diff --git a/tutorial_code/embed_loops.py b/tutorial_code/embed_loops.py
index 4b54e85..98ca7bf 100644
--- a/tutorial_code/embed_loops.py
+++ b/tutorial_code/embed_loops.py
@@ -22,10 +22,11 @@
 from minorminer.utils.parallel_embeddings import (
     find_sublattice_embeddings,
     embeddings_to_array,
+    lattice_size,
 )
 
 from minorminer.utils.feasibility import (
-    lattice_size,
+    embedding_feasibility_filter,
     lattice_size_lower_bound,
 )
 
@@ -75,6 +76,8 @@ def embed_loops(
     G = dimod.to_networkx_graph(bqm)
     A = sampler.to_networkx_graph()
 
+    if not embedding_feasibility_filter(S=G, T=A):
+        raise ValueError(f"Embedding {G} on {A} is infeasible")
     sublattice_size = kwargs.pop(
         "sublattice_size",
         min(lattice_size_lower_bound(S=G, T=A) + 1, lattice_size(T=A)),
@@ -90,7 +93,7 @@ def embed_loops(
         "\nTo accelerate the process a smaller lattice (L) might be "
         "considered and/or the search restricted to max_num_emb=1."
     )
-    max_num_emb = kwargs.pop("max_num_emb", float("Inf"))
+    max_num_emb = kwargs.pop("max_num_emb", G.number_of_nodes()//A.number_of_nodes())
     embedder_kwargs = {"timeout": kwargs.pop("timeout", 10)}
     embeddings = embeddings_to_array(
         find_sublattice_embeddings(
@@ -123,7 +126,7 @@ def embed_loops(
     return embeddings
 
 
-def main(max_num_emb=float("inf")):
+def main(max_num_emb=None):
     from time import perf_counter
 
     L = 2048  # L=2048 anticipate ~ 2.5 seconds on i7
diff --git a/tutorial_code/embed_square_lattice.py b/tutorial_code/embed_square_lattice.py
index eadd2bb..3c8c94c 100644
--- a/tutorial_code/embed_square_lattice.py
+++ b/tutorial_code/embed_square_lattice.py
@@ -22,11 +22,11 @@
 from minorminer.utils.parallel_embeddings import (
     find_sublattice_embeddings,
     embeddings_to_array,
+    lattice_size,
 )
 
 from minorminer.utils.feasibility import (
     embedding_feasibility_filter,
-    lattice_size,
     lattice_size_lower_bound,
 )
 
@@ -102,7 +102,7 @@ def embed_square_lattice(
             "\nTo accelerate the process a smaller lattice (L) might be "
             "considered and/or the search restricted to max_num_emb=1."
         )
-        max_num_emb = kwargs.pop("max_num_emb", float("Inf"))
+        max_num_emb = kwargs.pop("max_num_emb", 1)  # many can be slow
         embedder_kwargs = {"timeout": kwargs.pop("timeout", 10)}
         embeddings = embeddings_to_array(
             find_sublattice_embeddings(
@@ -135,7 +135,7 @@ def embed_square_lattice(
     return embeddings, bqm
 
 
-def main(max_num_emb=float("inf")):
+def main(max_num_emb=1):
     from time import perf_counter
 
     L = 10  # L=2048 anticipate ~ 14 seconds on i7
diff --git a/tutorial_code/example0_1_orbits.py b/tutorial_code/example0_1_orbits.py
index 4d0b09a..57c290a 100644
--- a/tutorial_code/example0_1_orbits.py
+++ b/tutorial_code/example0_1_orbits.py
@@ -11,12 +11,12 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
-import dimod
-import networkx as nx
 
+import networkx as nx
 from matplotlib import pyplot as plt
 
+import dimod
+
 from helpers.helper_functions import get_coupler_colors, get_qubit_colors
 from helpers import orbits
 
diff --git a/tutorial_code/example1_1_fm_loop_balancing.py b/tutorial_code/example1_1_fm_loop_balancing.py
index a705716..1d073e4 100644
--- a/tutorial_code/example1_1_fm_loop_balancing.py
+++ b/tutorial_code/example1_1_fm_loop_balancing.py
@@ -11,18 +11,19 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
 
-import dimod
+from typing import Optional
+
+from tqdm import tqdm
 import numpy as np
 
