Fibonacci Numbers Trick
William Simon was probably the first to employ a property of Fibonacci's recursion
X_{n+1} = X_{n} + X_{n1}
as a professional magic trick. This is best described in Hilton, Holton, Pedersen:


The property that underlies the trick is simple:
For a sequence X_{1}, X_{2}, X_{n+1} = X_{n} + X_{n1}, n ≥ 2,
X_{1} + X_{2} + ... + X_{10} = 11X_{7}.
For a young audience adding up 10 numbers may be an unnecessary hard. A similar magic trick may be performed based on a simpler property:
For a sequence X_{1}, X_{2}, X_{n+1} = X_{n} + X_{n1}, n ≥ 2,
X_{1} + X_{2} + ... + X_{6} = 4X_{5}.
Returning to the above example, 3 + 2 + 5 + 7 + 12 + 19 = 48 = 4×12, as you can check.
How do you prove those properties? Stepbystep:
X_{3} = X_{2} + X_{1},
X_{4} = X_{3} + X_{2} = 2X_{2} + X_{1},
X_{5} = X_{4} + X_{3} = 3X_{2} + 2X_{1},
X_{6} = X_{5} + X_{4} = 5X_{2} + 3X_{1},
X_{7} = X_{6} + X_{5} = 8X_{2} + 5X_{1},
X_{8} = X_{7} + X_{6} = 13X_{2} + 8X_{1},
X_{9} = X_{8} + X_{7} = 21X_{2} + 13X_{1},
X_{10} = X_{9} + X_{8} = 34X_{2} + 21X_{1}.
For the sum X_{1} + X_{2} + ... + X_{10} we then have
X_{1} + X_{2} + ... + X_{10} = 88X_{2} + 55X_{1} = 11X_{7},
and similar for the shorter sum. Simplest in this series will probably be the trick based on
References
 R. Grimaldi, Fibonacci and Catalan Numbers: an Introduction, Wiley, 2012
 P. Hilton, D. Holton, J. Pedersen, Mathematical Reflections in a Room with Many Mirrors, Springer, 1997
 W. Simon, Mathematical Magic, Dover, 1964
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September 24th, 2012 at 8:42 pm