I was using the beginner method to deliberately solve a 3X3 cube with one edge piece flipped. But, by the time I got to the top I discovered that I couldn't solve it with only one edge flipped. I had to also flip an edge on the top layer. Why is that?
12 edges on a cube. 24 positions an edge piece can be in once you consider flipping. If you consider one move to be a single quarter turn, then you can move an edge to exactly 12 of those 24 positions in an even number of moves, and to the other 12 only in an odd number of moves. You should be able to convince yourself of this by trial and error, but if you want proof you could come up with names for the 24 positions and check that each possible move only moves edges from a position in the even set to one in the odd set or vice versa.
Now consider what a single move does. It moves each of four edge pieces from one set of positions to the other. This means if you started with (say) 6 in one position-set and 6 in the other, you now have 2-10 or 4-8 or 6-6 or 8-4 or 10-2, but you can't have an odd number of pieces that changed position-sets.
In order to flip a single edge piece, you need to move a single piece to the other position set without moving any of the others. But you can only switch the position-sets of an even number of pieces at a time.