Condition for Valid Time Indication

Theorem

Consider an analogue clock $C$ with an hour hand $H$ and a minute hand $M$.

Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock.

Let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock.

$12 \phi \mod 360 = \theta$

Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock.

Let $\rho \degrees$ be the angle made by the hour hand with respect to the hour just past.

$\rho = \dfrac \theta {12}$

Let $T$ be a time of day specified in hours $h$ and minutes $m$, where:

$1 \le h \le 12$ is an integer

$0 \le m < 60$ is a real number

whether a.m. or p.m. is immaterial.

From Speed of Minute Hand, $M$ travels $6 \degrees$ per minute.

So at time $m$ minutes after the hour, $\theta = 6 m$.

From Speed of Hour Hand, $H$ travels $\dfrac 1 2 \degrees$ per minute.

The hour marks are at $30 \degrees$ intervals.

So at time $m$ minutes after the hour, $\phi = 30 h + \dfrac m 2 \degrees$ past hour $h$.

That is:

$\phi = 30 h + \dfrac 1 2 \dfrac \theta 6$

or:

$12 \phi = 360 h + \theta$

where $h$ is an integer.

Thus:

$12 \phi \mod 360 = \theta$

$\blacksquare$