# We Got A Problem #2: The current time is… 120 degrees

June 27, 2011 3 Comments

The hour, minute, and second hands of a continuously-moving clock all point the same direction at 12 noon. But when are they equally spaced around the clock face?

In attacking this problem, Kathy from Pittsburgh pointed out that at 12:20:40, the hands almost form three 120-degree angles. But they don’t, because the hour hand has moved a little passed 12 on its way to 1, and the minute hand has moved a little too.

So, when do the three clock hands form three 120-degree angles?

*We’d love for readers to be able to explore this problem, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to approach the problem, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!*

A slight variation of this problem (perhaps an easier solution too) comes from the question ‘At what exact time(s) do the two hands meet?’ Or the extension, ‘Do the 3 hands meet at the same spot at a certain time other than 12:00?)

I’m pretty convinced that this never happens. They get close, but never exact. Has anyone found a solution?

The problem has definitely been “solved” but that doesn’t mean there is a time when the hands make 120-degree angles!