Coincidence of Hour and Minute Hands

Problem

 * The hour and minute hands of a watch coincide at noon.


 * When will they once again coincide, during the next $12$ hours?

Solution
The hands coincide at the approximate times, in hours, minutes and seconds:


 * $13:05:27$
 * $14:10:55$
 * $15:16:22$
 * $16:22:49$
 * $17:27:16$
 * $18:32:44$
 * $19:38:11$
 * $20:43:38$
 * $21:49:05$
 * $22:54:33$
 * Midnight

Proof
We have that:
 * the minute hand takes $1$ hour to go $360 \degrees$ around the dial.
 * the hour hand takes $1$ hour to go $30 \degrees$ around the dial.

Let the minute hand and hour hand be coincident at some point in time.

Let $t$ hours be the time elapsed when they are next coincident.

Let $\theta$ degrees be the angle the hour hand has moved in time $t$.

During $t$, the minute hand has moved $\theta + 360 \degrees$.

Hence we have:

That is, the hour hand and minute hand coincide every $1 \tfrac 1 {11}$ hours.

Hence that will be at:
 * $1 \tfrac 1 {11}, 2 \tfrac 2 {11}, 2 \tfrac 3 {11}, \ldots$

The result follows by converting the fractional hours into minutes and seconds.