Exam 2 Review

Readings/Viewings:

This reading on number systems
Chapter 10: Working Your Quads;
A short description of the Fibonacci numbers
Is Geometry a Language That Only Humans Know? (here's the source -- the New York Times). Neuroscientists are exploring whether shapes like squares (regular rectangles) and and our ability to recognize them are part of what makes our species special.

Topics/Concepts:

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.
Fibonacci numbers and spirals
- Know your Fibonacci numbers: 1,1,2,3,5,8,13,21,34,55,89,....
- Fibonaccis within Pascal's triangle
- attaching squares to squares to make a spiral. Here are some beautiful spirals that another class made for me:
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.
The golden ratio and rectangle
- ratios of Fibonaccis lead to gold: 3/2, 5/3, 8/5, 13/8, ...., $\phi=\frac{1+\sqrt{5}}{2}\approx{1.618}$
- Removing squares is how we think about deriving the value of $\phi$. But you can also spiral
- Quadratic formula -> $\phi$ (You finally needed that! Your high school teacher told you that you'd need it someday -- except for Bradley's teacher, who said it would never come in handy. Shame!)
Egyptian Multiplication
- Powers of 2 -- doubling
- Binary decomposition of numbers (the Fraudini trick)
Egyptian Division
- Powers of 2 -- doubling/halving
- multiplication table backwards
- unit fraction table
- Why did they do it this way? Fair division?
Graph Rules:
- Euler paths -- can you trace the graph without picking up your pencil, and without going over any edge twice?
- Euler's formula for planar graphs (ran-to: R-A+N=2)
- Pascal's triangle applications for counting edges in the complete graph of $n$ vertices; or types of each type of simple Facebook graph -- those with each different possible number of edges.

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