# Clock

Source: https://www.techinterview.org/post/521425053/clock/
Updated: 2026-04-16 · techinterview.org

Part I: what is the angle between the minute hand and the hour hand at 3:15 on an analog clock? no, its not 0.

Part II: how often does the minute hand pass the hour hand on an analog clock?

### Solution

part I: 12 hours on the clock make 360 deg. so one hour is 30 deg. the hour hand will be directly on the 3 when the minute hand is at 12 (3:00). after 15 minutes or 1/4 of an hour, the hour hand will be 1/4 * 30 deg = 7.5 deg. away from the minute hand.

part II: if you just think about it, intuitively you’ll see the minute hand passes the hour hand 11 times every 12 hours, so it must pass it every 1 1/11 hours. but this doesn’t make sense to me. i need to prove it.

if x is our answer then every x hours, the minute hand and the hour hand will be right on top of each other. every hour the hour hand travels 5 units. so between every time that the minute and the hour hand meet, the hour hand will go 5*x units. every hour the minute hand travels 60 units, so it will have gone 60*x units.

what we’re trying to find is the distance traveled by the minute hand to reach the hour hand, once the minute hand has looped around once. consider its 12:00. both hands in the same position. after an hour, minute hand is on 12, hour hand on 1 (its traveled 5 units). now in the time it takes the minute hand to catch up to the hour hand it will travel a little bit further.

we only need to find x where 5*x = 60*(x-1), since the real distance traveled by the minute hand, from where it started to where it ends, is 60*(x-1). the first hour just puts it back where it started, so we’re only concerned with the extra part it traveled to reach the hour hand.


```
5x = 60(x-1)5x = 60x - 6060 = 55x60/55 = x
```

there it is. the answer is 60/55 hours, or every 1 and 1/11 hours.

i apologize that this is horribly confusing, but if you stare at it long enough it will make sense.

## 2026 Update: Clock and Angle Puzzles in Interviews

Clock puzzles ("at what times between 3:00 and 4:00 do the hour and minute hands overlap?") remain asked at quant firms and in Goldman Sachs / Morgan Stanley analyst interviews. They test mathematical reasoning and attention to detail under pressure.

**The formula approach (what interviewers want to see):** The minute hand moves at 6°/min; the hour hand at 0.5°/min. At time T minutes after 12:00, the minute hand is at 6T degrees and the hour hand at 0.5T degrees. For overlap: 6T = 0.5T + 360k, solving: T = 720k/11. The 11 overlaps per 12 hours occur at T = 0, 65.45, 130.9, ... minutes.


```
def clock_overlaps():
    """Calculate all times when hour and minute hands overlap."""
    times = []
    for k in range(11):
        minutes = 720 * k / 11
        hours = int(minutes // 60)
        mins = minutes % 60
        times.append(f"{hours:02d}:{mins:05.2f}")
    return times

for t in clock_overlaps():
    print(t)
```


**2026 modern variant:** "At what time do the hands of an analog clock form a right angle? Write a function that returns all such times in a 12-hour period." Tests the same reasoning but asks for code.

**Still asked at (2026):** Goldman Sachs, Morgan Stanley, quantitative trading firms (Jane Street, Citadel) in first-round screening interviews.