Day 8, MAT115

• Announcements:
  • Reminder: turn off your rectangles!
  • Sorry, haven't gotten to your Mayan tables yet.
  • You do have a new homework, however, due Thursday next.
  Question of the Day: How do I win cash and other goodies by mastering Nim?

• Review of Egyptian division:
  • Let's begin with that other problem, just to get warmed up:
    1. 21*79 (= 1659)
    2. Then 1659/79 (= 21)

    In the division problem we know the dividend, so we reverse the process: we find numbers on the right that sum to the dividend, and then add up the corresponding numbers on the left to give the quotient, which is what we're after.

  • Last time I showed that we can do division using the Fraudini trick and the unit fractions table. So let's try those same examples but using the unit fraction table rather than our doubling tables.
    1. 35/8
    2. Here's a relatively easy one: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?
    3. A little trickier:
       1. How would you divide 31 by 13? Let's use the multiplication method backwards first.
       2. How can we use the unit fraction table to get the same answer?

  • How would you like to do story problems like this one?!

• Today we move on to the Nim Game: I've told you that I like to give you ways to make money, and this is the best one I've got. So learn this game! Play it for keeps, and you'll make a lot of money over your lifetime.
  • 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.

  • Okay, so what's the strategy? How does one win at Fibonacci Nim?

    To answer that question, we must first retreat to 13th century Italy, to study 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?

  • 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)

• Unit Fraction Table
• Unit Fraction Table

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