Move all disks to the right peg. A larger disk can never sit on a smaller one.
Three pegs hold a stack of 5 disks (large to small) on the leftmost peg.
Tap a peg to lift its top disk, then tap another peg to drop it there.
A disk can only land on an empty peg or on a larger disk.
Goal: move all disks to the rightmost peg, as fast as possible.
The Tower of Hanoi has a perfectly mechanical solution, so once you see the pattern you never have to think again. The smallest disk moves on every other turn, and it should always travel in the same circular direction — for an odd number of disks send it right-to-left wrapping around, and the puzzle solves itself if you simply never make an illegal move in between.
Between each move of the smallest disk there is exactly one other legal move available, and you should always take it. If you ever feel stuck, you have almost certainly moved the small disk in the wrong direction at some point. Since the score rewards speed and the minimum number of moves, settling into that "small disk, only-other-move, small disk" rhythm beats hesitating over every choice.