CTK Insights

02 Jul

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:

Consider the following number trick — try it out on your friends. You ask them to write down the numbers from 1 to 10. Against 1 and 2 they write any two numbers (we suggest two fairly small positive integers just to avoid tedious arithmetic, but all participants should write the same pair of numbers). Then against 3 they write the sum of the entries against 1 and 2; against 4 they write the sum of the entries against 2 and 3; and so on. Once they have completed the process, producing entries against each number from 1 to 10, you suggest that, as a check, they call out the entry against the number 7. Thus their table (which, of course, you do not see) might look like the table in the margin. You now ask them to add all the entries in the second column, while you write 341 quickly on a slip of paper.

 
1     3
2     2
3     5
4     7
5     12
6     19
7     31
8     50
9     81
10     131

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 X1 + X5 = 3X3.

References

  1. R. Grimaldi, Fibonacci and Catalan Numbers: an Introduction, Wiley, 2012
  2. P. Hilton, D. Holton, J. Pedersen, Mathematical Reflections in a Room with Many Mirrors, Springer, 1997
  3. W. Simon, Mathematical Magic, Dover, 1964

One Response to “Fibonacci Numbers Trick”

  1. 1
    Wilder: Elementary Math and Science - fibonacci and the 11′s Says:

    [...] Bogomolny, Fibonacci numbers trick from CTK Insights http://www.mathteacherctk.com/blog/2012/07/fibonacci-numbers-trick/ Accessed 24 September [...]

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