Converting between Radian, Degree, and DMS Maze Activity - Classful - Free Printable

To solve this math maze, we need to convert degrees to radians and follow the path where the conversion is correct.

The Rule:
To convert from Degrees to Radians, multiply by $\frac{\pi}{180}$.
Formula: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Let's trace the path step-by-step starting from Start.

Step 1: Start at $30^\circ$
* Convert $30^\circ$ to radians:
$$30 \times \frac{\pi}{180} = \frac{30\pi}{180} = \frac{\pi}{6}$$
* Look at the arrows coming out of $30^\circ$:
* Arrow down to $15^\circ$ is labeled $\frac{\pi}{6}$. (This matches!)
* Arrow right to $150^\circ$ is labeled $\frac{2\pi}{3}$. (Incorrect, $150^\circ = \frac{5\pi}{6}$)
* Arrow diagonal to $60^\circ$ is labeled $\frac{\pi}{3}$. (Incorrect connection for the start)
* Path: Go to $15^\circ$.

Step 2: At $15^\circ$
* Convert $15^\circ$ to radians:
$$15 \times \frac{\pi}{180} = \frac{15\pi}{180} = \frac{\pi}{12}$$
* Look at the arrows coming out of $15^\circ$:
* Arrow left to $110^\circ$ is labeled $\frac{\pi}{13}$. (Incorrect)
* Arrow up to $30^\circ$ is labeled $\frac{\pi}{6}$. (Backwards)
* Arrow right to $60^\circ$ is labeled $\frac{\pi}{4}$. (Incorrect)
* Arrow diagonal down-right to $105^\circ$ is labeled $\frac{7\pi}{12}$. Let's check if the *next* step works from here, but usually, the label on the arrow indicates the value of the box you are leaving or the relationship. Wait, looking at the diagram structure, the label on the line seems to be the radian measure of the *source* box. Let's re-verify Step 1.
* Re-evaluating Step 1: The arrow from $30^\circ$ to $15^\circ$ has the label $\frac{\pi}{6}$. Since $30^\circ = \frac{\pi}{6}$, the label represents the radian value of the box you are *leaving*.
* Let's test this hypothesis on Step 2. We are at $15^\circ$. $15^\circ = \frac{\pi}{12}$.
* Arrows leaving $15^\circ$:
* To $110^\circ$: Label $\frac{\pi}{13}$? No.
* To $60^\circ$: Label $\frac{\pi}{4}$? No.
* To $105^\circ$: Label $\frac{7\pi}{12}$? No.
* Wait, let me look closer at the connections.
* From $15^\circ$, there is an arrow going DOWN to $105^\circ$? No, the arrow goes from $15^\circ$ to $105^\circ$? Let's check the label. The label between $15^\circ$ and $105^\circ$ is $\frac{7\pi}{12}$. That doesn't match $15^\circ$.
* Let's check the arrow from $15^\circ$ to $60^\circ$. The label is $\frac{\pi}{4}$. That doesn't match $15^\circ$.
* Let's check the arrow from $15^\circ$ to $110^\circ$. The label is $\frac{\pi}{13}$? No, it looks like $\frac{11\pi}{18}$? No, that's below $110$.
* Let's re-read the first step. $30^\circ \rightarrow \frac{\pi}{6}$. Correct. The arrow points to $15^\circ$.
* From $15^\circ$, where can we go?
* There is a path to $105^\circ$. The label on the segment is $\frac{7\pi}{12}$. This is the radian value for $105^\circ$ ($105 \cdot \frac{\pi}{180} = \frac{21\pi}{36} = \frac{7\pi}{12}$). So maybe the label is the value of the *destination*?
* Let's test "Label = Destination Radians" on Step 1. Start $30^\circ$. Arrow to $15^\circ$ has label $\frac{\pi}{6}$. But $15^\circ = \frac{\pi}{12}$. So that hypothesis fails.
* Let's test "Label = Source Radians" on Step 1. Start $30^\circ$. Arrow to $15^\circ$ has label $\frac{\pi}{6}$. $30^\circ = \frac{\pi}{6}$. This works.
* Let's re-examine the arrows from $15^\circ$.
* Arrow to $60^\circ$: Label $\frac{\pi}{4}$. Source is $15^\circ (\frac{\pi}{12})$. Mismatch.
* Arrow to $105^\circ$: Label $\frac{7\pi}{12}$. Source is $15^\circ$. Mismatch.
* Arrow to $110^\circ$: Label $\frac{\pi}{13}$? It's blurry. Let's look at other paths from Start.

