Day 9, MAT115

• Announcements:
  • Reminder: turn off your rectangles!
  • Return of the Mayan tables...
  • You do have a new homework, however, due Thursday.
  Question of the Day: How do I win cash and other goodies by mastering Nim?

• 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.

  • 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?

  • Sample games:
    • Let's play a sample game together: Suppose that there are 27 counters on the table. I will show you my strategy, and your job is to learn it.
    • You play a sample game with a partner: Suppose that there are 43 counters on the table.

  • So here's the winning strategy. It comes down to two things:
    • If the number of counters starts as a non-Fibonacci (more common), try to go first. Otherwise, if Fibonacci, go second if you can. Then you're in the driver's seat.
    • If at any time you can apply this strategy, you are guaranteed to win:

      Write the number of counters as a sum of non-consecutive Fibonacci numbers, and take the smallest in the sum.

    If you play with this strategy, you will win -- provided it's possible. That is, provided you're not playing against someone else who knows the strategy, and also uses it, and starts in a better position than you.

  • Fibonacci Numbers and Nature:
    • Let's look at some pine cones....
    • dandelions (Here's a scanned dandelion -- a scandelion)
    • 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

fish fish fish
fish fish fish