Henry Ernest Dudeney/Puzzles and Curious Problems/66 - Meeting Trains/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $66$

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?

$9$ trains.

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.

$\blacksquare$