Day 09: Heart of Mathematics

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Announcements:
- Return Estimation Assignment
  - When you estimate, you need to talk about/using
    - Questions: what are the issues to consider?
    - Data -- where you got it, quality, etc.
    - Units
    - How you do your calculations
    - Bounds
  - Be neat, and use complete sentences to answer your questions.
- Your problems, due next Friday, will be due Monday instead (due to the missed day):
  - Show how to express 1729 in base 2.
  - Explain why we can't express 1729 using base 1.
  - Show how to add 147 and 173 (both numbers expressed in base 8)
  - Bring one thing from nature that illustrates Fibonacci numbers
  - Problems from section 2.2: pp. 60-61, #16, 18, 20, 22
- For Friday: read section 2.3: Prime Cuts of Numbers
For starters, I hear that the government has committed $1 trillion dollars in the last week to shore up failing private companies (E.g. Freddie Mac, AIG -- an $85 billion loan last night): how much is that to you and me?
Today, we continue using Fibonacci numbers.
- 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... How do we calculate the n^th Fibonacci number?
- Fibonacci number decomposition and Fibonacci Nim
  - Rules:
    - Two players, n sticks/coins/bottlecaps/etc.
    - First player must take anywhere from 1 to (n-1) sticks
    - From then on, the next player may take from 1 up to twice what the previous player took.
    - Winner is the player who picks up the last stick
  - Is there a winning strategy?
  - Is there a perfect defensive strategy?
- We "decompose" natural numbers using Fibonacci numbers and sums:
  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?

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