How many times are the hands of a clock at the right angle in a day?

Concept: The minute hand moves at a constant 6 degrees per minute and the hour hand at a constant 0.5 degrees per minute, so the minute hand gains on the hour hand at a fixed relative rate of 5.5 degrees per minute.

This constant relative-speed gain means the minute hand completes one full 360-degree relative lap around the hour hand once every 720/11 minutes, and within every such relative lap the angle between the hands passes through exactly 90 degrees twice - once while opening past a right angle on one side, once while closing back through a right angle on the other side.

1. In 12 hours (720 minutes), the number of complete relative laps the minute hand makes over the hour hand is 720 divided by (720/11) = 11 laps - not 12, because the hour hand keeps advancing too, so the minute hand needs slightly more than an hour to lap it each time.

2. Each of these 11 relative laps contains exactly 2 moments where the hands are at a right angle, so the right-angle count for any 12-hour half of the clock face is 11 times 2 = 22.

3. A full day covers two such 12-hour halves (00:00-12:00 and 12:00-24:00), so the right-angle count for the entire day is 22 times 2 = 44.

Cross-check: the same 11-relative-lap reasoning is the standard basis for the two companion clock-hand facts - the hands coincide (0 degrees) 11 times and are exactly opposite (180 degrees) 11 times in every 12-hour half, since a coincidence and an opposite-alignment each occur once per relative lap while a right angle occurs twice per lap. This consistent 11-lap structure across all three cases confirms 44 as the correct count of right angles across a full day.