Day 16, MAT115

Last Time

Next Time

Announcements:
- Homework is returned.
- I need to receive extra credit contest stories by today.
- Our first exam is Friday. It will cover everything up to rock groups.
Question of the Day: How do I win cash and other goodies by mastering Nim?
I'll start with questions about the material to be covered on the exam, however:
1. Fun and games (Probability)
2. Historical mathematics (Mayan, Babylonian, Egyptian, Pascal, Chinese)
3. Don't forget to do your reading!
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.
Fibonacci:
- 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?
- The Fibonacci numbers as a recurrence relation
  
  ${F_{1}=1$
  ${F_{2}=1$
  ${F_{n}=F_{n-1}+{F_{n-2}}}$
  
  (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:
  - 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)

Website maintained by Andy Long. Comments appreciated.

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