How Many Trains Meet?

Date: 09/17/2001 at 19:16:07
From: Eric
Subject: Trains leaving every hour on the hour

Every hour, on the hour, a train leaves Tallahassee for Jacksonville, 
while another train leaves Jacksonville for Tallahassee. The trip 
between the two cities takes exactly two hours. How many trains going 
in the opposite direction will a Tallahassee train to Jacksonville 
meet?

Please help.

Date: 09/18/2001 at 10:46:32
From: Doctor Ian
Subject: Re: Trains leaving every hour on the hour

Hi Eric,

I find it hard to think about something like this without a picture, 
so let's try drawing one. At any given hour, we have one train ready 
to leave both cities:

             []\
 Tallahassee ========================================= Jacksonville
             =========================================
                                                   /[]
                                
We also have a train arriving at both cities (i.e., the ones that 
departed two hours ago):

             []\                                   []\
 Tallahassee ========================================= Jacksonville
             =========================================
             /[]                                   /[]
             
   
Finally, we have a train halfway between, going in each direction 
(i.e., the ones that departed an hour ago):

             [1]\             [2]\                [3]\
 Tallahassee ========================================= Jacksonville
             =========================================
             /[4]             /[5]                /[6]
               

So this is the situation at any time t=0.  Now, let's pick a train 
leaving from Tallahassee (#1), and see what happens to it.  

It's not clear from the description of the problem whether train 1 
'passes' train 4 as it pulls out of the station, so I'll leave that 
for you to work out with your teacher.

But a few minutes after the hour, we have this situation:

                 [1]\             [2]\                
 Tallahassee ========================================= Jacksonville
             =========================================
                          /[5]                /[6]        
          

After one hour, we have this situation:

             [7]\                [1]\             [2]\                
 Tallahassee ========================================= Jacksonville
             =========================================
             /[5]               /[6]              /[8]


Train 1 has passed train 5, and is just about to pass train 6.  
Another hour later, we have this situation:

             [9]\               [7]\              [1]\              
 Tallahassee ========================================= Jacksonville
             =========================================
             /[6]              /[8]               /[9]


So train 1 has passed train 8, and again, it remains to be negotiated 
whether train 1 also passes train 9.

Can you take it from here? 

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