Let's restart and look at ALL options from Start ($30^\circ$).
1. Path to $150^\circ$: Label $\frac{2\pi}{3}$. $30^\circ = \frac{\pi}{6}$. Mismatch.
2. Path to $60^\circ$: Label $\frac{\pi}{3}$. $30^\circ = \frac{\pi}{6}$. Mismatch.
3. Path to $15^\circ$: Label $\frac{\pi}{6}$. $30^\circ = \frac{\pi}{6}$. Match.

Okay, so we are definitely at $15^\circ$. Now, what is the correct exit from $15^\circ$?
The rule must be consistent. The label on the outgoing arrow equals the radian measure of the current box.
Current Box: $15^\circ = \frac{\pi}{12}$.
We need an arrow leaving $15^\circ$ with the label $\frac{\pi}{12}$.
Looking at the diagram:
- Arrow to $110^\circ$: Label looks like $\frac{\pi}{13}$ or something. Not $\frac{\pi}{12}$.
- Arrow to $60^\circ$: Label is $\frac{\pi}{4}$.
- Arrow to $105^\circ$: Label is $\frac{7\pi}{12}$.

Is it possible I misidentified the start?
Start points to $30^\circ$.
Maybe the label is the *difference*? No.
Maybe the label is the radian value of the *target*?
Let's re-test "Label = Target Radians".
Start $30^\circ$.
- To $15^\circ$: Label $\frac{\pi}{6}$. Target $15^\circ = \frac{\pi}{12}$. Mismatch.
- To $150^\circ$: Label $\frac{2\pi}{3}$. Target $150^\circ = \frac{5\pi}{6}$. Mismatch.
- To $60^\circ$: Label $\frac{\pi}{3}$. Target $60^\circ = \frac{\pi}{3}$. MATCH!

Let's pursue this path: Start $\rightarrow$ $60^\circ$.
Hypothesis: The number on the arrow is the radian measure of the box you are entering.

Step 1: Start at $30^\circ$. Move to $60^\circ$.
Check: Arrow label is $\frac{\pi}{3}$. $60^\circ = \frac{60\pi}{180} = \frac{\pi}{3}$. Correct.

Step 2: From $60^\circ$, where do we go?
We need an arrow leaving $60^\circ$ with a label equal to the radian measure of the destination.
Options from $60^\circ$:
1. To $150^\circ$: Label $\frac{5\pi}{6}$.
Check: $150^\circ = \frac{150\pi}{180} = \frac{5\pi}{6}$. Match.
2. To $-20^\circ$: Label $\frac{\pi}{2}$.
Check: $-20^\circ = -\frac{\pi}{9}$. Mismatch.
3. To $330^\circ$: Label $\frac{11\pi}{6}$.
Check: $330^\circ = \frac{11\pi}{6}$. Match.
4. To $15^\circ$: Label $\frac{\pi}{4}$.
Check: $15^\circ = \frac{\pi}{12}$. Mismatch.

We have two potential matches: $150^\circ$ and $330^\circ$. Let's trace both to see which one leads to the finish.

Path A: Via $150^\circ$
Current: $150^\circ$.
Options from $150^\circ$:
1. To $-954^\circ$: Label $-4.4\pi$? Or $-5.3\pi$? Let's calculate $-954^\circ$.
$-954 / 180 = -5.3$. So $-954^\circ = -5.3\pi$.
The arrow from $150^\circ$ to $-954^\circ$ has label $-5.3\pi$? No, the label between them is $-5.3\pi$? Let's look at the labels near $-954^\circ$.
Arrow from $150^\circ$ to $-954^\circ$: Label is $-5.3\pi$? Actually, looking at the layout:
$150^\circ \xrightarrow{-5.3\pi} -954^\circ$?
Let's check the label on the arrow connecting $150^\circ$ and $-954^\circ$. It says $-5.3\pi$? No, that's between $-954$ and $-756$.
Between $150^\circ$ and $-954^\circ$, the label is $-5.3\pi$? Wait, $150^\circ$ is connected to $-954^\circ$ via an arrow pointing right? No, $150^\circ$ connects to $-954^\circ$?
Let's look at the grid.
Row 1: $30, 150, -954, -756$.
Arrow $150 \rightarrow -954$. Label is $-5.3\pi$?
Calculation: $-954^\circ = -5.3 \pi$.
If the rule is "Label = Target Radians", then the label should be $-5.3\pi$.
Let's assume we go to $-954^\circ$.

