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 the twelve o'clock.

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

Then $C$ displays a valid time indication :


 * $12 \phi \mod 360 = \theta$

Proof
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:
 * $12 \phi \mod 360 = \theta$