Fibonacci Numbers Trick
William Simon was probably the first to employ a property of Fibonacci's recursion
Xn+1 = Xn + Xn-1
as a professional magic trick. This is best described in Hilton, Holton, Pedersen:
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The property that underlies the trick is simple:
For a sequence X1, X2, Xn+1 = Xn + Xn-1, n ≥ 2,
X1 + X2 + ... + X10 = 11X7.
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 X1, X2, Xn+1 = Xn + Xn-1, n ≥ 2,
X1 + X2 + ... + X6 = 4X5.
Returning to the above example, 3 + 2 + 5 + 7 + 12 + 19 = 48 = 4×12, as you can check.
How do you prove those properties? Step-by-step:
X3 = X2 + X1,
X4 = X3 + X2 = 2X2 + X1,
X5 = X4 + X3 = 3X2 + 2X1,
X6 = X5 + X4 = 5X2 + 3X1,
X7 = X6 + X5 = 8X2 + 5X1,
X8 = X7 + X6 = 13X2 + 8X1,
X9 = X8 + X7 = 21X2 + 13X1,
X10 = X9 + X8 = 34X2 + 21X1.
For the sum X1 + X2 + ... + X10 we then have
X1 + X2 + ... + X10 = 88X2 + 55X1 = 11X7,
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