On the town hall tower, there are clocks that have maintenance doors near the center of the dial. However, these doors open outward, which is impractical - for example, at exactly 12:09, the minute hand will cover the doors, making them impossible to open until exactly 12:21.

![](https://cdn.mathpix.com/cropped/2024_04_17_3f052d529d7f6d768bd1g-3.jpg?height=366&width=388&top_left_y=1756&top_left_x=868)

How many minutes per day are the doors unable to be opened?

(Do not forget that the hour hand can also cover the doors; the entire door lies within the circle traced by the hour hand.)
Every hour, the door is blocked by the minute hand from the 9th to the 21st minute, i.e., 12 minutes. We need to determine exactly when the hour hand covers the door: at 9 minutes past the hour, the minute hand points to where the hour hand is at 1:48, and at 21 minutes past the hour, the minute hand points to where the hour hand is at 4:12. From 1:48 to 4:12, the door is covered by the hour hand, and similarly in the afternoon. Counting from midnight to the hours when the door cannot be opened:

- 0:09-0:21, i.e., 12 minutes,
- 1:09-1:21 and 1:48-2:00, i.e., 24 minutes,
- 2:00-3:00, i.e., 60 minutes,
- 3:00-4:00, i.e., 60 minutes,
- 4:00-4:21, i.e., 21 minutes,
- 5:09-5:21, i.e., 12 minutes,
- 6:09-6:21, i.e., 12 minutes,
- etc., up to noon, i.e., another $5 \cdot 12$ minutes.

From midnight to noon, the door cannot be opened for $8 \cdot 12 + 24 + 2 \cdot 60 + 21 = 261$ minutes, so for the entire day it is $2 \cdot 261 = 522$ minutes.