[dj_mc_merlin]

Both. Intuition tells you which moves are even possible. It's like having a voice in your head that tells you "sacrifice the knight here", since the pattern of the pieces looks like it "has that move" in it. But you don't know how. This is where the algorithmic part comes in, you brute-force the most likely ways to do it by playing your best move, then your opponent's best move and so on. Your ability to win is defined by how many moves you see and how many moves ahead you can think.

Of course, it's more fuzzy than this. Intuition can be wrong. Sometimes the messages are along the lines of "there's some good move here", or "my opponent has mode some sort of mistake", rather than something definite. But when it all comes together, it feels like you're a pattern-matching wizard machine.