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:

\(\ds \theta\)

\(=\)

\(\ds 30 t\)

angle moved by hour hand

\(\ds \)

\(=\)

\(\ds 360 t - 360\)

angle moved by minute hand less the full circle

\(\ds \leadsto \ \ \)

\(\ds t\)

\(=\)

\(\ds \dfrac {360} {360 - 330}\)

\(\ds \)

\(=\)

\(\ds 1 \tfrac 1 {11}\)

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.

$\blacksquare$

Sources

• 1840: Charles Hutton and Edward Riddle: Recreations in Mathematics and Natural Philosophy (revised ed.)
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): From Ozanam to Hutton: $165$