Henry Ernest Dudeney/Puzzles and Curious Problems/59 - Mistaking the Hands/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $59$

Mistaking the Hands
"Between two and three o'clock yesterday," said Colonel Crackham,
"I looked at the clock and mistook the minute hand for the hour hand,
and consequently the time appeared to be fifty-five minutes earlier than it actually was.
What was the correct time?"


Solution

The time was $02:05 \tfrac 5 {11}$, but it appeared to be $01 : 10 \tfrac 5 {11}$.

Dudeney-Puzzles-and-Curious-Problems-59-Solution.png


Proof

Let $T$ be the time in question.

Let $m$ be the number of minutes after $2:00$ at $T$.

The time mistaken is $T - 55$ minutes, that is $m + 5$ minutes after $1:00$.

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 $T$.

At $T - 55$ minutes, the angle made by the minute hand with respect to twelve o'clock is $\theta + 30 \degrees$.

But this is the angle of the hour hand at $T$.

Hence from Condition for Valid Time Indication, we have that:

$12 \paren {\phi - 60} = \theta$

as $T$ is between $2:00$ and $3:00$.

Hence:

\(\text {(1)}: \quad\) \(\ds \phi\) \(=\) \(\ds \theta + 30\) $55$ minutes is $12 \times 5 = 30 \degrees$
\(\text {(2)}: \quad\) \(\ds 12 \paren {\phi - 60}\) \(=\) \(\ds \theta\) Condition for Valid Time Indication
\(\ds \leadsto \ \ \) \(\ds 12 \paren {\theta + 30 - 60}\) \(=\) \(\ds \theta\) substituting for $\phi$ in $(2)$ from $(1)$
\(\ds \leadsto \ \ \) \(\ds 11 \theta\) \(=\) \(\ds 360\)
\(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds \dfrac {360} {11}\)
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds \dfrac {360} {11} \times \dfrac 1 6\) Speed of Minute Hand
\(\ds \) \(=\) \(\ds 5 \dfrac 5 {11}\)

Hence the time was $02:05 \tfrac 5 {11}$.

$\blacksquare$


Sources

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