This is because the difference between two consecutive perfect squares is always odd. If you have a number n, then n+1 squared is: (n+1)^2 = n^2+2n+1. If you subtract n^2, our original number, you get n^2+2n+1-n^2 = (n^2-n^2)+ 2n + 1 = 2n+1. Any number of the form 2n+1 is odd, so the difference must always be an odd number.

Timeout error for my Python code.

I mean, my code is like 27 lines long so I'm not proud of it or anything,
but I do think I solved it

Because doing it that way you skip values. Use a tool like http://www.pythontutor.com/visualize.html#mode=edit to see it.

In the second one you're comparing a number to a list.

Whoa. Is there a name for the rule that this is expressing?