(Game: locker puzzle) A school has 100 lockers and 100...

(Game: locker puzzle) A school has 100 lockers and 100 students. All lockers are closed on the first day of school. As the students enter, the first student, denoted S1, opens every locker. Then the second student, S2, begins with the second locker, denoted $\mathrm{L} 2,$ and closes every other locker. Student $\mathrm{S} 3$ begins with the third locker and changes every third locker (closes it if it was open, and opens it if it was closed). Student S4 begins with locker L4 and changes every fourth locker. Student S5 starts with L5 and changes every fifth locker, and so on, until student S100 changes L100. After all the students have passed through the building and changed the lockers, which lockers are open? Write a program to find your answer. (Hint: Use a list of 100 Boolean elements, each of which indicates whether a locker is open (True) or closed (False). Initially, all lockers are closed.)

Key Concepts

Boolean Array Representation

This concept involves representing a collection of elements using a list or array where each element can be in one of two states, typically true or false. In algorithmic problems like this one, a Boolean array is used to model whether lockers are open or closed, allowing efficient updates and queries to the state of each element.

Iterative Algorithms

An iterative algorithm is used to repeatedly execute a set of operations over a range or collection of items. In the locker problem, multiple iterations are performed where each student changes the state of selected lockers according to a specific sequence, illustrating how repetitive processes can lead to a predictable final state.

Toggle Operations

A toggle operation refers to the process of switching an element's state from one value to another (for example, from closed to open or vice versa). It is a key concept in problems that involve state changes based on a set of conditions or events, and serves as the fundamental action performed by each student in the puzzle.

Divisor Counting

This concept involves counting the number of factors (divisors) a number has, which in the context of the locker problem determines how many times each locker is toggled. A locker ends up open only if it has been toggled an odd number of times, which occurs precisely when the locker number is a perfect square.

Perfect Squares

A perfect square is an integer that is the square of another integer. In the locker puzzle, only the perfect square numbered lockers remain open at the end because they have an odd number of divisors, resulting in an odd number of state changes (toggles).

Transcript

0:00 Hello.

00:01 For problem 21, you're solving a sort of locker puzzle where you're told that all of the lockers start closed on the first day.

00:11 The first student comes in and opens every locker.

00:17 And then the second student comes in and closes every other locker.

00:22 And then the third student comes in and changes every third locker and so on and so forth.

00:28 And you want to write a program that will.

00:31 Tell us what the locker states are at the end of 100 students going through.

00:39 So first we need to create our lockers.

00:42 So we will create a list lockers, which is 100 elements long, and every element in that list is just false to start with because all of the lockers start close.

00:56 And then we're going to iterate through each student.

00:59 So for s in range 100, because there are 100 students, we then need to iterate through each locker to determine how each student will affect each locker.

01:10 So for l in the range, s to length of lockers, and we start at s because the student will start at their own locker.

01:22 So the second student starts at locker two, the third student starts at locker three, and so on and so forth.

01:28 But they then all go through to the final locker.

01:32 And then we have our conditional statements here indicating what to do.

01:36 So if the locker is true and because the lockers, the student is also only going to interact with lockers that are multiples of their number...

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