Henry Ernest Dudeney/Puzzles and Curious Problems/58 - When did the Dancing Begin/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $58$

When did the Dancing Begin?

"The guests at that ball the other night," said Dora at the breakfast-table,

"thought that the clock had stopped,

because the hands appeared in exactly the same position as when the dancing began.

But it was found that they had really only changed places.

As you know, the dancing commenced between ten and eleven o'clock.

What was the exact time of the start?"

Solution

The answer given by Dudeney is:

The dancing must have begun at $59 \tfrac {83} {143}$ minutes past ten,

and the hands were noticed to have changed places at $54 \tfrac {138} {143}$ minutes past eleven.

However, this is merely one of $11$ possible solutions, although some are less feasible than others.

This is a list of all the ambiguous times between $10:00$ and $11:00$:

$10 : 04 \tfrac {28} {143}$	$12 : 50 \tfrac {50} {143}$
$10 : 09 \tfrac {33} {143}$	$01 : 50 \tfrac {110} {143}$
$10 : 14 \tfrac {38} {143}$	$02 : 51 \tfrac {27} {143}$
$10 : 19 \tfrac {43} {143}$	$03 : 51 \tfrac {87} {143}$
$10 : 24 \tfrac {48} {143}$	$04 : 52 \tfrac 4 {143}$
$10 : 29 \tfrac {53} {143}$	$05 : 52 \tfrac {64} {143}$
$10 : 34 \tfrac {58} {143}$	$06 : 52 \tfrac {124} {143}$
$10 : 39 \tfrac {63} {143}$	$07 : 53 \tfrac {41} {143}$
$10 : 44 \tfrac {68} {143}$	$08 : 53 \tfrac {101} {143}$
$10 : 49 \tfrac {73} {143}$	$09 : 54 \tfrac {18} {143}$
$10 : 59 \tfrac {83} {143}$	$11 : 54 \tfrac {138} {143}$

Of all of these, the period of time between $10 : 59 \tfrac {83} {143}$ and $11 : 54 \tfrac {138} {143}$ is the shortest:

Proof

See Ambiguous Times for the analysis of this.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $58$. -- When did the Dancing Begin?
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $50$. When did the Dancing Begin?