Day 10, MAT115

Last Time

Next Time

Announcements:
- Welcome back from part of your "Spring Break" -- funny that it occurred while we were experiencing particularly vicious winter weather!
- It's about time for your first exam, wouldn't you say? I hope that you say "Yes", because it is indeed time for that. So the exam is written, and today we'll do a little review.
- Before we do that, however, let me tell you about the logistics.
  1. You may take it between 5:00 pm today and 11pm tomorrow. You should be prepared to spend about 75 minutes on it (although I will allow a little more time on each end). I'm thinking of this as a twice-a-week, 75 minute per day class -- so that would be what you would have received ordinarily.
  2. The exam will be "half and half" -- half taken by IMath, and half via Canvas and the Lockdown Browser.
  3. You will take them back-to-back, the IMath part first. At the last IMath question, you will be given a couple of randomly generated numbers: you will be using those on the Canvas exam. Write them down!
  4. You should have your own paper to write your answers to the Canvas exam. You will be staring at a pdf of the exam, and will need to write the answers on your paper. Then, when you're finished, you'll need to upload your exam as a pdf. So make sure that you're ready to do that before you start.
  5. In terms of material to be covered, I mentioned these things last time:
    - You will be responsible for the readings too, of course. Material in the readings is fair game for the exam. I might ask you summarize, or describe, or explain a figure, or....
      Nothing too complicated, but I trust you will have done the readings -- and really read them (not just skimmed over).
    - You should know how to write counting numbers in
      - Binary
      - Babylonian
      - Mayan
      - Prime factorization
      - Fibonacci factorization
      - As trees in any one of primitive counting (binary), primes, or fibonaccis.
- Here's a video covering today's review material.
Today we review. We'll go from most recent to oldest.
1. 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 (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.
2. 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.
3. 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.
4. 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.
5. 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)
6. We talked about prime and composite numbers.
  - Prime numbers are those 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 noted above -- 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!).

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