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1. 

   Becca said, 9 months ago

   For those who don’t know… Fibonacci Series: The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself, yielding the sequence 0, 1, 1, 2, 3, 5, 8, etc.

2. 

   treblemaker said, 9 months ago

   Why’s Peter complaining? He gets his three squares a day.

3. 

   tiger1tt said, 9 months ago

   neat method of informing d masses …i for one did not catch d pun until i read sappha 1958 comment…i have been enriched

4. 

   kfaatz925 said, 9 months ago

   Only squares, treblemaker? In Peter’s case I’d have said cubes.

5. 

   m_ortal said, 9 months ago

   13, 21…

6. 

   Dypak said, 9 months ago

   I always appreciate how Amend expects us to know and understand the joke, and doesnt have to explain it. The ‘Fibonachos’ punchline was great though! Amend gives his readers credit for being smart enough to get his jokes. Laugh, a search for fibonacho’s on Wiki actually refers you to fibonaci!

   In the book/movie, “The Phantom Tollbooth” there is a well done sequance on the Fibonaci series. Tiger itt mentioned informing the masses, getting kids to read that book would do it. Or at least see the movie.

7. 

   Mowgli-Chiara said, 9 months ago

   Thanks for made my brain to work on a Sunday

8. 

   Boxknight_Jace said, 9 months ago

   Dypak, The Phantom Tollbooth is one of the best novels out there. Everyone should read it as a kid and then again as an adult.

9. 

   Hugh B. Hayve said, 9 months ago

   I guess when they’re older they’ll be playing Fibeernacci.

10. 

    Geekologist said, 9 months ago

    Whats sad is I didn’t know what “Fibonachos” are till I read the comments…

11. 

    bigmitchperez said, 9 months ago

    luckily i play video games ,like “the da vinci code”,otherwise i may seem undereducated!

12. 

    LateToTheGame said, 9 months ago

    Hugh, awesome!!! Though I imagine at whatever age, Jason and Marcus would barely be able to hold a single draught, at most.

13. 

    cwreenactor said, 9 months ago

    Arrrrgggghhhhh!!!!

14. 

    johndrake said, 9 months ago

    I am fairly sure that the fibonacc1 numbers do not start with 0:

    1,1,2,3,5,8,13,…

    I believe that the wikipedia entry is in error. I am 99% sure of this but I could be 100% wrong. I will check.

    When I first encountered this sequence it was presented this way:

    Fibonacci had one pair of baby rabbits. At the end of two months they were still too young to mate. At the end of the third month the orginal pair had two babies. At the end of the fourth month the original pair had another set while the first litter were too young. And so it went. If he started with 0 pair this would not work.

    The Fibonacci numbers occur in all
    sorts of areas of mathematics.

    Yup, in “An Introduction to Number Theory” , Edward Burger states that the Fibonacci series starts 1,1.

15. 

    farren said, 9 months ago

    Johndrake: they’re talking about the Alternative Minimal Fibonacci series: 0, 0, 0, 0, 0, 0, …

16. 

    circuit7 said, 9 months ago

    ROFL! What a great laugh!

    And The Phantom Tollbooth should be required reading for every 12-year-old in the civilized world.

17. 

    Mickeysnotadog said, 9 months ago

    Yulk! They are double dipping! Too many germos here. They could Fibonacci themselves into the ICU.

18. 

    LateToTheGame said, 9 months ago

    drake, it sounds to me like the version you heard was drafted for some youngsters – I can’t imagine Fibonacci using bunny wabbits for part of his original thesis. As for the bunnies, 0 was probably left off since not too many spontaneously reproduce. Now if it were frogs or sharks perhaps…

19. 

    johndrake said, 9 months ago

    farrenPro says:

    Johndrake: they’re talking about the Alternative Minimal Fibonacci series: 0, 0, 0, 0, 0, 0, …

    —————————-
    hmmm, is that the AMF series or the Zen series? the Bertrand Russell series? {0} {{0}} {{{0}}}…

    ——————-
    LateToTheGame says:

    drake, it sounds to me like the version you heard was drafted for some youngsters – I can’t imagine Fibonacci using bunny wabbits for part of his original thesis. As for the bunnies, 0 was probably left off since not too many spontaneously reproduce. Now if it were frogs or sharks perhaps…

    ——————
    maybe zero was included - spontaneous generation was probably a respected (if not respectable) idea then. ;-]

    whether Fibonacci would use bunny wabbits, I cannot say, but I did have a college linear algebra instructor who suggested that I use maple syrup to solve a Steiner point problem in a transportation problem! of course, he said any sufficiently viscous liquid would do.

    I know that the idea of wabbits sounds like an idea some mathemagician pulled out of his hat to explain it to children, but consider this:

    if you stipulate that it takes two months before a pair of rabbits can reproduce and you start with newborns, then the pattern fits with the numbers produced (or is that re-produced?) starting with 1.

    Barbie may be right, but math can also be fun!

    I couldn’t imagine Fibonacci coming up with a series that pops up in so many places in mathematics and physics.

    (BTW, I have authorities to back me up on the wabbit question: Bugs Bunny and Crusader rabbit! Of course Daffy Duck claims rabbits had nuttin to do with it.)

20. 

    Liowatcher said, 9 months ago

    According to Wikipedia (http://en.wikipedia.org/wiki/Fibonachi):
    —–
    Liber Abaci also posed, and solved, a problem involving the growth of a hypothetical population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. The number sequence was known to Indian mathematicians as early as the 6th century, but it was Fibonacci’s Liber Abaci that introduced it to the West.
    —–
    So he did use rabbits, and in Latin!

