From: cliff@watson.ibm.com (cliff)
Subject: Cliff Puzzle 4
Date: Tue, 15 Sep 1992 20:55:32 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: A

Title: Cliff Puzzle 4: Time in a Bottle                                         
From: cliff@watson.ibm.com                                                      
                                                                                
If you respond to this puzzle, if possible please include your name,            
address, affiliation, e-mail address.  If you like, tell me a little bit        
about yourself.  PLEASE ALSO directly mail me a copy of your response           
in addition to any responding you do in the newsgroup.  I will assume it        
is OK to describe your answer in any article or publication I may write         
in the future, with attribution to you, unless you state otherwise.             
Thanks, Cliff Pickover                                                          
                                                                                
* * *                                                                           
                                                                                
Consider a chain of bottles (B) each connected to one another by a thin         
tube. A marble is placed in bottle 1.                                           
Each tube contains a one-way valve so marbles can only                          
go from left to right in the tubes which are symbolized with "-" marks:         
                                                                                
1   2   3   4                                                                   
B - B - B - B -                                                                 
                                                                                
                                                                                
The tubes are thin so it takes                                                  
1 hour of constant random shaking to get the marble from B1 to B2.              
Likewise for each bottle.                                                       
                                                                                
I have not fully described the bottle collection.  Each bottle                  
has a backward 1-way tube to bottle 1.  I've tried to diagram these             
with "*" symbols.  Each time the marble enters bottle B(N) it has               
a 50% probability of going back to bottle 1 via these tubes.                    
                                                                                
                                                                                
****<********                                                                   
*           *                                                                   
***<*****   *                                                                   
*       *   *                                                                   
* * *   *   *                                                                   
1   2   3   4                                                                   
B - B - B - B -                                                                 
                                                                                
Stop And Think                                                                  
                                                                                
1.  In how many hours will you expect to get the marble out of bottle 10        
after placing the marble in bottle 1?                                           
                                                                                
2. Is there a general formula for the amount of time                            
required to get the ball out of bottle N into bottle N+1 given                  
a probability P of backwards motion (given as 50% in this problem)?             
                                                                                
3.  In how many hours will you expect to get the marble out of bottle 10        
after placing the marble in bottle 1 given two backward tubes for each          
bottle instead of one backward tube?