Question

How many different ways can you write the number 6 as the sum if natural numbers? with proof.

Answer 1

you just neeed to organise your results in such a way that its impossible you missed any   
possibilities with:  
six 1's is max ones   
five 1's  
four 1's  
three 1's  
two 1's  
one 1   
number of 2's with zero 1's max is three  
number of 3's with zero 1's and zero 2's   
there are no results with a 4 or a 5 without a 1 or a 2  
there is one result with a six.   
no result with any higher number

Answer 2

0 doesn't count, right?

Let's call N(x) the number of ways to write x

* N(1) = 1
   * trivial case
* N(2) = 1 + N(1)
   * the 2 itself
   * sum a 1, then add 1
      * = N(1) + N(1) = 2·N(1)
* N(3) = 1 + N(1) + N(2)
   * the 3 itself
   * sum a 1, then add 2
   * sum a 2, then add 1
      * = N(2)+N(2) = 2·N(2)
* N(4) = 1 + N(1) + N(2) + N(3)
   * the 4 itself
   * sum a 1, then add 3
   * sum a 2, then add 2
   * sum a 3, then add 1
     * = N(3)+N(3) = 2·N(3)
* N(5) = 1 + N(1) + N(2) + N(3) + N(4)
   * the 5 itself
   * sum a 1, then add 4
   * sum a 2, then add 3
   * sum a 3, then add 2
   * sum a 4, then add 1
      * = N(4)+N(4) = 2·N(4)
* etc.

So N(x) = 2·N(x-1), or N(x) = 2^(x-1)

It's not a proof but maybe it'll help