Why the Full Moon Never Rises at Midnight
The sun always illuminates half of the moon. The phase of the moon that we see depends on how much of the illuminated surface is visible from earth. When the sun illuminates the same half of the moon that we see, the moon looks full.
That happens when the sun, earth, and moon are approximately lined up, with the earth between the sun and moon. So the moon and the sun are on opposite sides of our sky. When the moon is coming up, the sun is going down.
At midnight, the sun is approximately straight down, shining on the opposite side of the earth. If the moon is full, then at midnight it is straight up. It's not just starting to rise over the horizon.
According to H. D. Hoover, principal author of the Iowa Test of Basic Skills for 40 years, this question would not work in a high school standardized test because so few students would get it. About 25 percent of those who picked their answer randomly would make the right choice, and the total percentage right would probably not be much higher than that.
[Thanks to Noreen Grice of the Hayden Planetarium at the Museum of Science in Boston for helping with this explanation.]