Day 26, MAT115

• This session will be captured on Zoom, if I remember to turn it on, and record it.

  Our page of zooms and the play-by-plays.

• On Friday I struck for the climate, in association with Fridays for Future.

  I was the 4th floor atrium, near the central elevators, from 9am-3pm, with forty dollars in one dollar bills in my pocket. I didn't lose one of them! I had hoped to lose them at Fibonacci Nim to each of my students, but alas.

  I hope that I won't take any of your dollars during the exam, but, if history is any judge, I'll take a fair number. Don't let me take YOUR dollar! Learn how to play.

• Your Egyptian multiplication and division homework has been graded. Multiplication seems to be going quite well; division, not so much!:)

  There are lots of tricks, and I want to show off a couple that students discovered.

  1. nice tricks on #3
  2. Three ways to do 3a (including this, the cowboy way!)

• I hope that you got to your reading assignment (regarding Fibonacci numbers):

  Chapter 10: Working Your Quads; and also a short description of the Fibonacci numbers

  Today we'll focus on the second reading; next time, back to your Quads.

• You have an IMath assignment, due Monday (4/4), which will check to see if you're able to win at Fibonacci Nim.

  But there are also some throwback questions, to make sure that you're getting ready for our next exam (not this Friday, but the following Friday).

• The third and last great factorization (into Fibonacci numbers)

• How to use the Fibonacci factorization to win at nim.

(and, unlike most of our questions, I don't have the answer yet!)

Before we go there, however, I have a little story to tell: Margaret Atwood (author of "The Handmaid's Tale"), had this exchange with Ezra Klein in a recent podcast (about 8m24s):

Margaret Atwood:

Well, stories, by their very nature, have central characters, unless they're history stories dealing just with statistics, but we know that they're much more likely to be able to remember a story that is about a person or people, not one that is just about numbers, unless we make the numbers themselves into actors in the story. [Let me] tell you a story about the number nine, a very heroic number.

Ezra Klein

I watch a lot of "Sesame Street" these days.

...

Margaret Atwood

Yeah, so you make the numbers into sort of entities and then we can be interested in them. But if they're just numbers, not so much. We didn't develop math until pretty late in our human history, whereas we developed language and music very, very early. So stories come naturally to small kids. You know this yourself. So this happens, then this happens and then this happens. They understand that there's a plot, and that there are actors in the plot, like their teddy bear. So it's really built in.

And I think what kids do before the age of two is pretty indicative of what comes with the toolkit. They already are doing little dances. They have a sense of rhythm. They're very interested in music. And they're very interested in words and facial expressions. But they're not interested in nine times nine at that age, if ever.

Ezra Klein

To my father's enduring disappointment -- he's a mathematician, and I was never that interested in nine times nine.

(I'll use a tree to tell this story, of course!)

Let's find the gcd of 40 and 12, and then 40 and 7 -- but without using the prime factorization! Using rectangles....

(Next time we're going to be talking about the golden rectangle.)

1. Recall the Fibonacci factorization:

   ---

   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. 89 is Fibonacci
   2. 24 = 21 + 3
   3. 67 = 55 + 8 + 3 + 1

2. 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
      • arrange to 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 the unique sum of non-consecutive Fibonacci numbers; call the smallest Fibonacci in the sum $m$.
      • 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 will find that you are not allowed to take even the smallest of the Fibonaccis -- it will be more than twice what your opponent picked up, so that would be illegal.
   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.

3. 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.