The following two identities serve as the basis for two mathematical games/tricks.
Theorem 13.4: Any positive integer N can be expressed as a sum of distinct Fibonacci numbers, no two of which are consecutive; that is,
where 
and 
for 
(the Zeckendorf representation).
This theorem is useful in playing a two-person game called ``Fibonacci Nim'', in which players take turns picking up a number of sticks.
- Starting with N sticks, the first player may pick up any number of sticks x from 1 to N-1;
- The second player may then pick up any number of sticks y from 1 to 2x.
- The first player may then pick up any number of sticks from 1 to 2y.
- Iterate.
- The last player to pick up a stick wins.
The successful strategy uses the Zeckendorf representation (if one starts, and chooses the initial number of sticks, then the strategy is guarenteed to win); if one goes second, a typical player will make a mistake along the way and allow one in the know to win anyway! A very nice bar game....
Let's play a round and see if you can figure out the strategy!
This identity is the basis of a well-known puzzle apparently showing that area can be created from nothing:
