Although Wikipedia does a great job with the explanation, it was bothering me that I remembered so little about this problem of dubious use. So, I thought I would give it a more elaborate treatment as far as beating up on the recursive algorithm that is normally expressed. For this exercise, I will avoid the intellectual words used by many others on the topic, like "ordinal" and "canonical" and "parity" (technical people like to feel smart by obfuscation, if you haven't noticed!)...

public class Hanoi
{
	static void moveTopmostDisk(int from, int to)
	{
	  System.out.println("move disk: peg "+from+"--> peg "+to);
	}
	static void solve(int numDisks, int source, int spare, int destination)
	{
	  System.out.println("solve called with "+numDisks+" source="+source+" spare="+spare+" destination="+destination);
	  if(numDisks>0)
	  {
	    solve(numDisks-1,source,destination,spare);
	    moveTopmostDisk(source,destination);
	    solve(numDisks-1,spare,source,destination);
	  }
	}
	public static void main(String[] args)
	{
	  // toy problem for 3 disks
	  solve(3,1,2,3);
	}
}

// let's indent and start from the inside.
1.1 solve called with 3 source=1 spare=2 destination=3
	2.1 solve called with 2 source=1 spare=3 destination=2
		3.1(i.e. 2.1.1) solve called with 1 source=1 spare=2 destination=3
			4.1 solve called with 0 source=1 spare=3 destination=2
			// this one is trivial, it exits immediately.
			3.1.2 ***moveTopmostDisk from 1 to 3***
			3.1.3 call solve again with 0, 2, 1, 3
			// will exit immediately
		3 Iteration Done
		2.1.2 ***moveTopmostDisk from 1 to 2***
		2.1.3 call solve again with 1, 3, 1, 2
			2.1.3.1 call solve with 0, 3, 2, 1
			// this exits immediately
			2.1.3.2 ***moveTopmostDisk from 3 to 2***
			2.1.3.3 call solve with 0, 1, 3, 2
			// this exits immediately
	2 Iteration Done (we now have the small and middle disks on peg 2)
1.1.2 ***moveTopmostDisk from 1 to 3*** (we now have the bottom of the final stack)
1.1.3 call solve with 2, 2, 1, 3 // note that "2" is now the "source"
	1.1.3.1 call solve with 1, 2, 3, 1
		1.1.3.1.1 call solve with 0, 2, 1, 3
		// this exits immediately
		1.1.3.1.2 ***moveTopmostDisk from 2 to 1***
		1.1.3.1.3 call solve with 0, 3, 2, 1
		// ...which exits immediately
	1.1.3.2 ***moveTopmostDisk from 2 to 3*** (middle disk moves to final spot)
	1.1.3.3 call solve with 1, 1, 2, 3
		1.1.3.3.1 call solve with 0, 1, 3, 2
		// this exits immediately
		1.1.3.3.2 ***moveTopmostDisk from 1 to 3***
		1.1.3.3.3 call solve with 0, 2, 1, 3
		// ...which exits immediately
1 Iteration Done