The Mystery of Open Lockers: Solving the Locker Problem
What is the locker problem and how can it be solved programmatically?
Analysis:
Design:
The major steps for solving the locker problem programmatically are as follows:
1. Create an array of 100 boolean elements, initialized to false (closed lockers).
2. Iterate through the students from 1 to 100.
3. For each student, iterate through the lockers from 1 to 100.
4. If the locker's index is divisible by the student's position number, change its state (open if closed, closed if open).
5. After all the students have passed through the building and changed the lockers, output the indices of the open lockers.
Coding:
```
// Initialize an array of 100 lockers
boolean[] lockers = new boolean[100];
// Iterate through students and lockers to change locker states
for (int student = 1; student <= 100; student++) {
for (int locker = student - 1; locker < 100; locker += student) {
lockers[locker] = !lockers[locker];
}
}
// Output the indices of open lockers
for (int i = 0; i < 100; i++) {
if (lockers[i]) {
System.out.println("Locker " + (i + 1) + " is open");
}
}
```
Testing:
To test this program, you can run it and check if the output matches the expected result. You can also manually verify the correctness of the program by simulating the locker opening/closing process for a smaller number of lockers and students.
Final answer:
The lockers that are open after all the students have passed through the building and changed the lockers can be determined by following a step-by-step process. Initially, all lockers are closed. Each student takes turns changing the state of the lockers based on their position number. If a locker is changed an odd number of times, it will end up open, and if it is changed an even number of times, it will end up closed. By iterating through the students and lockers, we can determine the indices of the open lockers.