Solution
The Quarter-Turn Hands
Show the problem again
On an ordinary analogue clock the hour and minute hands move at constant speeds. Counting a full 24-hour day starting at midnight (00:00 inclusive, the following midnight excluded), how many times are the hour hand and the minute hand exactly perpendicular? Give an integer.
Worked solution
Measure angles in degrees from 12. The minute hand moves 6° per minute and the hour hand 0.5° per minute, so the minute hand gains 5.5° per minute on the hour hand. The hands are perpendicular whenever the gap is 90° or 270° modulo 360°. Each lap of the gap (360°) therefore contains exactly two perpendicular moments. The minute hand laps the hour hand 11 times in 12 hours, giving 22 perpendicular moments per 12 hours, hence **44** per day. An exhaustive count over t = (90 + 180k)/5.5 minutes with 0 ≤ t < 1440 confirms 44.
Source: Classic clock-hand counting exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.