-from helpers.sampler_wrapper import ShimmingMockSampler
+import dimod
 from dwave.system.samplers import DWaveSampler
-from tqdm import tqdm
 
 from embed_loops import embed_loops
 from helpers.helper_functions import load_experiment_data, save_experiment_data
 from helpers.paper_plotting_functions import paper_plots_example1_1
+from helpers.sampler_wrapper import ShimmingMockSampler
 
 
 def make_fbo_dict(param: dict, shim: dict, embeddings: list) -> dict:
@@ -244,7 +245,7 @@ def main(
     coupling: float = -0.2,
     num_iters: int = 100,
     num_iters_unshimmed_flux: int = 10,
-    max_num_emb: float = float('Inf'),
+    max_num_emb: Optional[int] = None,
     use_cache: bool = True,
 ) -> None:
     """Main function to run example.
@@ -264,9 +265,9 @@ def main(
             of flux_biases. Defaults to 100.
         num_iters_unshimmed_J (int): Number of iterations without shimming of
             couplings. Defaults to 200.
-        max_num_emb (float): Maximum number of embeddings to use per programming.
-            Published tutorial data uses the maximum number the process can
-            accommodate.
+        max_num_emb (optional, int): Maximum number of embeddings to use per
+            programming. Published tutorial data uses the maximum number the
+            process can accommodate (defaults to max_num_emb=None).
         use_cache (bool): When True embeddings and data are read from
             (and saved to) local directories, repeated executions can reuse
             collected data. When False embeddings and data are recalculated on
diff --git a/tutorial_code/example1_2_fm_loop_correlations.py b/tutorial_code/example1_2_fm_loop_correlations.py
index c84f8fe..851294f 100644
--- a/tutorial_code/example1_2_fm_loop_correlations.py
+++ b/tutorial_code/example1_2_fm_loop_correlations.py
@@ -11,17 +11,19 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
-import dimod
+
+from typing import Optional
+
+from tqdm import tqdm
 import numpy as np
 
-from helpers.sampler_wrapper import ShimmingMockSampler
+import dimod
 from dwave.system.samplers import DWaveSampler
-from tqdm import tqdm
 
 from embed_loops import embed_loops
 from helpers.helper_functions import load_experiment_data, save_experiment_data
 from helpers.paper_plotting_functions import paper_plots_example1_2
+from helpers.sampler_wrapper import ShimmingMockSampler
 
 
 def make_fbo_dict(param: dict, shim: dict, embeddings: list) -> dict:
@@ -243,7 +245,7 @@ def main(
     num_iters: int = 300,
     num_iters_unshimmed_flux: int = 100,
     num_iters_unshimmed_J: int = 200,
-    max_num_emb: float = float('Inf'),
+    max_num_emb: Optional[int] = None,
     use_cache: bool = True,
 ) -> None:
     """Main function to run example.
@@ -259,9 +261,9 @@ def main(
         num_iters (int): Number of sequential programmings.
         num_iters_unshimmed_flux (int): Number of sequential programmings without flux shimming.
         num_iters_unshimmed_J (int): Number of sequential programmings without J shimming.
-        max_num_emb (float): Maximum number of embeddings to use per programming.
-            Published tutorial data uses the maximum number the process can
-            accommodate.
+        max_num_emb (optional, int): Maximum number of embeddings to use per
+            programming. Published tutorial data uses the maximum number the
+            process can accommodate (defaults to max_num_emb=None).
         use_cache (bool): When True embeddings and data are read from
             (and saved to) local directories, repeated executions can reuse
             collected data. When False embeddings and data are recalculated on
diff --git a/tutorial_code/example2_1_frustrated_loop_orbits.py b/tutorial_code/example2_1_frustrated_loop_orbits.py
index 0b37b2b..001994b 100644
--- a/tutorial_code/example2_1_frustrated_loop_orbits.py
+++ b/tutorial_code/example2_1_frustrated_loop_orbits.py
@@ -11,13 +11,13 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
-import dimod
+
 import matplotlib
 import networkx as nx
 import numpy as np
+import matplotlib.pyplot as plt
 
-from matplotlib import pyplot as plt
+import dimod
 
 from helpers import paper_plotting_functions, orbits
 
diff --git a/tutorial_code/example2_2_frustrated_loop_anneal.py b/tutorial_code/example2_2_frustrated_loop_anneal.py
index 33e09d7..dba60ff 100644
--- a/tutorial_code/example2_2_frustrated_loop_anneal.py
+++ b/tutorial_code/example2_2_frustrated_loop_anneal.py
@@ -11,18 +11,19 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
-import dimod
+
+from typing import Optional
+
+from tqdm import tqdm
 import numpy as np
 
