Besides striking the hour number at every full hour, this clock strikes once at every half hour. How many strikes does it make in one full day of 24 hours?

In 12 hours the hour strikes are 1 + 2 + … + 12 = 78, and there are 12 half-hour strikes: 90. A day has two such halves: $2\times 90=180$. (156 forgets the half-hours; 168 counts the half-hour strikes for only one half-day.)