During nighttime operations, a vessel that has a mechanical failure and is unable to maneuver must show __________.

In nighttime operations, a vessel that experiences a mechanical failure and is unable to maneuver must display an all-round white light. This is essential for alerting other vessels to its condition, indicating that it is anchored or not under command. The presence of an all-round white light serves a dual purpose: it signals to others that the vessel is unable to safely operate and helps to ensure that it is visible, reducing the risk of collision.

The other options do not meet the requirements for signaling a vessel that cannot maneuver. For instance, a green light is typically used to indicate a vessel that is under way and is a specific type of navigation light. Similarly, one red light is not a standard distress signal, and two all-round green lights would suggest a vessel engaged in fishing or other specific activities, which could mislead oncoming traffic. The all-round white light is the established and recognized signal for a vessel in this scenario, making it the appropriate choice.