Apparently ambiguous times

My spring semester schedule is such that on certain days I have a four-hour break between my classes, which is an ideal time to get research done, although in practice most of that time is wasted on the web. The other day I glanced up at my wall clock to gauge the time, and was surprised when I saw this:

I first I though the hour hand had fallen off the clock (which had happened to a wristwatch I owned) and was momentarily panicked at the prospect of missing my afternoon class (or rather, being 11 minutes late to it). However, a few seconds later the minute hand moved a smidgen more to reveal that it had simply obscured the hour hand entirely, indicating a time of 2:11.

That got me to thinking… What are the other such “apparently ambiguous times,” times at which the hour hand and the minute hand both point in the same direction? Generalizing a bit, what are the “really apparently ambiguous times” at which the hour, minute, and second hands all point in the same direction?

It’s a little fun to figure out, and I won’t deprive you of it until after the jump.

Let’s deal with the hour and minute hand question first. Let $h(t)$ and $m(t)$ denote the clockwise angles (in degrees) the hour-hand and the minute-hand, respectively, make with “high noon” at hour $t$ . In particular, the minute hand completes a full revolution in one hour, so

$m(t) = 360 cdot t$ ,

while the hour hand only makes one-twelfth a revolution, so

$displaystyle h(t) = 360 cdot frac{t}{12} = 30 cdot t$ .

The two hands will point in the same direction whenever there respective angle functions differ by a multiple of 360, so we need only solve the equation $m(t) = h(t) + 360 k$ . Solving this yields

$displaystyle t = frac{12}{11} k$ ,

which means the two hands point in the same direction every $1 frac{1}{11}$ hours. Note that one-eleventh of an hour is $5 frac{5}{11}$ minutes, or five minutes and $27 frac{3}{11}$ seconds.

Hence, starting from 12 o’clock, the two hands will realign every 1 hour, 5 minutes, and 27.2727… seconds, doing this a grand total of eleven times before reaching 12 o’clock again. To the nearest second, these times are:

12:00:00

1:05:27

2:10:55