Day 07 in Mat115

Today:

• 2.1 problems returned.
  • 14 of 29 -- almost half -- didn't turn one in. Many of those turned in were incomplete.
  • The solution that I ask to have typed should be on a separate page. Please double space those, so that I can make comments between lines, if necessary.
  • I.2:
    • Read carefully: "How could you estimate the number that would fill the box?"
    • One good approach
    • Another
    • "The ball used in table tennis has a diameter of 40 mm...." (formerly 38 mm). 40mm is about 1.6 inches: hence you can get 7 balls (but not 8) along the edge of the box. Therefore, you should be able to get about __________ balls in the box.
  • II.14:
    • Three solutions:
      • making use of the pigeonhole principle -- good!
      • Two solutions,
        • making use of good reasoning
        • making use of good reasoning
    • Watch the writing!
    • You need to do some analysis: it's not enough to simply write down some sock numbers to pick (especially if they're wrong!).

• Hand in problems 2.2 (if you didn't bring a Fibonacci pine cone or pineapple last week).

• FYI (and for Nick, especially): Solution to the Mindscape 1.4.15

• Questions/Comments on old stuff?
  • Last time, we learned
    • How to win at Fibonacci Nim;
    • What primes are,
    • How to find them using the Sieve of Eratosthenes,
    • How to decompose a number into its prime factors, and
    • That there are an infinite number of primes (although we didn't see a proof).

• Section 2.4: Crazy Clocks and Checking Out Bars:
  • What is "clock arithmetic" (i.e. modular arithmetic)?
    • What time will it be 27 hours from now?
    • What time will it be 27*24 hours from now?
    • What time will it be 25²⁴ hours from now?
    • What time will it be 30²⁴ hours from now?
    • What is 27 mod 24? What does "mod" mean?

  • Not all "clocks" have 12 hours:
    • Today is Wednesday, February 6th: on what day of the week will February 6th fall next year? How about Christmas, 2008?

    • A related problem: if your birthday this year is on Monday, what day will your birthday fall on next year?

    • Problem #8, p. 90

  • Here's a sneaky one:
    What is 2³⁰⁰⁰ mod 7? Why is this important? (Secret codes!)

  • What are UPCs? "Universal Product Codes" (a sort of "secret code")
    • How do they work?
      • First of all, they identify manufacturer and item
      • "3 1 3 1 3 1 3 1 3 1 3"
      • mod 10
      • What is a "check digit"?
      • Example: Kroger's Crunchy Peanut Butter
        0 11110 73933 9
    • It was my birthday Monday, and I enjoyed one of the following beers:
      • Moerlein OTR Ale: 8 97232 00070 7
      • Great Lakes Commodore Perry India Pale Ale: 7 04361 94001 1
      • Sierra Nevada Porter: 0 83783 47521 0
      (but just one -- I had to prepare for my classes!;). The incorrect UPC will tell you which I had last night. Which was it?

    • Why is correctly reading a UPC important?
    • What if a single digit is wrong in the UPC -- will a computer discover the error?

  • Let's break into groups to
    • Check samples.
    • Work on homework problems: Mindscapes pp. 90-94, I.2, 4; II.6, 7, 9, 10; III.32. Please type up Mindscape II.10.