Quiz

How many times a day does the hour hand form a 120-degree angle with the minute hand?

That is, there are 20 minutes on the clock face between them, in one direction or the other. Each hour equals five minutes. The speed of the hour hand is v1 = 5 minutes/hour.
The speed of the minute hand is v2 = 60 minutes/hour.
The hour hand is 120 degrees behind => (v2*t - v1*t) mod 60 = 20 => (55t) mod 60 = 20
Conversely => (v1*t - v2*t) mod 60 = 20 => (55t) mod 60 = 40
t ranges from 0 to 24 hours.
We can see that t must be a multiple of four (or eight in the second case), but not a multiple of three.
t = 4 => 220 mod 60 = 40
t = 8 => 440 mod 60 = 20
t = 16 => 880 mod 60 = 40
t = 20 => 1100 mod 60 = -100 mod 60 = 20
I don't know, something doesn't quite seem right here, but for now, the answer is 4.