Saturday, December 3, 2011

Fibonacci Day: Fibonacci Magic Trick

I don’t know if you noticed, but today is a Fibonacci Day! It is December third, and three is a Fibonacci number. We’ve looked at some cool patterns in Fibonacci numbers, but it’s time we learn how to do some magic with them! Let’s look at the Fibonacci numbers, but this time in lines.

Line 1: 1
Line 2: 1
Line 3: 2
Line 4: 3
Line 5: 5
Line 6: 8
Line 7: 13
Line 8: 21
Line 9: 34
Line 10: 55
Line 11: 89
Line 12: 144
Line 13: 233
Line 14: 377
Line 15: 610
Line 16: 987
Line 17: 1597
Line 18: 2584
Line 19: 4181
Line 20: 6765

What do you think the sum is of all of the numbers up until line, say thirteen? I can tell you immediately that it is 609. How?

Wait, we know this! Remember when we were adding Fibonacci numbers? The answer was always two ahead minus one! But we can take it one step further.

Let’s make our own Fibonacci sequence this time, starting with any two numbers we want. If you are doing it on pencil and paper, you might want to stick 1 – 10, or you can make an excel or numbers spreadsheet and do it as high or little as you want. In this case, we’ll start with 4 and 7.

Line 1: 4
Line 2: 7
Line 3: 11
Line 4: 18
Line 5: 29
Line 6: 47
Line 7: 76
Line 8: 123
Line 9: 199
Line 10: 322
Line 11: 521
Line 12: 843
Line 13: 1364
Line 14: 2207
Line 15: 3571
Line 16: 5778
Line 17: 9349
Line 18: 15127
Line 19: 24476
Line 20: 39603

What is the grand total up to line eight, you can do any line. The answer is 315. It’s still the same exact principle, except for one little thing.

We move up two lines, and then subtract line two. It is the easiest of all things to do! No matter how gigantic the numbers are, you can still pull it off. What’s even cooler is that there is no specific line you are adding up to, unlike other methods that only go up to line ten.

Bonus Trick: Make one of these sequences yourself, and make sure you have at least ten lines. Now, divide the last line by the one before it. In this case, we would be doing 39603 ÷ 24476. You should have 1.61, right?

This is the same thing as the golden ratio appearing in the Fibonacci sequence. To prove it, we will actually do something a little different than usual. We will add fractions “badly.” If you were a young kid, how would you guess adding fractions works?

I’d say add the numerators, then add the denominators. Like ½ + ¼ should be 2/6, or 1/3. This doesn’t give you the right answer, but it does assure that the answer is in between the two fractions. In this case, we are using line ten and line nine.

Line nine has its own formula: 13x + 21y, assuming that line 1 is x and line 2 is y. Line ten has formula 21x + 34y. So, we have:

21x + 34y/13x + 21y

This is the same as adding fractions badly. This says that this ratio is between 21x/13x and 34y/21y.

21x/13x = 21/13 = 1.61538…
34y/21y = 34/21 = 1.61904…

Both of these numbers begin with 1.61, meaning any number in between them will begin with 1.61. This proves that line ten over line nine is always 1.61, a great bonus prediction effect to the trick.

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