In this clock the hands meet every 64 minutes of correct time. How much does the clock gain in a day?

In a correct clock the hands meet every 720/11 = 65 5/11 min. Here they meet after only 64 true minutes, so the clock gains 65 5/11 − 64 = 16/11 min every 64 min. In 1440 min: $16/11\times 1440/64=360/11=32$ 8/11 min. (1 5/11 min is the gain in one 64-minute stretch.)