Prove that:

$Q(n)=\{\begin{array}{cc}1,& n=1\end{array}$

$2R(n-1)+1,n>1$

$R(n)=\{\begin{array}{cc}2,& n=1\end{array}$

$Q(n)+Q(n-1)+1,n>1$

Hint $1$: To provide the proof you may need to first come up with a corresponding algorithm under the new clockwise condition. That should give you the mathematical recurrence shown above. Please illustrate your algorithm in the form of a figure following the example of the figure shown above in the question, for the original version of the problem.

Hint: If $R(n)$ is the number of moves required to move $n$ disks from peg A to peg C $($using B$),$ then how many moves will be needed to move $n$ disks from peg B to peg A $($using C$)?$