Addition of Fibonacci numbers. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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Addition of Fibonacci numbers. Focus secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

This trick consists in almost instantaneous addition of any ten consecutive Fibonacci numbers (the so-called series of numbers in which each, starting with the third, is the sum of the previous two). This trick is demonstrated as follows: the demonstrator asks someone to write down any two numbers he wants under each other. Let's say for example that 8 and 5 were chosen.

The viewer then has to add up these numbers. The third number found in this way is added to the second (standing above it), and the fourth number is obtained.

This process is repeated until there are ten numbers in the vertical column:

8
5
13
18
31
49
80
129
209
338

Focus secret:

While writing down the numbers, the demonstrator stands with his back turned to the audience. When all the numbers are written down, he turns, draws a line under the column of numbers and, without hesitation, signs the sum of these numbers. To get this sum, he just needs to take the fourth number from the bottom and multiply it by 11 - an operation that can easily be done mentally).

A series similar to the Fibonacci series, but starting not with 1 and 1, but with any numbers a and b, has the form a, b, a + b, a + 2b, 2a + 3b, Za + 5b, 5a + 8b, 8a + 13b, 13a + 21b, 21a + 34b ... Its coefficients are the Fibonacci numbers, and the sum of the ten terms written out is, as it is easy to calculate, 55a + 88b - one b less than the second of the numbers following the written series.

In our case, the fourth number will be 80, so the answer will be the number 80, taken 11 times, i.e. 880.

Tricks with predicting the results of operations on numbers and tricks with guessing numbers are easily reversible; by this it is meant that a trick with a prediction of a number can be shown as a trick with guessing this number, and vice versa. Suppose, for example, that the shower knows in advance the result of a calculation, which, as the viewer assumes, cannot be known to him. Then the demonstrator can arrange the focus in the form of a prediction, writing down the result of the future calculation known to him on a piece of paper; in this case, the focus should be considered as a predictive focus. But he can arrange the same trick as "reading the thoughts" of the viewer - after the viewer has completed his calculations - in this case, the trick should be classified as tricks with guessing the number.

A third option could be to design the focus as a lightning-fast calculation.

Author: M.Gardner

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