In a faulty clock, the hands are found to overlap every 64 minutes of correct time, as noted beside it. How much does the clock gain in a day?

In a correct clock the hands meet every $\frac{720}{11}=65\frac{5}{11}$ min. Here 64 true minutes pass while the dial runs $65\frac{5}{11}$ min, so it gains $\frac{16}{11}$ min per 64 min. In 1440 min: $\frac{16}{11}\times \frac{1440}{64}=32\frac{8}{11}$ min. (24 min uses a 65-minute gap of 1 min per meeting.)