Henry Ernest Dudeney/Puzzles and Curious Problems/60 - Equal Distances/Solution

by : $60$

 * Equal Distances

Solution

 * $23 \tfrac 1 {13}$ past $3$.


 * Dudeney-Puzzles-and-Curious-Problems-60-Solution.png

Proof
Let $T$ be the time in question.

Let $m$ be the number of minutes past $3:00$ that $T$ is.

Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock at time $T$.

Let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock at time $T$.

Between $3:00$ and $4:00$ we have that $90 < \phi < 120$.

Hence from Condition for Valid Time Indication, we have that:
 * $12 \paren {\phi - 90} = \theta$

Then we are told that:
 * $\size {240 - \theta} = \phi$

This can mean that:


 * $240 - \theta = \phi$

in which case:

Or it can mean that:


 * $\theta - 240 = \phi$

in which case:

But in this case $T = 4:00$, which is not (strictly) between $3:00$ and $4:00$.

Hence the result.