They're definitely not Turing machines, but they are Turing complete, they use tapes, and you can write to the tape. Some of the later puzzles ask you to solve

They're definitely not Turing machines, but they are Turing complete, they use tapes, and you can write to the tape. Some of the later puzzles ask you to solve problems that are impossible for a DFA (such as accept X blues followed by X reds (though a PDA could also do this)). Once you do the bottom branch you get the ability to read/write green and yellow, which largely increases what you can do.