21. 

    tobybartels said, 9 months ago

    It's arbitrary where you start the sequence. In fact, it can go infinitely far in either direction. As you go forwards, you add two numbers to get the next. But as you go backwards, you subtract them instead. So you get something like this:

    
    

    …, 8, −5, 3, −2, 1, −1, 0, 1, 1, 2, 3, 5, 8, …

    
    

    It is true that Fibonacci himself started with 1, 1, 2, …, whereas nowadays most people prefer to start with 0, 1, 1, …. But in the end, where you start is up to you.

    You can also get fun results starting with any two numbers, even if they don't appear in the Fibonacci sequence. For example, Édouard Lucas started with 1, 3, 4, …, so those are called the Lucas numbers. Bonus points for spotting where the Lucas numbers are hiding within the Fibonacci sequence!

22. 

    johndrake said, 9 months ago

    1,1,2,3,5,8 Fibo

    instead of adding contiguous numbers, skip one and you get Lucas:

    1, 1+2, 1+3, 2+5, 3+ 8,…

23. 

    tobybartels said, 9 months ago

    We have a winner! Johndrake gets a cookie, or whatever the kids are giving each other on the Internet these days.

24. 

    LateToTheGame said, 9 months ago

    Would that be a virus?

25. 

    slavesofspeigel said, 9 months ago

    1;1;2;3;5;8;13;21;34;55;89;144;233;377;610;
    987;1,597;2,584;4181;6,765

    The first 20 numbers of fibonacci!
    (Ithink

26. 

    johndrake said, 9 months ago

    beating a dead horse:

    start with -2, -1

    -2 -1 -3 -4 -7 -11 -18 …

    After the initial -2 you have the Lucas series but negative.

    What can you do with the Fibonacci or Lucas series?

    Suppose you take any four consecutive numbers of the Fibonacci series. Take the product of the first and last - call it x.
    Take the product of the second and third, double it, and call the result y. Take the square of the second and the square of the third, add the two squares together and call it z. You now the lengths of the legs of a right triangle (the x and the y) and the legnth of the hypotenuse - z. I.E.:
    xx + yy = zz.

    Same is true for the Lucas series.

    aren’t you glad you asked?

27. 

    zev.farkas said, 9 months ago

    I love when Amend gets scientific/mathematical! Thanks to all who commented. Particular thanks to johndrake for the comment about the relationship between the Fibonacci and Lucas series and Pythagorean triples! This is a fascinating result that I had not known about.
    I actually went to the trouble of working it out algebraically (tedious, but not too difficult) and it works out! Cool!
    It turns out that it works for any series where p(n)=p(n-2)+p(n-1).

    as for the following:

    “tobybartels says:

    It’s arbitrary where you start the sequence. In fact, it can go infinitely far in either direction. As you go forwards, you add two numbers to get the next. But as you go backwards, you subtract them instead. So you get something like this:

    …, 8, −5, 3, −2, 1, −1, 0, 1, 1, 2, 3, 5, 8,
    ”

    please check the transition from alternating negatives and positives to all positives… -1+0=-1, not 1.

    There is a closed-form solution to the Fibonacci series that involves the square root of 5, among other things… see what happens when you plug in negative arguments in that function (sorry, but it’s too late at night, and I’m too lazy, for figuring that one out right now - you’re on your own, math fans…)

    Finally, as to whether the Fibonacci series starts with zero or one, I think it’s just a matter of taste or the particular problem you’re working on.

    Reminds me of an old story. A couple comes to a Rabbi for help in settling a marital dispute. The wife tells her side, and the Rabbi says, “You’re right”. Then the husband gives his side, and the Rabbi says, “You’re right”. This seems to satisfy the couple, and they leave smiling. The Rabbi’s wife, who overheard the whole session says, “It can’t be possible that they’re both right!”, and the Rabbi says, “You’re also right!” :)

28. 

    johndrake said, 8 months ago

    pythagorean triples are hiding all sorts of places:
    take your age and double it (this is x).

    take your age, square it and subtract 1 (this is Y).

    square your age and add1 (this is z).

    (x,y,z) constitutes a Pythagorean triple.

    +++++++++++++
    another use of the Fibonacci numbers is a quick and dirty kilometer to miles converter. Once you get past three in the sequence, the ratio of a number to its predecessor is roughly 1.6 to 1. Now 1 mile is 1.59 Km. Suppose you want to know how many miles 50 Km equals?

    fibo: 1 1 2 3 5 8 13 21 34 55 89

    now 50 is close to 55 and 55’s predecessor is 34; so, 50Km roughly equals 34 miles (actually 31.1 miles). If you want a closer approximation you can notice that 50 = 34 + 13 + 3 .
    All the numbers on the right are Fibonacci numbers and their respective predecessors are: 21,8,2 whose sum is 31. So 50 Km approximately equals 31 miles.

    Not a bad approximation and you can do it on the back of a napkin.

29. 

    zev.farkas said, 8 months ago

    Thanks, johndrake -

    As an American transplanted to Israel, that mile to kilometer converter can be pretty convenient!

    Your comment on the relationship between the Fibonacci series and pythagorean triples helped me find a result I’ve been trying to come up with for some years now… a simple relationship that takes any two numbers and gives a pythagorean triple - like the age game you just described. (I’m sure I could have found it on the web, but it’s more fun to crank out the algebra yourself…)

    Thanks again!

30. 

    johndrake said, 8 months ago

    zev.farkasPro, you are welcome.
    I am just a conduit; see

    http://www.teach12.com/ttcx/coursedesclong2.aspx?cid=1495

    Be sure to check out the “two for one” special at the bottom of the page.