Sequence generation/ breadth first searching

Essentially what I'm doing is trying to solve a Rubik's cube with a breadth first search of all possible moves. I know this isn't the best way to solve the cube, but I only need it for very short sequences (so the depth of the search is unlikely to be deeper than 3), and I don't need to store anything other than the current sequence.

I'm trying to find a way of printing out ever increasing strings of number (0,1,2,00,01,02...), so I can just plug each one into a function to check if that particular sequence of moves solves the cube, but I'm having trouble finding a way of continuing the sequence indefinitely.

So far all I've managed is nested for loops, but there needs to be another loop each time the search gets deeper. Does anyone have any idea how I can approach this problem?

Sorry if I've been too vague, I could write an essay on what I'm trying to do but thought I'd try and keep it simple.

I'm not very familiar with what's in the Java libraries, so apologies if this is implementing something that is already there, but if I were writing this from scratch, I'd probably do something like this:

public class Enumerator {
    private int maxDepth;
    private int currentDepth;
    private int currentPerm;
    private String alphabet;

    public Enumerator(String alphabet, int d) {
        this.maxDepth = d;
        this.currentDepth = 1;
        this.currentPerm = 0;
        this.alphabet = alphabet;
    }

    public boolean next() {
        int numPermutations = (int) Math.pow(alphabet.length(), this.currentDepth);
        boolean res=false;

        // finished if
        if ((this.currentDepth == this.maxDepth) && 
            (this.currentPerm == numPermutations - 1)) {
            res = false;
        }
        // next perm at this depth
        else if (this.currentPerm < numPermutations - 1) {
            this.currentPerm++;
            res = true;
        }
        // next depth
        else if (this.currentDepth <= this.maxDepth) {
            this.currentDepth++;
            this.currentPerm = 0;
            res = true;
        }
        return res;
    }

    public String getPermutation() {
        int tmpPerm = this.currentPerm;
        String res = "";
        for (int i=0; i<this.currentDepth; i++) {
          int ind = tmpPerm % this.alphabet.length();
          res = this.alphabet.charAt(ind) + res;
          tmpPerm /= this.alphabet.length();
        }
        return res;
    }

    public static void main(String args[]) {
        int depth = 3;
        String alphabet = "012";
        Enumerator e = new Enumerator(alphabet, depth); 
        do {
            System.out.println(e.getPermutation());
        } while (e.next());
    }
}

That way you can enumerate sequences from alphabets of arbitrary symbols to an arbitrary depth. This also does what you want insofar as it iterates over depth and for each depth generates the full set of possible sequences. It could also be done with recursion, as Gian says, which might be more elegant. In Python I'd use a generator function for this, but I'm not familiar with anything similar in Java.