19 Odd Number Puzzle

Date: 10/22/2001 at 22:17:38
From: Jesse Vermillion
Subject: Adding odd to get an odd

What four odd numbers add up to 19?

Date: 10/23/2001 at 00:21:55
From: Doctor Ian
Subject: Re: Adding odd to get an odd

Hi Jesse,

An even number can always be drawn as a rectangle:

  6  * * * 
     * * * 

  12 * * * * * *
     * * * * * *

But an odd number will always have one guy sticking out:

  5  * * *
     * * 

  11 * * * * * *
     * * * * *

Note that if you combine two of these 'almost rectangles', the odd 
guys match up to make one more pair:

     5    +      11              16
   * * *     * * * * *  = * * * * * * * *
   * *     * * * * * *    * * * * * * * *

That is, two odd numbers always add up to an even number. 

Now, if you have four odd numbers to add up, you can add them two at a 
time... 

  a + b + c + d = (a + b) + (c + d)

which means that the sum of four odd numbers is also the sum of two 
even numbers... which is always an even number. (Think about putting 
two rectangles together.)

So, apart from 'trick' answers (for example, switching bases, using 
Roman numerals, and so on), the answer to your question is:  

There _are_ no such numbers. 

Does this help? 

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