int start = aux[0];
	int goal = aux[1];

	if (n_dsks == 1)
		moves.push_back(VI { aux[0], aux[1] });

	else {
	    int k = k_hanoi[n_twrs][n_dsks];

	    // moves top k disks from start to sparing tower
	    int sparing = aux[n_twrs-1];
        aux[1] = sparing;
        aux[n_twrs-1] = goal;
	    hanoi(moves, n_twrs, k, aux);

	    // moves bottom n-k disks from start to goal
	    aux[0] = start;
        aux[1] = goal;
	    hanoi(moves, n_twrs - 1, n_dsks - k, aux);

	    // moves original k disks from sparing tower to goal
        aux[0] = sparing;
        aux[n_twrs-1] = start;
	    hanoi(moves, n_twrs, k, aux);
	}