Puzzle

As a math professor, I often receive e-mail from makers of educational software. I usually just ignore them, as they're often aimed at classes I don't teach. One recent e-mail, though, posed an interesting little problem. I began thinking about it in the shower - I do some of my best thinking there - and was able to work out the answer before getting out of the stall. I'd like to share the problem today; I'll post my answer, probably, sometime tomorrow. Anyone mathematically inclined is hereby challenged to work out their own answer.

Here's the problem. Suppose you have a properly functioning twelve-hour clock, ordinary in all respects except that the hour and minute hands are the same length. (There is no second hand.) How often, looking at the clock, will it be impossible to be sure what time it is? That is, how often will it be impossible to tell which is the hour hand and which the minute? (It's certainly possible to tell most of the time. For example, if one - and only one - of the hands is pointing exactly at "12", it's the minute hand.)

Take your - heh - time.