Exam 1 Review

Readings/Viewings:

"From Fish to Infinity" in your text (and make sure you watch the video from Sesame Street!)
"Rock Groups" (p. 7)
Location, Location, Location (p. 35)
this article on One-to-one correspondence.
Vi Hart teaching us how to Binary hand-dance

Topics/Concepts: Let's start at the very beginning: a very good place to start.

We talked about prime and composite numbers: rock groups!
- Prime numbers are numbers whose "rock groups" are just single file: you can't block them in nice ways. The exception, perhaps, is 1: that's not prime (because it would ruin a theorem -- the prime factorization theorem -- that says that there's only one way to write a counting number as a product of primes (order aside).
- The prime numbers can be found using the Sieve of Eratosthenes, and a process of cancellation.
- To find prime factors of a number $n$, start small and work your way up to the square root of $n$. If you haven't found any prime numbers by then, the number $n$ must be prime.
- We learned some interesting facts about primes, on Day 5 (a prime numbered day). Like there's an infinite number of them, and that they protect your credit card! That there are "twin primes", like 29 and 31 (even triplets, like 3, 5, 7!).
Although we didn't explore the Chinese bamboo counting of Yanghui's triangle, we did explore the triangle itself -- often called "Pascal's Triangle" -- and discovered several of these interesting types of numbers within it.
- Counting numbers (second diagonals)
- Triangular numbers (third diagonals)
- Tetrahedral numbers (fourth diagonals)
- Powers of 2 (sums of rows)
Trees proved useful to carry out many of our calculations: in particular
- primitive counting relied on dividing sheep, and we used a tree to keep track of the sheep in their pens.
- We used trees to break down numbers in different bases and number representations (e.g. binary, Mayan, Babylonian).
- We used a tree to break down a number by its prime factorization.
We studied a variety of different cultures' counting schemes, and their representations of numbers. Some were very primitive; some were very advanced. Some were described by muppets....
- Counting by cairns, tallies, knots, body parts (counting via one-to-one relationships)
- primitive counting -- equivalent to binary representation (base 2)
- Babylonian representation -- base 60 (no zero)
- Mayan representation -- base 20 (with 360 as one place)
All of our representations can be thought of as using different denominations of bills to make change.
- decimal (base 10); 10 digits; places of 1=10⁰, 10=10¹, 100=10², etc.
- binary: 2 digits (0 and 1); bills (and places) of powers of 2: 1, 2, 4, 8, 16, 32, ....
  This representations is at the heart of the binary card trick.
- Babylonian: 59 digits (no zero) are represented using 1s (martinis) and 10s (boomerangs); places by 60: 1=60⁰, 60=60¹, 3600=60², ....
- Mayan; 20 digits made up of 1s (dots) and 5s (bars); places of 1=20⁰, 20=20¹, 360=18*20¹, 7200=18*20², etc.
Factorizations or Decompositions of Counting numbers: there are three ways of representing counting numbers uniquely, as combinations of special collections of numbers: we can now represent a counting number uniquely as
1. a prime or a product of primes (except for the number 1);
2. a power or two or a sum of distinct powers of two;
3. a Fibonacci number or a sum of non-consecutive Fibonacci numbers.
We ended up on Fibonacci numbers, and in particular, on a game called "Fibonacci Nim". Fibonacci numbers popped out as the solution of an interesting story problem about rabbit pairs back in Italy, around 1200, and are easily calculated: you need the first two Fibonacci numbers (1 and 1), and then add the first two to get the third, and then successive pairs to get the pattern 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.
The winning strategy of Fibonacci Nim. Suppose there are $n$ counters on the table to start.
1. First or Second? To be "in the driver's seat", you should
  - go second if $n$ is Fibonacci;
  - go first otherwise.
2. Your move? If you're in the driver's seat, then you'll be able to do this:
  - First of all, if you can legally take all the counters, do so! Then you win. Otherwise,
  - Write $n$ as a sum of non-consecutive Fibonacci numbers; call the smallest Fibonacci $m$.
  - Take $m$ counters.
  You'll be guaranteed a victory.
3. Now here's another key point when you're on the defensive: when you're not in the driver's seat, take 1 -- the smallest legal move. That's because you want to slow down the game, and let your opponent make a mistake and put you back in the driver's seat.

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