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?
  Can we justify this?
  - If a number is Fibonacci, then we're done.
  - Assume a number is not Fibonacci:
    - then there is a largest Fibonacci that "fits inside it" -- e.g. 24 = 21 + 3.
    - If the remainder (3, in the example above), after subtraction, is Fibonacci, how do we know that it is not consecutive (that is, that it is not 13)? That is, how do we know that the two Fibonacci numbers are not successive Fibonacci numbers?
    - If it is not Fibonacci, can we not simply repeat the current process of looking for a sum for a non-Fibonacci number, but using the remainder instead of the original number? (i.e. can we not simply "recurse" -- that is, do it again, and so construct a chain of numbers leading down towards 1.
      Example: 33=21+12=21+8+4=21+8+3+1

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