system
Given the following post, respond as if you're commenting on Reddit:
human
Does the clock arms (seconds, minutes, hours) ever split the clock face in three equal areas?

This is a hypothetical clock where the arms move around the middle axis with constant speed, so no tick tocks. If you extend the lines of the clock arms, do they perfectly split the clock face in three thirds at any point during the 12 hours?
gpt
No. Consider a clock that rotates counterclockwise at exactly the rate the hour hand moves, so that the hour hand always points up. Now the minute hand goes at a rate of 1/60min - 1/12hr = 11/720min and the second had at a rate of 1/1min - 1/12hr = 719/720min. Any perfect split will happen when the second hand has gone around a multiple of 1/3 times, say n/3 for some integer n. At that time, the minute hand will have gone around 11/719\*n/3 times. We need this to also be a multiple of 1/3, so 11n/719 must be an integer. Since 11 and 719 are coprime (11 is prime and doesn't divide 719; in fact, 719 is prime), this only happens when n is a multiple of 719. When n=3, the second hand has gone around 719 times, which takes 720 sec = 12 hr, a complete cycle. So we only need to consider n=1 or n=2. For n=1, the second hand has gone 719/3 = 239+2/3, so the second hand is pointing 2/3 of the way around the circle (i.e. 8 o'clock, but the 8 will have moved). The minute hand has gone 11/719 * 719/3 = 11/3 = 3+2/3, so the minute hand is also pointing at 2/3, so this is not the split you want. Similarly, when n=2, both hands point at 1/3. So there is no such perfect split.