Henry Ernest Dudeney/Puzzles and Curious Problems/66 - Meeting Trains/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $66$
- Meeting Trains
- At Wurzeltown Junction an old lady put her her head out of the window and shouted:
- "Guard! how long will the journey be from here to Mudville?"
- "All the trains take five hours ma'am, either way," replied the official.
- "And how many trains shall I meet on the way?"
- This absurd question tickled the guard, but he was ready with his reply:
- "A train leaves Wurzletown for Mudville, and also one from Mudville to Wurzletown, at five minutes past every hour. Right away!"
- The old lady induced one of her fellow passengers to work out the answer for her.
- What is the correct number of trains?
Solution
- $9$ trains.
Proof
Let $W$ and $M$ denote Wurzeltown and Mudville respectively.
Each train takes $5$ hours for the journey.
At the point of departure from $W$, there will be:
- $1$ train just arriving at $W$ from $M$
- $1$ train just starting out from $M$
- Another $4$ trains already in transit from $M$ to $W$.
During the course of the journey from $W$ to $M$, $4$ more trains start off from $M$ to $W$.
At the arrival at $M$, another train is just about to leave $M$ for $W$.
So, not counting the two trains at either end (just arriving at $W$ and just about to leave $M$), there are $9$ trains met along the way.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $66$. -- Meeting Trains
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $80$. Meeting Trains