From $-954^\circ$:
Options:
1. To $-756^\circ$: Label $-4.2\pi$.
Check: $-756 / 180 = -4.2$. So $-756^\circ = -4.2\pi$. Match.
2. To $-20^\circ$: Label $-6.3\pi$? No.

So Path A continues: $-954^\circ \rightarrow$ $-756^\circ$.

From $-756^\circ$:
Options:
1. To $-240^\circ$: Label $-4.2\pi$? No, label is $-4.2\pi$ on the incoming arrow. Outgoing arrow to $-240^\circ$ has label $-4\pi/3$?
Let's calculate $-240^\circ$.
$-240 \times \frac{\pi}{180} = -\frac{4\pi}{3}$.
The arrow from $-756^\circ$ to $-240^\circ$ has label $-\frac{4\pi}{3}$. Match.

So Path A continues: $-756^\circ \rightarrow$ $-240^\circ$.

From $-240^\circ$:
Options:
1. To $-420^\circ$: Label $-\frac{7\pi}{3}$?
Calculate $-420^\circ$: $-420 / 180 = -2.333... = -7/3$. So $-420^\circ = -\frac{7\pi}{3}$.
The arrow label is $-\frac{7\pi}{3}$? The image shows $-\frac{7\pi}{3}$? Or maybe $-2.3\pi$?
Let's look at the arrow from $-240^\circ$ to $-420^\circ$. The label is $-\frac{7\pi}{3}$?
Actually, looking at the bottom right, there is a Finish sign.
Let's check the other option from $-240^\circ$.
2. To $-20^\circ$? No connection.
3. To $-97^\circ 30'$? No.

Let's check the arrow from $-240^\circ$ to $-420^\circ$.
Label: $-\frac{7\pi}{3}$.
Target: $-420^\circ = -\frac{7\pi}{3}$. Match.

So Path A continues: $-240^\circ \rightarrow$ $-420^\circ$.

From $-420^\circ$:
Options:
1. To Finish? There is an arrow pointing to the Finish guy.
2. To $-97^\circ 30'$?

Let's look at the Finish. The Finish box is pointed to by an arrow from $-420^\circ$?
The label on the arrow from $-420^\circ$ to Finish is $-2.3\pi$?
Wait, the Finish isn't a degree value. The problem says "Follow the flow... to 'finish'".
Usually, the last step lands on a box that connects to Finish.
Let's check the arrow from $-420^\circ$ to the Finish figure.
The label is $-2.3\pi$?
$-420^\circ = -2.33\pi$.
Is there another path?

Let's re-evaluate Path B from Step 2 to ensure Path A is the unique solution.

Path B: Via $330^\circ$
Recall Step 2: From $60^\circ$, we could go to $330^\circ$.
Check: Arrow label $\frac{11\pi}{6}$. Target $330^\circ = \frac{11\pi}{6}$. Match.

Current: $330^\circ$.
Options from $330^\circ$:
1. To $-97^\circ 30'$: Label $\frac{52\pi}{180}$? Or something else.
Let's calculate $-97^\circ 30'$.
$30' = 0.5^\circ$. So $-97.5^\circ$.
$-97.5 \times \frac{\pi}{180} = -\frac{97.5}{180}\pi = -\frac{195}{360}\pi = -\frac{39}{72}\pi = -\frac{13}{24}\pi$.
The label on the arrow from $330^\circ$ to $-97^\circ 30'$ is $\frac{52\pi}{180}$? No, that simplifies to $\frac{13\pi}{45}$. Doesn't match.
Also, the sign is wrong. $330$ is positive, target is negative.
2. To $630^\circ$: Label $\frac{7\pi}{2}$.
Calculate $630^\circ$: $630 / 180 = 3.5 = 7/2$. So $630^\circ = \frac{7\pi}{2}$. Match.
3. To $105^\circ$: Label $\frac{11\pi}{12}$?
Calculate $105^\circ$: $\frac{7\pi}{12}$. Mismatch.
4. To $52^\circ 30'$: Label $\frac{7\pi}{24}$?
Calculate $52.5^\circ$: $52.5 / 180 = 105 / 360 = 7 / 24$. So $52.5^\circ = \frac{7\pi}{24}$. Match.

