How many times in a day do the hands of a clock coincide, and how many times are they at right angles?
They coincide 22 times in 24 hours (11 times every 12 hours, since the coincidence between 11 and 1 happens only once, at 12). They are at right angles 44 times a day and opposite each other 22 times.
Explain it simply
In 12 hours the minute hand goes round 12 times and the hour hand once, so the minute hand laps the hour hand 12 - 1 = 11 times. That is why the hands meet every 12/11 hours, about every 65 minutes, rather than exactly every hour. Double everything for 24 hours.
Example
Question: In 24 hours, how many times do the hands coincide, form a right angle, and point in opposite directions?
- Coincide: 11 times in 12 hours, so 22 a day. They are together at 12:00 and then roughly 1:05, 2:11, ..., 10:55, but not between 11 and 12 separately, because the next meeting is exactly 12:00.
- Right angle: in each lap there are two right-angle positions (the minute hand 90 degrees behind and 90 degrees ahead), so 2 x 22 = 44 a day.
- Opposite (180 degrees): once per lap, so 22 a day.
Interval between coincidences = 720/11 = 65 5/11 minutes.
console.log((720 / 11).toFixed(2)); // Output: 65.45
console.log([22, 44, 22]); // Output: [22,44,22]What to say in the interview
"In 12 hours the minute hand laps the hour hand 11 times, not 12, so they coincide 22 times a day, every 65 5/11 minutes. Each lap has two right angles and one straight line, so 44 and 22."
Common mistakes
- Answering 24, assuming one meeting every hour.
- Answering 48 for right angles by doubling 24.