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Every day at noon a ship leaves San Francisco for Tokyo, and at the same instant
a ship leaves Tokyo for San Francisco. Each trip lasts exactly eight days.
How many Tokyo ships will each San Francisco ship meet?
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For simplicity, let's assume that today is July 1 of some year.
We are in San Francisco, on a ship which set sail at noon.
As soon as the ship sails, another ship sails from Tokyo, and two ships
meet at noon. So, our ship meets 1st ship from Tokyo at noon, July first.
+----------------------------------------------------------+
| This ship sailed from Tokyo exactly 8 days ago, |
| i.e. on June 23, using the calendar of San Francisco. |
+----------------------------------------------------------+
Your ship will arrive to Tokyo 8 days after June 1 on the San Francisco calendar, i.e. on July 9.
Thus, your ship will meet ALL THE SHIPS that sailed from Tokyo starting from June 23
till July 9, inclusive. Count all these days from June 23 till July 9: you will get 17 days.
Hence, from noon July 1 till noon July 9 you will meet 17 ships in opposite direction.
ANSWER. During the journey, each San Francisco ship will meet 17 Tokyo ships.
Solved.
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There is another way to make your logical reasoning.
Since each one way trip is 8 days, it means that on every noon there are 9 ships
equally distributed on the way from San Francisco to Tokyo.
One ship arrived at noon, while the other
ship leaves the port at this time.
Similarly, every day at noon, there are 9 ships equally distributed on the way from Tokyo to San Francisco.
One ship arrived at noon, while the other
ship leaves the port at this time.
So, in all, there are 9+9 = 18 ships on the line.
Your ship will meet all other 18-1 = 17 ships (all except your own ship).
So, you got two different solutions from me, for your better understanding.
. . . It is a classic entertainment problem . . .