So from $330^\circ$, we have two matches: $630^\circ$ and $52^\circ 30'$.

Path B1: Via $630^\circ$
Current: $630^\circ$.
Options from $630^\circ$:
1. To $315^\circ$: Label $\frac{7\pi}{4}$.
Calculate $315^\circ$: $315 / 180 = 1.75 = 7/4$. So $315^\circ = \frac{7\pi}{4}$. Match.
2. To $52^\circ 30'$: Label $\frac{7\pi}{24}$? No, that's from $330$.

So Path B1 continues: $630^\circ \rightarrow$ $315^\circ$.

From $315^\circ$:
Options:
1. To $20^\circ 15'$: Label $\frac{9\pi}{80}$?
Calculate $20^\circ 15' = 20.25^\circ$.
$20.25 / 180 = 81 / 720 = 9 / 80$. So $20.25^\circ = \frac{9\pi}{80}$. Match.
2. To Finish? No direct arrow.

So Path B1 continues: $315^\circ \rightarrow$ $20^\circ 15'$.

From $20^\circ 15'$:
Options:
1. To $22^\circ 30'$: Label $\frac{7\pi}{80}$?
Calculate $22.5^\circ$.
$22.5 / 180 = 1/8$. So $\frac{\pi}{8}$.
Label is $\frac{7\pi}{80}$? Mismatch.
2. Any other exit?
Looking at the box $20^\circ 15'$, it seems to be a dead end or I missed an arrow.
There is an arrow from $22^\circ 30'$ TO $20^\circ 15'$?
Let's check the arrow direction.
Arrow from $22^\circ 30'$ down to $20^\circ 15'$ has label $\frac{7\pi}{80}$?
$22.5^\circ = \frac{\pi}{8} = \frac{10\pi}{80}$. Mismatch.

Let's re-read the arrow labels around $20^\circ 15'$.
Arrow from $315^\circ$ to $20^\circ 15'$: Label $\frac{9\pi}{80}$. Correct.
Is there an arrow OUT of $20^\circ 15'$?
It looks like there is no arrow leaving $20^\circ 15'$ towards the finish. The finish is on the right side. $20^\circ 15'$ is on the bottom left. This path seems to lead to a dead end.

Path B2: Via $52^\circ 30'$
Recall from $330^\circ$, we could go to $52^\circ 30'$.
Current: $52^\circ 30'$.
Location: Bottom row, second from right.
Options from $52^\circ 30'$:
1. To Finish: There is an arrow pointing directly to the "Finish" sign.
Let's verify the label on this arrow.
The label is $\frac{7\pi}{24}$.
Does this match the rule "Label = Target Radians"?
Target is "Finish". "Finish" is not a degree value.
However, in these mazes, sometimes the last arrow's label corresponds to the *source* box's radian value, or the path just ends.
Let's check the previous step's logic.
In Path A, the last step was $-420^\circ \rightarrow$ Finish.
Label was $-2.3\pi$?
$-420^\circ = -2.33\pi$.

Let's look at the arrow from $52^\circ 30'$ to Finish.
Label: $\frac{7\pi}{24}$.
Source: $52^\circ 30' = 52.5^\circ$.
$52.5 \times \frac{\pi}{180} = \frac{7\pi}{24}$.

So, for the final step, the label matches the Source box's radian value.
Let's re-evaluate Path A's final step with this new insight.
Path A ended at $-420^\circ$.
Arrow to Finish label: $-2.3\pi$? Or $-7\pi/3$?
$-420^\circ = -7\pi/3 \approx -2.33\pi$.
If the label is $-7\pi/3$, it matches the source.

So we have two competing paths that reach the Finish area:
1. Path A: $30 \rightarrow 60 \rightarrow 150 \rightarrow -954 \rightarrow -756 \rightarrow -240 \rightarrow -420 \rightarrow$ Finish.
2. Path B2: $30 \rightarrow 60 \rightarrow 330 \rightarrow 52^\circ 30' \rightarrow$ Finish.

Which one is correct?
Let's check the intermediate steps of Path A again to ensure no errors.

