Numbers as a sum of Fibonacci's:

When we write a number as a sum of Fibonacci's, we often have several choices:

10=2+3+5
10=1+1+8
10=2+8****best one

The best way to do it is the one that uses the fewest Fibonaccis. This way never has two consecutive Fibonaccis (because then you could add them tegether to get a better way).

One way to get the best sum is to always use the biggest Fibonaccis you can. 8 is the biggest Fibonacci less than 10, so we use it: 10=8+2, and 2 is a Fibonacci number, so we're done.

Example 1: Find 17 as a sum of Fibonaccis:

1   1   2   3   5   8   13   21   34   55  
                          17            

The biggest Fibonacci number that is less than 17 is 13, so we will have 17=13+???

Next, Take 17-13=4, and look for the best Fibonacci to use with 4:

1   1   2   3   5   8   13   21   34   55  
              4                        

The biggest Fibonacci number that is less than 4 is 3, so we will have 17=13+3+?

Next, Take 4-3=1. 1 is a Fibonacci number, so that is where we end: 17=13+3+1

Example 1: Find 50 as a sum of Fibonaccis:

1   1   2   3   5   8   13   21   34   55  
                                  50    

The biggest Fibonacci number that is less than 50 is 34, so we will have 50=34+???

Next, Take 50-34=16, and look for the best Fibonacci to use with 16:

1   1   2   3   5   8   13   21   34   55  
                          16            

The biggest Fibonacci number that is less than 16 is 13, so we will have 50=34+13+???

Next, Take 16-13=3. 3 is a Fibonacci number, so that is where we end: 50=34+13+3