Day 23, MAT115

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Announcements:
- 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.
- No mask day! Wear one if you like, however....
- Your Egyptian multiplication and division homework is due this Wednesday (submit on Canvas)
Last time:
- We wrapped up Egyptian division with the use of the Unit Fraction Table. We'll do one more example today, just to make sure you're headed in the right direction for the homework.
Today's Question of the Day:

How can I win a million dollars playing Nim?
- Before we win a million dollars, however, let's begin with an Egyptian division problem (and we'll do it by both the "multiplication table backwards" and unit fraction method).
  What's 45/13?
  - Answer's on the left!
    
    1 13 *
    2 26 *
    1/13 1
    2/13 2 *
    4/13 4 *
    
    45 = 26 + 13 + 4 + 2
    
    Division:
    
    Answers on the left Doubling in the middle Build on the right
  - Now there's more to do, because we know we can't leave the answer that way. And there is another way to get the same answer, using the Fraudini trick and the Unit Fraction Table. So let's try those same examples but using the unit fraction table rather than our doubling tables.
    First of all, the Egyptians knew enough to distribute out all the pies they could at the outset:
    \[ \frac{45}{13} =\frac{39+6}{13} =3 + \frac{6}{13} \] Now we have to deal with that fraction on the right (using the Fraudini trick): \[ \frac{45}{13} =3 + \frac{6}{13} =3+\frac{4}{13}+\frac{2}{13} =3+2\left(\frac{2}{13}\right) + \frac{2}{13} \] We look up $\frac{2}{13}$ in our unit fraction table, and \[ \frac{45}{13} =3+ 2\left(\frac{1}{8}+\frac{1}{52}+\frac{1}{104}\right) + \left(\frac{1}{8}+\frac{1}{52}+\frac{1}{104}\right) \] or \[ \frac{45}{13} =3+ \frac{1}{4}+\frac{1}{26}+\frac{1}{52}+ \frac{1}{8}+\frac{1}{52}+\frac{1}{104} \] Remember that you can't repeat fractions, so now you have a little more work to do: combine the two $\frac{1}{52}$ terms, to get one $\frac{1}{26}$ term: \[ \frac{45}{13}=3 + \frac{1}{4}+\frac{1}{26}+\frac{1}{26}+ \frac{1}{8}+\frac{1}{104} \] and then combine the two $\frac{1}{26}$ terms, to get one $\frac{1}{13}$ term: \[ \frac{45}{13}=3 + \frac{1}{4}+\frac{1}{13} + \frac{1}{8}+\frac{1}{104} \] And then maybe put them in order, \[ \frac{45}{13}=3 + \frac{1}{4} + \frac{1}{8} + \frac{1}{13} + \frac{1}{104} \] Finally, when you're done you can plug both sides into your calculator to make sure that you've done it right: \[ 3.461538 = 3.461538 \] We're good! Ready to move on to that next problem....
So now it's time to win that million bucks. Today we move on to the game I will call "Fraudini Nim".
I've told you that I like to give you ways to make money, and this is the best one I've got. So learn this game! Play it for keeps, and you'll make a lot of money over your lifetime. You'll also score points on your next exam....
- Rules:
  - There are two players, Player 1 and Player 2.
  - An arbitrary number of counters is thrown down (e.g. toothpicks, pennies).
  - Objective: The player who picks up the final counter wins.
  - Player 1 will go first. On the first play, Player 1 must pick up at least one counter, but may not pick up all the counters (otherwise this would be a silly game!).
  - Player 2 must then pick up at least one counter, and not more than twice the number of counters picked up on Player 1's turn.
  - The players then alternate turns, subject to these same conditions: pick up at least one, no more than twice what the preceding player took.
  - The player who legally picks up the final counter on their turn wins.
- So I'm going to play a game against someone, who can win my money. You don't have to put anything down, because I'm going to beat you anyway. Who wants to play, and thinks maybe they can win my money?
- Okay, so what's the strategy? How does one win at Nim? How could I be so confident that I was going to beat anyone?
- Let's try to figure out the strategy, by creeping up on it. What do I mean? Start with some really simple games....
  - If we look at some of the simplest cases, what can we conclude?
    
    Number of counters 1 2 3 4 5 6 7 8
    
    Winner (1 or 2? -- if they play right!) X
  - What's the winning strategy in Fraudini Nim?
Links:

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Number of counters	1	2	3	4	5	6	7	8
Winner (1 or 2? -- if they play right!)	X