Re-verifying Path A:
1. $30^\circ \rightarrow 60^\circ$. Label $\frac{\pi}{3}$. Target $60^\circ = \frac{\pi}{3}$. OK.
2. $60^\circ \rightarrow 150^\circ$. Label $\frac{5\pi}{6}$. Target $150^\circ = \frac{5\pi}{6}$. OK.
3. $150^\circ \rightarrow -954^\circ$. Label $-5.3\pi$. Target $-954^\circ = -5.3\pi$. OK.
4. $-954^\circ \rightarrow -756^\circ$. Label $-4.2\pi$. Target $-756^\circ = -4.2\pi$. OK.
5. $-756^\circ \rightarrow -240^\circ$. Label $-\frac{4\pi}{3}$. Target $-240^\circ = -\frac{4\pi}{3}$. OK.
6. $-240^\circ \rightarrow -420^\circ$. Label $-\frac{7\pi}{3}$. Target $-420^\circ = -\frac{7\pi}{3}$. OK.
7. $-420^\circ \rightarrow$ Finish. Label $-2.3\pi$?
Wait, look at the arrow from $-420^\circ$ to Finish.
The label is actually $-2.3\pi$?
$-420^\circ = -2.333\pi$.
$-2.3\pi$ is $-414^\circ$.
This is a slight mismatch. $-2.3 \neq -2.33$.

Let's re-read the label on the arrow from $-420^\circ$ to Finish.
It might be $-\frac{7\pi}{3}$ written as a decimal? $-7/3 = -2.333...$
Or is the label $-2.3\pi$ referring to something else?

Let's re-verify Path B2.
1. $30^\circ \rightarrow 60^\circ$. OK.
2. $60^\circ \rightarrow 330^\circ$. Label $\frac{11\pi}{6}$. Target $330^\circ = \frac{11\pi}{6}$. OK.
3. $330^\circ \rightarrow 52^\circ 30'$. Label $\frac{7\pi}{24}$. Target $52.5^\circ = \frac{7\pi}{24}$. OK.
4. $52^\circ 30' \rightarrow$ Finish. Label $\frac{7\pi}{24}$.
Here, the label matches the Source ($52.5^\circ$).

Why did the rule switch from "Label = Target" to "Label = Source"?
Let's look at the very first step again.
$30^\circ \rightarrow 60^\circ$. Label $\frac{\pi}{3}$.
Source $30^\circ = \frac{\pi}{6}$. Target $60^\circ = \frac{\pi}{3}$.
So Step 1 followed "Label = Target".

Let's look at the last step of Path A again.
$-420^\circ \rightarrow$ Finish.
If the rule is "Label = Target", what is the target? Finish.
If the rule is "Label = Source", Source is $-420^\circ = -7\pi/3 \approx -2.33\pi$.
The label looks like $-2.3\pi$. This is an approximation.

Let's look at the last step of Path B2 again.
$52^\circ 30' \rightarrow$ Finish.
Label $\frac{7\pi}{24}$.
Source $52.5^\circ = \frac{7\pi}{24}$.
So Step 4 follows "Label = Source".

Is it possible the rule changes? Or did I misinterpret an earlier step?

Let's check the step $330^\circ \rightarrow 52^\circ 30'$ in Path B2.
Label $\frac{7\pi}{24}$.
Source $330^\circ = \frac{11\pi}{6} = \frac{44\pi}{24}$.
Target $52.5^\circ = \frac{7\pi}{24}$.
So this step followed "Label = Target".

So Path B2:
Step 1: Label = Target ($60^\circ$)
Step 2: Label = Target ($330^\circ$)
Step 3: Label = Target ($52.5^\circ$)
Step 4: Label = Source ($52.5^\circ$) ??

This inconsistency is suspicious. Let's look at Path A's last step again.
Step 6: $-240^\circ \rightarrow -420^\circ$.
Label $-\frac{7\pi}{3}$.
Source $-240^\circ = -\frac{4\pi}{3}$.
Target $-420^\circ = -\frac{7\pi}{3}$.
So Step 6 followed "Label = Target".

Step 7: $-420^\circ \rightarrow$ Finish.
Label $-2.3\pi$ (approx).
Source $-420^\circ = -2.33\pi$.
If the rule holds as "Label = Target", does Finish have a value? No.
If the rule switches to "Label = Source" for the exit, then Path A works (with rounding).

Let's check if there is a path that consistently uses "Label = Source" throughout?
Start $30^\circ (\frac{\pi}{6})$.
Arrow to $15^\circ$ has label $\frac{\pi}{6}$. Matches Source.
Let's trace this "Source" path.

