1. The Hanoi Tower Problem
Problem: There are n disks of different diameters on three pegs, labeled as Peg 1, Peg 2, as Peg 3. At the beginning, all n disks are on Peg 1, arranged in an increasing order of their diameter from top down.
The question is to move all disks to Peg 3 according to the following rules:
1. Each time, only one disk on top of a peg is moved to the top of another peg.
2. No disk can be put on top of a smaller disk.

2. Start
Peg 1
Peg 2
Peg 3
End
Peg 1
Peg 2
Peg 3

3. Solving Idea
The base case: If there is no disk (n = 0), nothing has to be done.
Reduction: Now suppose the number of disks is at least one. Ignore the largest disk, we have n – 1 disks. A smaller case.
Recursion: The smaller case can be solved by a recursive call to the same algorithm since this reduction can eventually be reduced to the base case.
Building up the solution: Move n – 1disks from the source peg to the intermediate peg, move one disk from the source peg to the destination peg, and finally, move n – 1 disks from the intermediate peg to the destination peg.

4. The Algorithm GIVEN: N (a positive integer)
S, I, D (the source peg, intermediate peg, and the destination peg)
RESULT: Output the steps of movements
HEADER
Hanoi (N, S, I, D)
BODY
test: N > 0?
false true
Ø Hanoi (N - 1, S, D, I)
output: "S -> D"
Hanoi (N - 1, I, S, D)

5. Java Implementation
public static void hanoi (int n, int s, int i, int d) {
if (n > 0)
hanoi (n - 1, s, d, i);
System.out.println (s + " -> " + d);
hanoi (n - 1, i, s, d); }

6. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)

7. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2)

8. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

9. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

10. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

11. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

12. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

13. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

14. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

15. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

16. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

17. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

18. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)

19. How it works when n = 3, s = 1, i = 2, d = 3?
hanoi(3,1,2,3)
hanoi(2,1,3,2) > 3 hanoi(2,2,1,3)
hanoi(1,1,2,3) 1 -> 2 hanoi(1,3,1,2) hanoi(1,2,3,1) 2 -> 3 hanoi(1,1,2,3)
hanoi(0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3) hanoi(0,1,3,2)
1 -> > > > 3
hanoi(0,2,1,3) hanoi (0,1,3,2) hanoi(0,3,2,1) hanoi(0,2,1,3)