pardoned prisoners), indrawer (a shuffled deck of cards), and found (indicator if a prisoner finds their own card).
• Runs n simulations.
• Iterates through the prisoners in each simulation.
• For each prisoner, randomly reveals half the cards in the shuffled deck.
• If the prisoner's card is not revealed, sets found to true and breaks the loop.
• If all prisoners fail to find their cards, adds 1 to pardoned.
• Returns the percentage of simulations where at least one prisoner was pardoned.

3. optimalplay Function:

function optimalplay(n, numprisoners=100)
   ...
end

This function simulates the game with an optimal play strategy.

Function Body:

• Similar to randomplay but uses an optimal strategy for revealing cards.
• For each prisoner, starts revealing cards sequentially from their own card.
• If the prisoner's card is found, sets found to true and breaks the loop.
• Returns the percentage of simulations where at least one prisoner was pardoned.

4. Simulation Execution:

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.")

• Sets the simulation count to N.
• Calls randomplay and optimalplay functions and prints the percentage of simulations where at least one prisoner was pardoned for each strategy.

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.")