The $``$N Doors Puzzle$$ is a great challenge to explore various aspects of algorithmic thinkingand Python programming.

$$ Imagine you have N doors in a row that are all initially closed.

$$ You make N pass by each of these doors

$$ The first time through, visit every door and $$toggle$$ the door $($if the door is closed, open

it; if it is open, close it$).$

$$ The second time, only visit every $2$ nd door $($i$.$e$.,$ door #$2,$ #$4,$ #$6,...)$ and toggle it$.$

$$ The third time, visit every $3$ rd door $($i$.$e$.,$ door #$3,$ #$6,$ #$9,...),$ etc., until you only visit the

N th door on the N th pass.

Implement your algorithmic thinking using Python programming to determine the state of the

doors after the last pass. That is which doors are open and which are closed after the last pass?

Looking at the $$N doors puzzle$$ we can make the following observations:

$$ There are doors numbered $1$ through N that start closed. You can assume a list of

Boolean variables. Initially, they are all going to be set to False, with false representing a

closed door and true representing an open door.

$$ We will iterate over the range N times, toggling the doors open$/$closed $($represented by

true$/$false$)$

$$ Each iteration over the range the gap between toggles increases by a multiple,

incremented by one.

$$ On the last iteration, only the N th door will be toggled

Sample I$/$O:

Run $1$:

Enter the number of Doors $($N$)$: $10$

i: $1$; j:$1$; step size: $1.$ Toggling door number $1.$

i: $1$; j:$2$; step size: $1.$ Toggling door number $2.$

i: $1$; j:$3$; step size: $1.$ Toggling door number $3.$

i: $1$; j:$4$; step size: $1.$ Toggling door number $4.$

i: $1$; j:$5$; step size: $1.$ Toggling door number $5.$

i: $1$; j:$6$; step size: $1.$ Toggling door number $6.$

i: $1$; j:$7$; step size: $1.$ Toggling door number $7.$

i: $1$; j:$8$; step size: $1.$ Toggling door number $8.$

i: $1$; j:$9$; step size: $1.$ Toggling door number $9.$

i: $1$; j:$10$; step size: $1.$ Toggling door number $10.$

i: $2$; j:$2$; step size: $2.$ Toggling door number $2.$

i: $2$; j:$4$; step size: $2.$ Toggling door number $4.$

i: $2$; j:$6$; step size: $2.$ Toggling door number $6.$

i: $2$; j:$8$; step size: $2.$ Toggling door number $8.$

i: $2$; j:$10$; step size: $2.$ Toggling door number $10.$

i: $3$; j:$3$; step size: $3.$ Toggling door number $3.$

i: $3$; j:$6$; step size: $3.$ Toggling door number $6.$

i: $3$; j:$9$; step size: $3.$ Toggling door number $9.$

i: $4$; j:$4$; step size: $4.$ Toggling door number $4.$

i: $4$; j:$8$; step size: $4.$ Toggling door number $8.$

i: $5$; j:$5$; step size: $5.$ Toggling door number $5.$

i: $5$; j:$10$; step size: $5.$ Toggling door number $10.$

i: $6$; j:$6$; step size: $6.$ Toggling door number $6.$

i: $7$; j:$7$; step size: $7.$ Toggling door number $7.$

i: $8$; j:$8$; step size: $8.$ Toggling door number $8.$

i: $9$; j:$9$; step size: $9.$ Toggling door number $9.$

i: $10$; j:$10$; step size: $10.$ Toggling door number $10.$

Algorithm has finished.

Door number $1$ remains open.

Door number $4$ remains open.

Door number $9$ remains open.

Door number $2$ remains closed.

Door number $3$ remains closed.

Door number $5$ remains closed.

Door number $6$ remains closed.

Door number $7$ remains closed.

Door number $8$ remains closed.

Door number $10$ remains closed.

Bonus Part $(25$ points$)$:

For all the open doors, i$,$ after the last pass i$.$e$.,$ N th pass, return a list where:

doors$[$i$]==$ "FizzBuzz" if i is divisible by $3$ and $5.$

doors$[$i$]==$ "Fizz" if i is divisible by $3.$

doors$[$i$]==$ "Buzz" if i is divisible by $5.$

doors$[$i$]==$ i if none of the above conditions are true.

Run $1$: When N $=20,$

N Doors Puzzle's Fizz Buzz Implementation: $[1,4,$ 'fizz', $16]$

Run $2$: When N $=100,$

N Doors Puzzle's Fizz Buzz Implementation: $[1,4,$ 'fizz', $16,$ 'buzz', 'fizz', $49,64,$ 'fizz', 'buzz'$]$

Run $3$: When N $=257,$

N Doors Puzzle's Fizz Buzz Implementation: $[1,4,$ 'fizz', $16,$ 'buzz', 'fizz', $49,64,$ 'fizz', 'buzz',

$121,$ 'fizz', $169,196,$ 'fizzbuzz', $25$