Path C: Label = Source Rule
1. Start $30^\circ$. Arrow to $15^\circ$ has label $\frac{\pi}{6}$. Source $30^\circ=\frac{\pi}{6}$. Match.
2. At $15^\circ (\frac{\pi}{12})$.
Need arrow with label $\frac{\pi}{12}$.
- To $110^\circ$: Label $\frac{\pi}{13}$? No.
- To $60^\circ$: Label $\frac{\pi}{4}$? No.
- To $105^\circ$: Label $\frac{7\pi}{12}$? No.
- To $22^\circ 30'$? No direct link.
Wait, look at $15^\circ$ again.
Is there an arrow to $22^\circ 30'$? No.
Is there an arrow to $105^\circ$? Yes. Label $\frac{7\pi}{12}$.
Is there an arrow to $60^\circ$? Yes. Label $\frac{\pi}{4}$.
Is there an arrow to $110^\circ$? Yes. Label $\frac{\pi}{13}$?

Let's look really closely at the arrow from $15^\circ$ to $105^\circ$.
Label is $\frac{7\pi}{12}$.
Source $15^\circ = \frac{\pi}{12}$. Mismatch.

What about the arrow from $15^\circ$ to $60^\circ$?
Label $\frac{\pi}{4}$.
Source $15^\circ = \frac{\pi}{12}$. Mismatch.

What about the arrow from $15^\circ$ to $110^\circ$?
Label $\frac{11\pi}{18}$? No, that's the label for $110^\circ$ itself ($110 = 11\pi/18$).
The label on the arrow FROM $110^\circ$ TO $22^\circ 30'$ is $\frac{11\pi}{18}$?

Let's look at the box $110^\circ$.
$110^\circ = \frac{11\pi}{18}$.
Arrow leaving $110^\circ$ to $22^\circ 30'$ has label $\frac{11\pi}{18}$.
This matches "Label = Source".

So, IF we can get to $110^\circ$, the "Label = Source" rule works.
How to get to $110^\circ$?
From $15^\circ$? Label was $\frac{\pi}{13}$? No.
From $30^\circ$? No.

Let's check the arrow INTO $110^\circ$.
It comes from $15^\circ$.
Label is $\frac{\pi}{13}$?
If the rule is "Label = Target", then Target $110^\circ = \frac{11\pi}{18}$.
Label $\frac{\pi}{13} \neq \frac{11\pi}{18}$.

So Path C dies at Step 2.

Therefore, the "Label = Target" rule seems the most robust for the main path.

Let's compare Path A and Path B2 again.

Path A: Long path through negative numbers.
Path B2: Shorter path.

Let's check the label on the final arrow of Path B2 ($52^\circ 30' \rightarrow$ Finish).
Label: $\frac{7\pi}{24}$.
Source: $52.5^\circ = \frac{7\pi}{24}$.

Let's check the label on the final arrow of Path A ($-420^\circ \rightarrow$ Finish).
Label: $-2.3\pi$?
Source: $-420^\circ = -2.33\pi$.

In both cases, the final arrow label matches the Source.
In all previous steps, the arrow label matched the Target.

This implies the rule is:
- For moves between degree boxes: Label = Radians of the Destination Box.
- For the move to Finish: Label = Radians of the Source Box (since Finish has no degree value).

Both Path A and Path B2 satisfy this hybrid rule.
Which one is the intended path?
Usually, mazes have a single unique path. Did I miss a mismatch in one of them?

Let's re-check Path A Step 3: $150^\circ \rightarrow -954^\circ$.
Label: $-5.3\pi$.
Target: $-954^\circ$.
$-954 / 180 = -5.3$.
$-5.3 \times 180 = -954$.
Exact match.

Let's re-check Path A Step 4: $-954^\circ \rightarrow -756^\circ$.
Label: $-4.2\pi$.
Target: $-756^\circ$.
$-756 / 180 = -4.2$.
Exact match.

Let's re-check Path A Step 5: $-756^\circ \rightarrow -240^\circ$.
Label: $-\frac{4\pi}{3}$.
Target: $-240^\circ$.
$-240 / 180 = -4/3$.
Exact match.

Let's re-check Path A Step 6: $-240^\circ \rightarrow -420^\circ$.
Label: $-\frac{7\pi}{3}$.
Target: $-420^\circ$.
$-420 / 180 = -7/3$.
Exact match.

Path A is mathematically perfect until the finish.

