Day 7, MAT115

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Announcements:
- Your Mayan homework assignment is returned.
  - Mayan numbers:
    1. 57 = (2 17)
    2. 222 = (11 2)
    3. 817 = (2 4 17)
    4. 7581 = (1 1 1 1)
    5. 9432 = (1 6 3 12)
    6. 79420 = (11 0 11 0) -- a whole bunch of you neglected to use the Mayan symbol for zero. It's their big new thing, that the Babylonians didn't have. Make sure that you don't forget it!
  - Your Mayan Tables, too. For those I spot checked five values, and a couple of "calendar" dates (e.g. 6-16-0).
  - Each part was worth 6 points, for a total of 12.
- You have an assignment due today ("Fraudini" numbers).
- I've also assigned something for next time -- completion of your Yanghui triangle, including addition of the next row.
- The link for the stories didn't get into the assignment page properly, so let's put off the vote until next time:
  For next time, visit the Babylonian stories, and decide which is best. Then we'll vote!
Question of the Day: How do I win cash and other goodies by mastering Nim?
Last time we had a look at something I called "Yanghui's triangle":
What questions do you have? Last time we answered some of these:
- Mathematicians look for patterns. What do you see?
- What is this? What is its purpose?
- One of your homework problems will be to translate this triangle, so you need to learn its secrets!
Let me review that card trick from last time. Can anyone else do it? Do any of you have an explanation to turn in?
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.
  - Objective: The player who picks up the final counter wins.
- So I'm going to play a game against someone, who can win my money. You don't have to put anything down, because I'm going to beat you anyway. Who wants to play, and thinks maybe they can win my money?
- Okay, so what's the strategy? How does one win at 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
  
  ${F_{1}=1$
  ${F_{2}=1$
  ${F_{n}=F_{n-1}+{F_{n-2}}}$
  
  (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?
- 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:
  - 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