(Incorrect) UMD 2018 High School Mathematics Problem

The hands will not overlap during the $12:00$ hour after $12:01$ . They will overlap each subsequent hour, but they will not overlap during the $11:00$ hour. This is a total of $10$ overlaps.

The hands will be directly across from each other during the $12:00$ hour and each subsequent hour. However, since the hands are directly across from each other at exactly $6:00$ , they will never be across from each other during the $5:00$ hour. Therefore, they are only directly across from each other $11$ times.

The answer is $10+11=\boxed{21}$ .

The minute hand advances by $5.5^{o}$ per minute relative to the hour hand, so that the hands form a straight line every $\frac{180}{5.5}\approx 32.73$ minutes. On the interval (Noon, Midnight], this happens $\frac{720}{180/5.5}=22$ times, so that it happens $\boxed{21}$ times strictly between Noon and Midnight.