Project Locker Puzzle Problem Description

Project Locker Puzzleproblem Descriptiona School Has 100 Lockers And

A school has 100 lockers and 100 students. All lockers are initially closed. The first student opens every locker. The second student then begins with locker 2 and closes every other locker. The third student starts with locker 3 and changes every third locker (closes it if it was open, opens it if it was closed). This process continues until the 100th student, who changes only the 100th locker. After all students have completed their actions, the task is to determine which lockers remain open. The program should simulate this process and output the lockers that are open, formatted as: Locker x is open, Locker y is open, and so on.

Paper For Above instruction

The classic locker problem is a well-known illustrative example of the mathematical principles related to factors and perfect squares. In this problem, a school with 100 lockers and 100 students serves as a setting to explore the effects of repetitive toggling based on locker and student numbers. To simulate the problem, a program can use an array of boolean values, each representing the state of a locker: true for open, false for closed.

The initial state of all lockers is closed, represented by all false values in the array. The students sequentially toggle the lockers starting from their respective starting points, with student 1 toggling every locker, student 2 toggling every second locker, and so forth until student 100 toggles only the 100th locker. This iterative process effectively demonstrates a pattern based on the number of factors a locker number has: lockers with an odd number of factors remain open, which is characteristic of perfect squares.

Implementing this process involves nested loops: the outer loop models the students, and the inner loop models the lockers being toggled. Each toggle flips the current state of the locker. After all students have toggled lockers according to the specified pattern, the program iterates through the locker array to identify which lockers are open. These are then displayed in the specified format.

Understanding this problem helps students grasp important mathematical concepts such as factors, perfect squares, and their implications in algorithms. It also emphasizes the importance of simulation and using data structures like arrays to model real-world scenarios and solve combinatorial problems efficiently.

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