The puzzle: A well-known and simple puzzle is this: The hour and minute hands of a clock are superimposed at 12:00. When will they next be superimposed (I don't mean lined up as they are at 6:00)? When are the hands superimposed, roughly? 12:00, after 1:00, after 2:00, after 3:00, etc. We can be more accurate than that: 12:00, after 1:05, after 2:10, after 3:15, etc. The last time will be "after 11:55." I'll just bet that "after 11:55" is 12:00 exactly. And we have 11 events (starting with 1:05+, as the original 12:00 is time zero) equally spaced about a 12 hour period. And these events are 12/11 (1+1/11 or 1:05.4545...) hour apart.

How did I know it was 12/11 instead of 11/12? Well, 11/12 hour is before 1:00, which can't be right. Besides that, I can check my work: 1+1/11, 2+2/11, 3+3/11, ... 11+11/11. That last 11+11/11 is exactly 12:00. It checks out.

There is another, more general way to approach this problem. To illustrate this method, let me propose another puzzle: After 12:00, when is the first time that the angle between the hands is 30 degrees? You might want to solve this one, before reading on.

The angle between the top of the clock and the hour hand changes at the rate of 360 degrees in 12 hours, or 1/2 degree per minute. The angle between the top of the clock and the minute hand changes at a rate of 360 degrees in 1 hour, or 6 degrees per minute. The difference is 5.5 degrees per minute. In other words, the hands (starting together), separate at the rate of 5.5 degrees per minute. A little division shows that they reach 30 degrees in 5.454545... minutes. So the time when they first are 30 degrees apart would be 12:05.4545...

You probably can see how this method can be applied to our original problem. And we would, of course, get the same answer we did before (1:05.4545...).

Second puzzle:

There are many versions of this puzzle. Here is a setting that I just made up:

We sometimes are told that when we drive, we should place our hands on the steering wheel at ten minutes to two. But if the steering wheel were really a clock, then at 10 minutes to 2 the right hand (pretending that the two hands of the clock are the same length) would be slightly higher than the left hand. We might drive in circles, using that advice. A minute or so later, the hands will be at equal heights. What time would that be? That seems to be a tougher puzzle, doesn't it?