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

by : $58$

 * When did the Dancing Begin?

Solution
The answer given by 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 $11:00$:


 * {| border="1"


 * align="center" style = "padding: 2px 10px" | $10 : 04 \tfrac {28} {143}$
 * align="center" style = "padding: 2px 10px" | $12 : 50 \tfrac {50} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 09 \tfrac {33} {143}$
 * align="center" style = "padding: 2px 10px" | $01 : 50 \tfrac {110} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 14 \tfrac {38} {143}$
 * align="center" style = "padding: 2px 10px" | $02 : 51 \tfrac {27} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 19 \tfrac {43} {143}$
 * align="center" style = "padding: 2px 10px" | $03 : 51 \tfrac {87} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 24 \tfrac {48} {143}$
 * align="center" style = "padding: 2px 10px" | $04 : 52 \tfrac 4 {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 29 \tfrac {53} {143}$
 * align="center" style = "padding: 2px 10px" | $05 : 52 \tfrac {64} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 34 \tfrac {58} {143}$
 * align="center" style = "padding: 2px 10px" | $06 : 52 \tfrac {124} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 39 \tfrac {63} {143}$
 * align="center" style = "padding: 2px 10px" | $07 : 53 \tfrac {41} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 44 \tfrac {68} {143}$
 * align="center" style = "padding: 2px 10px" | $08 : 53 \tfrac {101} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 49 \tfrac {73} {143}$
 * align="center" style = "padding: 2px 10px" | $09 : 54 \tfrac {18} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 59 \tfrac {83} {143}$
 * align="center" style = "padding: 2px 10px" | $11 : 54 \tfrac {138} {143}$
 * }
 * align="center" style = "padding: 2px 10px" | $07 : 53 \tfrac {41} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 44 \tfrac {68} {143}$
 * align="center" style = "padding: 2px 10px" | $08 : 53 \tfrac {101} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 49 \tfrac {73} {143}$
 * align="center" style = "padding: 2px 10px" | $09 : 54 \tfrac {18} {143}$
 * align="center" style = "padding: 2px 10px" | $10 : 59 \tfrac {83} {143}$
 * align="center" style = "padding: 2px 10px" | $11 : 54 \tfrac {138} {143}$
 * }
 * align="center" style = "padding: 2px 10px" | $10 : 59 \tfrac {83} {143}$
 * align="center" style = "padding: 2px 10px" | $11 : 54 \tfrac {138} {143}$
 * }

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


 * Dudeney-Puzzles-and-Curious-Problems-58-Solution.png

Proof
See Ambiguous Times for the analysis of this.