Passing Ships

Date: 05/12/99 at 08:56:19
From: Mary Covington
Subject: "Till We Meet Again"

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?

This is a problem we have been working on. We have drawn it, acted it 
out, and come up with either 8 or 9. The answer that was given at a 
workshop last summer was 17. We are frustrated. Can you help us?

Thanks,
The 3rd Grade of The Raleigh School
Raleigh, NC

Date: 05/12/99 at 11:47:17
From: Doctor Peterson
Subject: Re: "Till We Meet Again"

Hi, Mary (and the rest of the class).

I like to approach this sort of problem with a time-distance graph, 
at least to get an idea of what's happening. The distance left to 
right is the position of the ship from San Francisco to Tokyo; each * 
on the vertical lines marks noon on one day, when a ship leaves from 
each end. The 8-day trip from SF to Tokyo is a diagonal line from top 
left to bottom right; the ships from Tokyo to SF that it meets are 
the diagonal lines that cross it at the X's:

    Day
      |/   /   /   /   /   /   /   /  |
    0 X   /   /   /   /   /   /   /   *
      |\ /   /   /   /   /   /   /   /|
      | X   /   /   /   /   /   /   / |
      |/ \ /   /   /   /   /   /   /  |
    1 *   X   /   /   /   /   /   /   *
      |  / \ /   /   /   /   /   /   /|
      | /   X   /   /   /   /   /   / |
      |/   / \ /   /   /   /   /   /  |
    2 *   /   X   /   /   /   /   /   *
      |  /   / \ /   /   /   /   /   /|
      | /   /   X   /   /   /   /   / |
      |/   /   / \ /   /   /   /   /  |
    3 *   /   /   X   /   /   /   /   *
      |  /   /   / \ /   /   /   /   /|
      | /   /   /   X   /   /   /   / |
      |/   /   /   / \ /   /   /   /  |
    4 *   /   /   /   X   /   /   /   *
      |  /   /   /   / \ /   /   /   /|
      | /   /   /   /   X   /   /   / |
      |/   /   /   /   / \ /   /   /  |
    5 *   /   /   /   /   X   /   /   *
      |  /   /   /   /   / \ /   /   /|
      | /   /   /   /   /   X   /   / |
      |/   /   /   /   /   / \ /   /  |
    6 *   /   /   /   /   /   X   /   *
      |  /   /   /   /   /   / \ /   /|
      | /   /   /   /   /   /   X   / |
      |/   /   /   /   /   /   / \ /  |
    7 *   /   /   /   /   /   /   X   *
      |  /   /   /   /   /   /   / \ /|
      | /   /   /   /   /   /   /   X |
      |/   /   /   /   /   /   /   / \|
    8 +   /   /   /   /   /   /   /   X
      |  /   /   /   /   /   /   /   /|
      SF                            Tokyo

I count 17 X's, though I'd question whether to call the first and last 
"meetings," since they represent a boat that arrives just as they 
depart, and one that departs just as they arrive. They meet each of 
the ships that left Tokyo in the last 8 days, as well as each ship 
that leaves Tokyo while they are on the way, making 8 + 8 + 1 = 17 
because we're including both ends of the 16 day interval. You don't 
need the picture to see this, but it helps a lot to convince me I'm 
right. Perhaps what you missed, and what I might have missed if I 
hadn't drawn it, was the ships that were already on the way when our 
ship left port.

Why is this more than the 8 or 9 you expect? Shouldn't they meet one 
ship per day? No; because they are going toward those ships as fast as 
the ships are coming toward them, it's as if the ships were coming 
twice as fast and twice as frequently. We meet all the ships that will 
arrive in San Francisco in the next 16 days, all in 8 days. This is 
related to the Doppler effect you may have heard of: if I'm moving 
toward a source of sound, it will seem as if the sound has a higher 
frequency. Here, I'm going at the speed of "sound," and I meet a 
"wave" twice as often as I would if I stayed at San Francisco, so the 
"sound" has twice the frequency.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/