-from helpers.sampler_wrapper import ShimmingMockSampler
+import dimod
 from dwave.system.samplers import DWaveSampler
-from tqdm import tqdm
 
 from embed_loops import embed_loops
 from helpers.helper_functions import load_experiment_data, save_experiment_data
 from helpers.paper_plotting_functions import paper_plots_example2_2
-
+from helpers.sampler_wrapper import ShimmingMockSampler
 
 def make_fbo_dict(param: dict, shim: dict, embeddings: list) -> dict:
     """Makes the FBO dict from the matrix of FBOs.
@@ -270,12 +271,12 @@ def run_experiment(
 
 
 def main(
-    solver_name: str = None,
+    solver_name: Optional[str] = None,
     coupling: float = -0.9,
     num_iters: int = 300,
     num_iters_unshimmed_flux: int = 100,
     num_iters_unshimmed_J: int = 200,
-    max_num_emb: float = float('Inf'),
+    max_num_emb: Optional[int] = None,
     use_cache: bool = True,
 ) -> None:
     """Main function to run example
@@ -296,9 +297,9 @@ def main(
             iteratrions that doesn't shim flux_biases. Defaults to 100.
         num_iters_unshimmed_J (int): option to specify number of iterations
             that doesn't shim alpha_J. Defaults to 200.
-        max_num_emb (float): Maximum number of embeddings to use per programming.
-            Published tutorial data uses the maximum number the process can
-            accommodate.
+        max_num_emb (optional, int): Maximum number of embeddings to use per
+            programming. Published tutorial data uses the maximum number the
+            process can accommodate (defaults to max_num_emb=None).
         use_cache (bool): When True embeddings and data are read from
             (and saved to) local directories, repeated executions can reuse
             collected data. When False embeddings and data are recalculated on
diff --git a/tutorial_code/example2_3_buckyball_orbits.py b/tutorial_code/example2_3_buckyball_orbits.py
index 2d4aea8..77ac311 100644
--- a/tutorial_code/example2_3_buckyball_orbits.py
+++ b/tutorial_code/example2_3_buckyball_orbits.py
@@ -14,12 +14,12 @@
 #
 from os.path import exists
 
-import dimod
 import matplotlib
 import networkx as nx
 import numpy as np
+import matplotlib.pyplot as plt
 
-from matplotlib import pyplot as plt
+import dimod
 
 from helpers import orbits
 from helpers.orbits import get_orbits
diff --git a/tutorial_code/example3_2_tafm_forward_anneal.py b/tutorial_code/example3_2_tafm_forward_anneal.py
index 5900e49..fa957b9 100644
--- a/tutorial_code/example3_2_tafm_forward_anneal.py
+++ b/tutorial_code/example3_2_tafm_forward_anneal.py
@@ -11,11 +11,13 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
-import dimod
-import numpy as np
+
+from typing import Optional
 
 from tqdm import tqdm
+import numpy as np
+
+import dimod
 
 from helpers.sampler_wrapper import ShimmingMockSampler
 from dwave.system.samplers import DWaveSampler
@@ -459,7 +461,7 @@ def main(
     num_iters: int = 800,
     num_iters_unshimmed_flux: int = 100,
     num_iters_unshimmed_J: int = 300,
-    max_num_emb: int = 1,
+    max_num_emb: Optional[int] = 1,
     use_cache: bool = True,
 ) -> None:
     """Main function to run example.
@@ -476,10 +478,10 @@ def main(
         num_iters (int): Number of sequential programmings.
         num_iters_unshimmed_flux (int): Number of sequential programmings without flux shimming.
         num_iters_unshimmed_J (int): Number of sequential programmings without J shimming.
-        max_num_emb (int): Maximum number of embeddings to use per programming.
-            Published tutorial data uses several parallel embeddings, but this
-            tutorial uses 1 (max_num_emb=1) by default to bypass the otherwise
-            slow search process.
+        max_num_emb (optional, int): Maximum number of embeddings to use per 
+            programming. Published tutorial data uses several parallel 
+            embeddings (value None, unbounded), but this refactored
+            defaults to 1 to bypass the otherwise slow search process.
         use_cache (bool): When True embeddings and data are read from
             (and saved to) local directories, repeated executions can reuse
             collected data. When False embeddings and data are recalculated on