Zeckendorf's theorem, named after the Belgian mathematician Edouard Zeckendorf, 
is a theorem about the representation of integers as sums of Fibonacci numbers.

Zeckendorf's theorem states that every positive integer can be represented uniquely 
as the sum of one or more distinct Fibonacci numbers
in such a way that the sum does not include any two consecutive Fibonacci numbers.

Based on this theorem, you can use the 9 'MAGIC' cards above
in order to find the secret number from 1 to 75 choosen by a friend.
Ask your friend to tell you on which cards his secret number appears.
The sum of the numbers in the left corner of the cards where his number appears,
will give you his secret number!

A proof is given on 

https://math.stackexchange.com/questions/338740/the-generating-function-for-the-fibonacci-numbers

A magic relation between π and the Fibonacci numbers.

In 2001 (100 years after Zeckendorf was born) I published a card game named 'PHI, the Fibonacci mystery'.

The game is based on Zeckendorf's theorem and played with a set of cards
with the Fibonacci numbers 1, 1, 2, 3, 5, 8 and 13.
The winner of the game is the first player to discard all his cards.

The game was edited by Van In (Wommelgem, Antwerp) 
and sponsored by Texas Instruments.

In 2015 I proved a new property of the Fibonacci numbers.
If the coordinates of the three corners of a triangle
are successive Fibonacci numbers (I call such a triangle a Fibonacci triangle),
the area of the triangle is constant.
Hereby you find the Dutch version of this property (followed by the proof)
and also two generalizations.