The number of times in a day the Hour-hand and the Minute-hand of a clock at right angles is

A naive count gives two right-angle positions each hour → 24×2 = 48 in 24 hours. However, in each 12-hour cycle there are two hours (around 3:00 and 9:00) where the hands are perpendicular only once instead of twice, reducing the count by 2 for that 12-hour period (and by 4 for a full day). Thus the actual total is 48 − 4 = 44 occurrences in 24 hours. This conclusion (44 times per day) and the explanation about the exceptional hours are documented in clock-angle discussions and solutions.