Day 15, MAT115

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Announcements:
- Homework is returned. I'll post the stories I've received -- at the moment, only two! So get in on the contest if you want.
- You have an assignment due to be handed in today.
- Our exam is this coming Friday. It will cover everything up to rock groups. Come Wednesday with questions about the material to be covered on the exam.
Question of the Day: How do I win cash and other goodies by mastering Nim?
Before we get to that, however, let's review what we did last time: Rock Groups.
- Every natural number is either prime, or can be expressed as a product of primes in a unique way.
- How do we say the formula suggested at the top of page 10? (By the way, add one more row to the top of the equations near the top of the page - what would the row be?
  How would you say what's happening in words? What do you think of the proof of the "theorem" at the bottom of the page?
  1. Do you see the theorem expressed in the rocks?
  2. Can you see the form "2n+1" in the rocks?
  3. I see a new theorem: T₅+T₄=5² (In words, "the sum of the fourth and fifth triangular numbers is the fifth square".) By playing with rocks we discover relationships. It's probably how primitive civilizations did it, too!
- The story of the "The Housekeeper and the Professor" is interesting, and modelled on a famous mathematical event -- the story of child prodigy Carl Friedrich Gauss:
  
  Once again, rock groups give us a formula for representing triangular numbers. So let's go through the reasoning for the housekeeper's answer.
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