From: cliff@watson.ibm.com (cliff)
Subject: Puzzle 3
Date: Mon, 14 Sep 1992 14:17:04 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: A

Title: Cliff Puzzle 3: Too many 3's                                             
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.  You might 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                                                          
                                                                                
* * *                                                                           
                                                                                
How many numbers have at least one digit -- a three?                            
                                                                                
In the first 10 numbers, 1,2,3,4,5,6,7,8,9,10 there is only one number          
which contains the digit 3.  This means that 1/10 or 10% of the numbers          
have the number 1 in the first 10 numbers.  In the first 100 numbers the        
occurrence of numbers with at least one three seems to be growing.  In          
fact there are 19 numbers:  3,13,23,33,43,53,63,73,83,93,                       
30,31,32,34,35,36,37,38,39.  This means that about 19% of the digits            
contain the number 3 in the first 100 numbers.                                  
                                                                                
We can make a table showing the percentage of numbers with                      
at least one 3-digit for the first N numbers.                                   
N        %                                                                      
10       1                                                                      
100      19                                                                     
1000     27                                                                     
10000    34                                                                     
                                                                                
The percentages rapidly increase to 100% indicating that almost all of          
the numbers have a 3 in them!  In fact, a formula describing the                
proportion of 3's can be written:  1-(9/10)**N.  The proportion gets            
very close to 1 as N increases.                                                 
                                                                                
Stop And Think                                                                  
                                                                                
1. How can it be that almost all of the numbers have a 3 in them?               
                                                                                
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