One example for door 1000:
(Prime factorisation)

1000 = 2³ × 5³

You can write 1000 as the product of two numbers in a few ways:

1 x 1000
2¹ x (2 x 5³) = 2 x 500
2² x (2 x 5³) = 4 x 250
5 x (2³ x 5²) = 5 x 200
2³ x 5³ = 8 x 125
(2 x 5) x (2² x 5²) = 10 x 100
(2² x 5) x (2 x 5²) = 20 x 50
5² x (2³ x 5) = 25 x 40

1st student opens door 1000
2nd student closes it
4th student opens it
5th student closes it
...
...
500th student opens it.
1000th student closes it.

When written as the product of two whole numbers, there is an even number of unique factors.

So for every time the door is open/closed, it will be reversed.

If a number has an even number of unique factors, it will be in the closed state in the end.

Only those numbers with an odd number of factors will remain open in the end. They are the square numbers.

Example: door 64

64

= 1 x 64
= 2 x 32
= 4 x 16
= 8 x 8 = 8²

We notice that 8 is repeated so it doesn't have another unique factor to pair with it.

So the door's state is changed an odd number of times. It will remain open in the end.


The doors that remain open are:

1² = 1
2² = 4
3² = 9
4² = 16
...
...
...
31² = 961

Lockers*