Day 13, MAT115

• Announcements:
  • Your exams are returned.
    • Curve: Add 10 points to your score.
    • The Exam
    • The Key

  • You have an assignment due today.

  • You have a reading assignment for next time: read the section on Fibonacci numbers (Idea #11, p. 44).

• Question of the Day:
  What is the secret to winning at Nim?

• The Nim Game:
  • Rules:
    • There are two players, Player 1 and Player 2. Player 1 will go first.
    • An arbitrary number of counters is thrown down (e.g. toothpicks, pennies).
    • Player 1 must pick up at least one counter, but may not pick up all the counters.
    • Player 2 must then pick up at least one counter, and not more than twice the number of counters picked up on Player 1's turn.
    • The players continue taking turns, subject to these same conditions: pick up at least one, no more than twice what the preceding player took.
    • The player who picks up the final counter wins.
  • Let's play a sample game: Suppose that there are 27 counters on the table.
  • You play a sample game with a partner: Suppose that there are 43 counters on the table.

• Now let's retreat to 13th century Italy for a moment: 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!

• Fibonacci number decomposition and Fibonacci Nim.
  • If you looked at some special cases, what can we conclude?

    Number of sticks	1	2	3	4	5	6	7	8
    Winner	X

  • What's the winning strategy in Fibonacci Nim?
  • Ratios of Fibonacci numbers

• The Fibonacci numbers as a recurrence relation (do it again, do it again, do it again!).

• More details on Fibonacci numbers and nim. 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?

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

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

• Mayan calendar conversions

Links:

• My unit fraction table handout
• Today's handout
• A good reference on Mayan mathematics