Henry Ernest Dudeney/Modern Puzzles/26 - What is the Time

by : $26$

 * What is the Time?
 * At what time are the two hands of a clock so situated that,
 * reckoning as minute points past $\textit {XII}$,
 * one is exactly the square of the distance of the other?

Proof
The minute hand by definition moves $1$ minute point per minute.

The hour hand moves $5$ minute points per $60$ minutes, or $\dfrac 1 {12}$ minute points per minute.

The question does not state whether the hand whose minute point is the square of the distance of the other is the hour hand or the minute hand, so the analysis will be done on both.

Let $t$ minutes be the time of the particular moment of interest.

We assume that:
 * $0 \le t \le 12 \times 60 = 720$

during which time the minute hand may have gone multiple times round the clock.

We have that:
 * $h = \dfrac t {12}$

and:
 * $m = t \pmod {60}$

Hour Hand Position is Square of Minute Hand Position
Let $t$ minutes be the time at which the position of the hour hand is the square of the distance of the minute hand.

We have that:
 * $h^2 = m$

giving that:
 * $\paren {\dfrac t {12} }^2 = t \pmod {60}$

Minute Hand Position is Square of Hour Hand Position
Let $t$ minutes be the time at which the position of the minute hand is the square of the distance of the hour hand.

We have that:
 * $h = m^2$

giving that:
 * $\dfrac t {12} = \paren {t \pmod {60} }^2$