stoutfellow

A couple of days ago, I posted a clock puzzle. As a reminder:

Suppose you have an ordinary, fully-functional twelve-hour clock, whose hour and minute hands are the same length. (There is no second hand.) How often will it be impossible to tell what time it is, by looking at the clock? That is, how often will it be impossible to tell which is the hour hand and which the minute?

I initially solved this problem by crank-and-grind methods, which gave the answer without explaining why that answer was correct. Later, I thought of a genuine explanation; that's what's under the cut.( Solution )