Project Locker Puzzle Problem Description: A School Has 100

Project Locker Puzzleproblem Descriptiona School Has 100 Lockers And

Analyze the classic locker problem: A school has 100 lockers and 100 students. Initially, all lockers are closed. The first student opens every locker. The second student then starts with the second locker and closes every other locker. The third student begins with the third locker and toggles every third locker (closes it if it is open, opens it if it is closed). This process continues until the 100th student changes the 100th locker. The task is to determine which lockers remain open after all students have completed their pass and to write a program that outputs these open lockers, such as "Locker x is open". The problem suggests using an array of booleans to simulate locker states, with true representing open and false representing closed, initially all set to false.

Paper For Above instruction

TheProblem of the 100 Lockers: Analyzing the Mathematical and Programming Aspects

The classic locker problem provides an excellent illustration of elementary number theory and programming concepts. In this problem, a series of students toggle lockers based on their position, ultimately leading us to ask which lockers remain open after all passes are complete. This problem can be approached through both mathematical reasoning and programming implementation, offering valuable insights into problem-solving and algorithm development.

Mathematical Explanation

The key to resolving this problem lies in understanding the number of factors each locker number has. Each locker is toggled once for every divisor it has. For example, locker 12 has divisors 1, 2, 3, 4, 6, 12, totaling six divisors, which is even, leading the locker to end up closed because it is toggled an even number of times. Conversely, a locker number with an odd number of divisors will remain open after all toggling, since it is toggled an odd number of times. Interestingly, only perfect squares have an odd number of divisors because their divisors come in pairs, except for the square root which pairs with itself. Hence, only lockers with numbers that are perfect squares remain open. Specifically, in this case, lockers numbered with perfect squares less than or equal to 100 (such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100) will be open at the end of the process.

Programming Implementation

The problem can be translated into a programming task using an array of booleans, each representing a locker. Initially, all booleans are set to false (closed). The simulation involves nested loops: the outer loop iterates through each student, and the inner loop toggles the state of lockers starting from the student’s corresponding locker number, incrementing by that number until the end of the array. After completing the toggling process, the program scans the array to identify which lockers are still open. The results are then displayed in a readable format, such as "Locker x is open". This simulation demonstrates core programming concepts such as array manipulation, control structures, and logical reasoning.

Implementation Code

function lockerPuzzle() {
const lockers = new Array(100).fill(false); // false indicates closed
for (let student = 1; student 
for (let locker = student; locker 
lockers[locker -1] = !lockers[locker -1]; // toggle state
}
}
const openLockers = [];
for (let i=0; i
if (lockers[i]) {
openLockers.push(i+1);
}
}
let result = '';
openLockers.forEach(function(locker) {
result += 'Locker ' + locker + ' is open';
});
document.write(result);
}
window.onload = lockerPuzzle;