Mathematics.Mathematical Jokes and Puzzles

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This message was updated on 9/5/2002 4:50:21 PM by Soroban

Fibonacci-type trick
posted on: 9/5/2002 4:26:53 PM

Give a friend a sheet of paper, a pencil, and a calculator.
(He/she will be more cooperative with a calculator.)

Simply give him these instructions. Do NOT look at his paper.

Have him number the lines from 1 to 20.

On the first two line, write any two positive integers.
(The numbers can be rational or irrational.)

Instruct him to add the two numbers and write the sum on the third line.
Thereafter, he is to add the last two numbers and write the sum on the
next line (a la Fibonacci) until he has completed the list of 20 numbers.

Here's a sample list (written in 4 columns to save space):

1) 17 ..... 6) 76 .... 11) 853 ... 16) 9459
2) 5 ...... 7) 125 ... 12) 1380 .. 17) 15305
3) 22 ..... 8) 291 ... 13) 2233 .. 18) 24764
4) 27 ..... 9) 326 ... 14) 3613 .. 19) 40069
5) 49 .... 10) 527 ... 15) 5846 .. 20) 64883

Part 1
Have him select a number from among the last three on the list
(#18, 19, or 20) and divide by the preceding number.

A little showmanship: Remind him that you do not know his two
starting numbers and you have not seen any of his numbers.

Have him concentrate on the display while you attempt to
determine (telepathically) the final quotient.

Slowly & dramatically say, "One-point-six-one-eight-zero-three-three."

Many of you will recognize 1.618033 as the Golden Mean.
This is true: consecutive ratios of any Fibonacci-type
sequence will converge to phi, the Golden Mean.

For more digits in the final answer, extend the list to 25 numbers.

Part 2
Have him draw a line under any number on his list (except the last two).
Tell him to add all the numbers above that line.

Give him a few moments, then ask "How are you doing?"
and casually glance at his list.

The required total is the second number after his line,
minus the second number on the list.

Example: If he drew a line under (#13) 2233, the sum of the
numbers above that line is: (#15) 5846 - (#2) 5 = 5841.

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