Editing Dynamic programming (section)

=== Tower of Hanoi puzzle ===
[[Image:Tower of Hanoi.jpeg|upright=1.2|thumb|A model set of the Towers of Hanoi (with 8 disks)]]
[[Image:Tower of Hanoi 4.gif|upright=1.2|thumb|An animated solution of the '''Tower of Hanoi''' puzzle for ''T(4,3)''.]]
The '''[[Tower of Hanoi]]''' or '''Towers of [[Hanoi]]''' is a [[mathematical game]] or [[puzzle]]. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

* Only one disk may be moved at a time.
* Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
* No disk may be placed on top of a smaller disk.

The dynamic programming solution consists of solving the [[Bellman equation|functional equation]]

: S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t)

where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and

: S(n, h, t) := solution to a problem consisting of n disks that are to be moved from rod h to rod t.

For n=1 the problem is trivial, namely S(1,h,t) = "move a disk from rod h to rod t" (there is only one disk left).

The  number of moves required by this solution is 2<sup>''n''</sup>&nbsp;&minus;&nbsp;1. If the objective is to '''maximize''' the number of moves (without cycling) then the dynamic programming [[Bellman equation|functional equation]]  is slightly more complicated and  3<sup>''n''</sup>&nbsp;&minus;&nbsp;1 moves are required.<ref>{{Citation |author=Moshe Sniedovich |title= OR/MS Games: 2. The Towers of Hanoi Problem |journal=INFORMS Transactions on Education |volume=3 |issue=1 |year=2002 |pages=34–51 |doi= 10.1287/ited.3.1.45 |postscript=.|doi-access=free }}</ref>