The best way to avoid false moves in this puzzle
is to move the smallest disk from one column to the next and then any
disk other than the smallest. Although such a recipe seems arbitrary,
it ensures that there will always be one legal move. And repeating the
pattern over and over will miraculously bring you to the solution.
There is some deep connection between the cyclical movements of the
disks and the mathematical underpinnings of this puzzle.
For Puzzle 1, which has the restriction against placing disk 1 on disk
4, nineteen moves are required.
For Puzzle 2, which has restrictions against placing disk 1 on disk 3,
and disk 2 on disk 4, the minimum number of moves required is only
fifteen - the same as if there were no restrictions.