We know that:

In case of iterative solution: Alternating between the smallest and the next-smallest disks, follow the steps for the appropriate case:

For an even number of disks:

make the legal move between pegs A and B
make the legal move between pegs A and C
make the legal move between pegs B and C
repeat until complete

For an odd number of disks:

make the legal move between pegs A and C
make the legal move between pegs A and B
make the legal move between pegs B and C
repeat until complete

In case of recursive solution: To move n discs from peg A to peg C:

move n-1 discs from A to B. This leaves disc n alone on peg A
move disc n from A to C
move n-1 discs from B to C so they sit on disc n

Now the questions are:

How did we get this two solutions?

Only by intuition?

Or by logical/mathematical computation?

If computation, how?