Day 9, MAT115

• The quiz this week will be over the Babylonian and Mayan number systems; also the class agendas, and the readings.
• You have an assignment, which is a reading.

• We had a quiz: it was a rough one, primarily because folks didn't seem to get the beautiful number patterns within the triangle.
  • Key
  • Median: 6.25
  • (Remember: I'll drop your three lowest quizzes.)
  So I want to make sure that we highlight those before we leave the triangle:
  

  Do you remember when we "built" the tetrahedral numbers with pennies? Those are the 1, 4, 10, 20 numbers.... built up of triangles (triangular numbers), etc. This is a three-dimensional version of stacks of triangles, which are two dimensional, made up of counting numbers stacked, which are made up of pennies stacked one on top of the next....

  Pascal's triangle contains the powers of 2 as sums of the rows (as seen in this version of the triangle):

  

  Finally, there was the use of the triangle for counting things:

  

  In terms of your quiz, the answer to the third question was contained in the 10th row of the triangle, which you might have noticed in the first problem; right in the center, in the middle position (notice that the position of \({10 \choose 5} = 252\) is the sixth entry in, because we start indexing the entries from the 0^th...).

• We wrapped up the Babylonians' and Mayans' number systems by writing some numbers in each system.

• Shall we do an example?

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 well, 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?

• Hexagonal Paper