Logic Challenge

Ugh, I'm not good at this type of thing. The only thing I'm sure of is that all the prime-numbered doors would be closed. But that's easy.

Okay... so basically if something has an even number of factors, it's closed. If it has an odd number of factors, it's open. Not sure how you could devise a way to do figure that out without having to systematically go through each door and figure out if it's open or closed; which is something I think the million is trying to discourage.

To have an odd number of factors, it needs all its components to be primes raised to even powers. In other words, they're all squares. So the answer is 1000 (1, 4, 9, ... , 1000000) - that is, 1 to 1000 squared.

Why? Because the number of divisors of a number can be found as follows:
1) divide into prime factors
2) take the exponents of these, add 1
3) multiply together the results

For example,
36
1) 2^2 * 3^2
2) 3, 3
3) = 9

In order for this final product (number of divisors) to be odd, each prime has to be raised to an even power - 0, 2, 4, etc. so the second step is all odd numbers, so the third step is an odd number as well.
In other words, they're squares.