Editing Tower of Hanoi (section)

=== Cyclic Hanoi ===
In Cyclic Hanoi, we are given three pegs (A, B, C), which are arranged as a circle with the clockwise and the counterclockwise directions being defined as A – B – C – A and A – C – B – A, respectively. The moving direction of the disk must be clockwise.<ref>{{cite journal |first=T. D. |last=Gedeon |title=The Cyclic Towers of Hanoi: An Iterative Solution Produced by Transformation |journal=The Computer Journal |volume=39 |issue=4 |year=1996 |doi=10.1093/comjnl/39.4.353 |pages=353–356}}</ref> It suffices to represent the sequence of disks to be moved. The solution can be found using two mutually recursive procedures:

To move ''n'' disks '''counterclockwise''' to the neighbouring target peg:

# move ''n'' − 1 disks '''counterclockwise''' to the target peg
# move disk #''n'' one step clockwise
# move ''n'' − 1 disks '''clockwise''' to the start peg
# move disk #''n'' one step clockwise
# move ''n'' − 1 disks '''counterclockwise''' to the target peg

To move ''n'' disks '''clockwise''' to the neighbouring target peg:

# move ''n'' − 1 disks '''counterclockwise''' to a spare peg
# move disk #''n'' one step clockwise
# move ''n'' − 1 disks '''counterclockwise''' to the target peg

Let C(n) and A(n) represent moving n disks clockwise and counterclockwise, then we can write down both formulas:
{|
| || {{nowrap|1=C(n) = A(n−1) n A(n−1)}} || and || {{nowrap|1=A(n) = A(n−1) n C(n−1) n A(n−1).}}
|-
| style="padding:1ex 3ex;"|
|-
| Thus || C(1) = 1 || and || A(1) = 1 1,
|-
| || C(2) = 1 1 2 1 1 || and || A(2) = 1 1 2 1 2 1 1.
|}
The solution for the Cyclic Hanoi has some interesting properties:

# The move-patterns of transferring a tower of disks from a peg to another peg are symmetric with respect to the center points.
# The smallest disk is the first and last disk to move.
# Groups of the smallest disk moves alternate with single moves of other disks.
# The number of disks moves specified by C(n) and A(n) are minimal.