Let's re-check Path B2 Step 3: $330^\circ \rightarrow 52^\circ 30'$.
Label: $\frac{7\pi}{24}$.
Target: $52.5^\circ$.
$52.5 / 180 = 0.291666...$
$7 / 24 = 0.291666...$
Exact match.

Path B2 is also mathematically perfect.

Is there a constraint I missed?
"Begin by answering the problem at start."

Let's look at the visual layout.
Path A goes: Right, Right, Down, Left, Down, Right, Down-Right.
Path B2 goes: Right, Down-Right, Down-Right, Right.

Often, these mazes cover more of the grid. Path A covers 7 boxes. Path B2 covers 4 boxes.
However, look at the arrow from $60^\circ$ to $330^\circ$.
And the arrow from $60^\circ$ to $150^\circ$.

Let's check if there is an error in my rejection of other branches.
From $60^\circ$, I rejected $-20^\circ$ (Label $\frac{\pi}{2}$, Target $-\frac{\pi}{9}$). Correct.
From $60^\circ$, I rejected $15^\circ$ (Label $\frac{\pi}{4}$, Target $\frac{\pi}{12}$). Correct.

From $330^\circ$, I rejected $-97^\circ 30'$ (Label $\frac{52\pi}{180}$? Target $-\frac{13\pi}{24}$). Correct.
From $330^\circ$, I rejected $105^\circ$ (Label $\frac{11\pi}{12}$? Target $\frac{7\pi}{12}$). Correct.

From $150^\circ$, I rejected $-20^\circ$ (Label $-6.3\pi$? No, label is on arrow from $-954$ to $-20$?).
Let's check the arrow from $150^\circ$ to $-20^\circ$.
Label is $\frac{\pi}{2}$? No, that's from $60$.
The arrow from $150^\circ$ goes to $-954^\circ$ and $-20^\circ$?
Looking at the lines:
$150^\circ$ connects to $-954^\circ$ (Right) and $-20^\circ$ (Down-Right)?
Label on $150 \rightarrow -20$: $\frac{\pi}{2}$?
Target $-20^\circ = -\frac{\pi}{9}$. Mismatch.

So both paths seem valid. Is there a subtle error?

Let's look at the Start again.
$30^\circ \rightarrow 60^\circ$.
What about $30^\circ \rightarrow 15^\circ$?
Label $\frac{\pi}{6}$.
If Rule = Target, Target $15^\circ = \frac{\pi}{12}$. Mismatch.
If Rule = Source, Source $30^\circ = \frac{\pi}{6}$. Match.

If the rule is "Label = Source", then Path C ($30 \rightarrow 15$) was the start.
But Path C died at $15^\circ$ because no outgoing arrow had label $\frac{\pi}{12}$.
UNLESS...
Look at $15^\circ$ outgoing arrows again.
To $105^\circ$: Label $\frac{7\pi}{12}$.
To $60^\circ$: Label $\frac{\pi}{4}$.
To $110^\circ$: Label $\frac{\pi}{13}$?

Wait, look at the arrow from $15^\circ$ to $105^\circ$.
Is it possible the label is $\frac{\pi}{12}$?
No, it clearly says $\frac{7\pi}{12}$.

So "Label = Source" is not the global rule.

Why would Path B2 be preferred over Path A?
Path A involves large negative numbers. Path B2 involves smaller positive numbers.
However, Path A is longer.

Let's check the very last arrow of Path B2 again.
$52^\circ 30' \rightarrow$ Finish.
Label $\frac{7\pi}{24}$.

Let's check the very last arrow of Path A again.
$-420^\circ \rightarrow$ Finish.
Label $-2.3\pi$?
$-420^\circ = -7\pi/3$.
$-7/3 = -2.333...$
Writing $-2.3\pi$ is an approximation. Writing $\frac{7\pi}{24}$ is exact.
Math problems usually prefer exact answers.
The approximation in Path A suggests it might be a distractor or I am misreading the label.
Could the label be $-\frac{7\pi}{3}$?
If it were $-\frac{7\pi}{3}$, it would be exact.
But visually, it looks like $-2.3\pi$.

Also, consider the complexity.
Path B2: $30 \rightarrow 60 \rightarrow 330 \rightarrow 52.5 \rightarrow$ Finish.
Conversions:
$60 = \pi/3$.
$330 = 11\pi/6$.
$52.5 = 7\pi