Day 9, MAT115

Last Time

Next Time

Announcements:
- This is the end of new material, prior to our exam (next week, 2/18-2/19, all or part taken via Lockdown browser).
  You'll need a webcam.
  There may be an IMath component as well.
- Here's a video covering today's material: The Secrets of Fibonacci Nim
- You have a new homework assignment to help make sure that you have got the game down. You'll need to be able to beat me on the exam.
  The assignment also includes some review of some of the other things we've done.
- By the way, 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.
Today you're going to get the strategy to win at Fibonacci Nim. The secret is, as you might guess, in the Fibonacci numbers.
- Our introduction began with a trip to 13th century Italy, and the bunnies of Leonardo Pisano Fibonacci
  - Start with a new-born pair of "immortal" bunnies (they never die).
  - A new-born pair requires a month to mature.
  - The following month, the pair produces a new pair, which is subject to the same rules.
  - Do it again, do it again, do it again!
  The resulting sequence of bunny pairs is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ....
- The Fibonacci numbers are best thought of via a recursive equation (the "do it again" part): \[ \begin{array}{c} {F_{1}=1}\cr {F_{2}=1}\cr {F_{n}=F_{n-1}+{F_{n-2}}} \end{array} \] (do it again, do it again, do it again!). So \[ \begin{array}{c} {F_{3}=F_{2}+F_{1}=1+1=2}\cr {F_{4}=F_{3}+F_{2}=2+1=3}\cr {F_{5}=F_{4}+F_{3}=3+2=5} \end{array} \] etc.
- "Fibonacci factorization": It turns out that
  
  Every natural number is either
  1. Fibonacci, or
  2. it can be written as a sum of non-consecutive Fibonacci numbers in a unique way.
  Examples?
  1. 34 is Fibonacci
  2. 38 = 34 + 3 + 1
  3. 98 = 89 + 8 + 1
- This is our third way 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
  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.
- Now, for the secrets of Fibonacci Nim. You need to learn this strategy, and you can't lose.
  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.
    You'll know that you're not in the driver's seat if you can't follow this strategy. The reason why? Remember that you can only take up to twice what your opponent took. Although you might write the number of counters remaining as a sum of Fibonacci numbers, you are not allowed to take even the smallest of the Fibonaccis.
  Let's look at some examples:
  - 29 counters, and you're first. You're in the driver's seat, since 29 isn't Fibonacci.
    29 = 21 + 8
    so you take 8.
    I'm up, and looking at 21. It's already Fibonacci. I can't take the smallest in the "sum" -- 21. So I'm out of luck. I can't follow "the strategy". Whatever I take, you'll be able to beat me.
    To slow things down, I take 1.
    It's now 20 to you. 20 = 13 + 5 + 2. So you take 2 (that's legal). And so on, and you're going to beat me. Argh!
  - 34 counters, and you're second. You're in the driver's seat, since 34 is Fibonacci.
    Suppose I go first, and take 4.
    Then it's 30 to you. 30 = 21 + 8 + 1, so you take 1. Now it's 29 to me.
    You might think "29 to Long, and he can pick first -- he's going to win!" But that's wrong. I'd like to take 8, as before, but I can't -- because 8 is more than twice what you took.
  Now here's another key point: 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.
- So basically, the whole things comes down to this. If you can choose whether to go first or second, do it and you're guaranteed a victory.
  1. Go first for non-Fibonacci, second for Fibonacci.
  2. Of course, take all of the counters is that's a legal move (this is what Fraudini could have done when playing Becky -- but he took one, to keep the game going).
  3. Whatever the number of counters on the table, write it as a sum of non-consecutive Fibonacci numbers;
    1. take the smallest Fibonacci, if you can; you're now in the driver's seat, and should win.
    2. if you can't take the smallest (because it's more than twice your opponent's move), take 1 -- it will slow the game down, and hopefully your opponent will make a mistake.
- Here's the pdf recap of my game with Becky, along with a new game, and the video recap.
- We talked last time about Fibonacci Numbers and Nature: turns out nature loves Fibonacci numbers!
  After our exam, we'll push forward with some of the beautiful aspects of Fibonacci numbers.
  As we move toward the so-called "golden mean" and "golden rectangle", we'll reconnect with that nice Fibonacci spiral (see this site), and I'll show you the relationship between the Fibonacci spiral and the Golden rectangle.
  But that's for a later date.
Links:
- Graph Paper

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Updated on 02/11/2021 06:26:23