C++ :: Algorithm For All Possible Combinations?

A few days ago I got a "bright idea" to see if I could match a string, with an arbitrary length from 1 to 12, to its formulated sequence by using an algorithm to find all possible combinations of the integer combinations from 0 to 9 at each length (1 to 12).

Example: Desired numerical combinations from integers 1 to 3:

At Length 1:

1, 2, 3

At Length 2:

11, 12, 13, 21, 22, 23, 31, 32, 33

At Length 3:

111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333

And so on until the nth length (in my case a length of 12).

First off, I would like to say that this is not as easy as I thought. I clearly underestimated the problem seeing as I've spent hours attempting to write a working algorithm, but feel like I've made no progress.

Here are a few of my attempts:

Attempt 1:

#include <iostream>
#include <algorithm>
#include <math.h>
#include <string>

[Code]....

I can't exactly explain this one. It works if the length is 2 or less; however, the order of the output is horrendous.

Attempt 3: I tried using recursion, but only found myself getting more and more lost the further I tried developing my function. Cannot find my work for this attempt.

I would really like to figure this out on my own, but I am very stuck as you can see. I also lack time that I can spend working on this since im a full time student.