Day 8, MAT115

• Here's a pair of video
  1. Part I
  2. Part II
  covering the material today.

• Your Babylonian and Mayan mathematics assignment was due, as of 8:00 am. I've chosen that weird due date/time because I divulge answers today, and I post stuff by 8:00 am. So you need to finish before that time!
  • Just the answers
  • First bunch o' numbers video
  • Second bunch o' numbers video
  • Here's the translation of the Mayan table (note the "leap months" -- the 177 increase isn't steady....)

• Last time you were introduced to a game, which you should have discussed with your classmates on this Canvas Discussion. Hopefully you did.

  A quick glance over there shows that five of you have. That's not a lot of engagement, which saddens The Great Fraudini greatly. And no one is particularly close to a solution.

  Rules:

  • This is a two-player game.
  • An arbitrary number of counters is thrown down (e.g. toothpicks, pennies, M&Ms). The winner is the player who picks up the final counter.
  • Player 1 must pick up at least one counter, but may not pick up all the counters.
  • Then players alternate taking counters, with the rule that you have to take at least 1, but no more than twice what your opponent took on the previous play.
  So our classmate Becky played a game or two of Nim against our friend The Great Fraudini; and, although she had a chance to win 20 dollars, alas! She did not take home the cash.

  This is a great game to have ready at all times. It's great to play against your nieces and nephews for their Halloween candy.

  The strategy is complicated enough that it's not likely that an ordinary human will figure it out in time. So you can generally always win, whatever the number of counters, and whether you start or go second.

  But don't play against someone who knows the strategy! Unless you can set things up in your favor. At the beginning of every game, once the counters go down, it's just a matter of who goes first -- and then whether they can stick with the proper strategy. The Great Fraudini made a misstep in his first game with Becky -- either planned, to let Becky back into the game, or just a careless mistake. And had Becky known the strategy (or had enough time to think about it), she could have taken that 20.

Today I will introduce you to the major players, which are known as the Fibonacci numbers.

• Our introduction begins with a trip to 13th century Italy, and the bunnies of Leonardo Pisano Fibonacci
  • Start with a new-born pair of "immortal" bunnies (they never die).
  • A new-born pair requires a month to mature.
  • The following month, the pair produces a new pair, which is subject to the same rules.
  • Do it again, do it again, do it again!

  Actually, this is how it was translated from Fibonacci's book Liber Abaci:

  A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

  The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ....

• The Fibonacci numbers as a recursive equation (the "do it again" part): \[ \begin{array}{c} {F_{1}=1}\cr {F_{2}=1}\cr {F_{n}=F_{n-1}+{F_{n-2}}} \end{array} \] (do it again, do it again, do it again!). So \[ \begin{array}{c} {F_{3}=F_{2}+F_{1}=1+1=2}\cr {F_{4}=F_{3}+F_{2}=2+1=3}\cr {F_{5}=F_{4}+F_{3}=3+2=5} \end{array} \] etc.

• It turns out that

  ---

  Every natural number is either

  1. Fibonacci, or
  2. can be written as a sum of non-consecutive Fibonacci's in a unique way.

  ---

  Examples?

• This is our third way of representing counting numbers uniquely, as combinations of special collections of numbers: we can now represent a counting number uniquely as
  1. a prime or a product of primes
  2. a power or two or a sum of distinct powers of two
  3. a Fibonacci number or a sum of non-consecutive Fibonacci numbers.

  Examples? Let's do a few....

• If the rabbit problem sounds a little artificial, with its incestuous bunnies and all, another mathematician (more modern) named Dudeney (1857 - 1930, pronounced Dude-knee) suggested this problem, which has precisely the same solution:

  If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

  (I might have said "how many female calves and cows are there after 12 years", because it's a mix of calves and cows -- all female of course. But hey, they're Dudeney's cows!)

• Fibonacci Numbers and Nature:
  • dandelions (Here's a scanned dandelion)
  • Spirals in daisies:
    • 21 spirals
    • 34 spirals
    • together
  • Let's look at some pineapples my wife bought today....
  • Similar things happen in artichokes, pinecones, etc.

• As we move toward the so-called "golden mean" and "golden rectangle", we'll start with a nice Fibonacci spiral (see this site)

• Graph Paper