The Locker Problem: Which Locker is Open?

What is the problem statement and algorithm to determine which lockers are open after all 100 students have passed through the building and changed the lockers? The problem states that there are 100 lockers and 100 students. The students follow a specific algorithm where the first student opens all lockers, the second student closes every other locker, the third student toggles every third locker, and so on until the 100th student toggles the 100th locker. To find out which lockers are open after all students have passed through, we can use an array of 100 boolean elements to represent the lockers. Initially, all lockers are closed (false). We can then simulate the process where each student changes the state of the lockers according to their specific rule. Finally, we can output the lockers that are left open.

The algorithm to solve the locker problem can be implemented using a simple program in Java. Here's the solution code:

Solution:

public class NewMain {
public static void main(String[] args) {
boolean[] locker = new boolean[101];
// Set all lockers to false (closed) initially
for (int i = 1; i < locker.length; i++) {
locker[i] = false;
}
// First student opens all lockers
for (int i = 1; i < locker.length; i++) {
locker[i] = true;
}
// Follow the students' rules to toggle the lockers
for (int S = 2; S < locker.length; S++) {
for (int k = S; k < locker.length; k += S) {
if (locker[k] == false) {
locker[k] = true;
} else {
locker[k] = false;
}
}
}
// Output the open lockers
for (int S = 1; S < locker.length; S++) {
if (locker[S] == true) {
System.out.println("Locker " + S + " Open");
}
}
}
}

This Java program follows the described algorithm where each student toggles the state of specific lockers. After all 100 students have passed through and changed the lockers, the program outputs the lockers that remain open. By running this program, you can determine which lockers are open based on the given rules.

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