An alarm clock runs 4 minutes slow every hour. It was set right $ 3 \frac{1}{2}$ hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
1. The alarm clock runs 4 minutes slow every hour. This means that for every 60 minutes of real time, the alarm clock only counts 56 minutes.
2. The clock was set correctly 3.5 hours ago. We need to convert this time into minutes:
   \[
   3.5 \text{ hours} = 3.5 \times 60 = 210 \text{ minutes}
   \]
3. Since the alarm clock runs slow, we need to determine how many minutes the alarm clock will show after 210 minutes of real time. The clock counts \(\frac{56}{60}\) minutes for every real minute:
   \[
   \text{Minutes counted by the alarm clock} = 210 \times \frac{56}{60} = 210 \times \frac{14}{15}
   \]
4. Calculate the above expression:
   \[
   210 \times \frac{14}{15} = 210 \times \frac{14}{15} = 14 \times 14 = 196 \text{ minutes}
   \]
5. The alarm clock will show 196 minutes after it was set correctly. Since it was set correctly at 8:30 AM, we need to add 196 minutes to 8:30 AM to find out what time the alarm clock shows when the correct clock shows noon.
6. Convert 196 minutes to hours and minutes:
   \[
   196 \text{ minutes} = 3 \text{ hours and } 16 \text{ minutes}
   \]
7. Add this time to 8:30 AM:
   \[
   8:30 \text{ AM} + 3 \text{ hours and } 16 \text{ minutes} = 11:46 \text{ AM}
   \]
8. Therefore, when the correct clock shows noon, the alarm clock will show 11:46 AM. To find out how many minutes until the alarm clock shows noon:
   \[
   12:00 \text{ PM} - 11:46 \text{ AM} = 14 \text{ minutes}
   \]

The final answer is \(\boxed{14}\) minutes.