Day 12 in Mat115H

Today:

• Announcement:
  • Your assignment for Crazy Clocks, etc. will be due Tuesday
  • Return of your bases homework
  • Our first exam is next week:
    • It will encompass both Lewis Carroll's story through Fit the Third
    • And the mathematics we have discussed, especially
      • Mobius bands
      • Fortunatus's Purse
      • The Tricks of p. 65
      • The binary card trick
      • Bases
      • Carroll's ciphers
      • Modular arithmetic and UPC codes

• Begin at the beginning and go on till you come to the end: then stop.
  -- Lewis Carroll

  The important thing is to begin at the beginning, and persist until you reach the end. Some things are hard (e.g. bases), but if we persist... we may make it to the end, where we can stop!;)

• Ciphers (pp. 76--)
  • his matrix cipher (1858)
    • How is modular arithmetic used in this cipher?
    • What's the message on p. 77?
    • Encode "The Jaws that bite, the claws that catch" using the matrix cipher.
  • his alphabet cipher (1868)
    • Can you use his alphabet cipher?
    • What's the message on p. 79?
    • Encode "The Jaws that bite, the claws that catch" using the alphabet cipher.
  • Find a partner, and exchange (short!) messages, using each of these codes. Make sure that you know how to use them!

• Crazy Clocks and Checking Out Bars (from The Heart of Mathematics)
  • 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?

  • Expanding binomials and Pascal's Triangle:

  • Not all "clocks" have 12 hours:
    • Today is Tuesday, September 9th: on what day of the week will September 9th fall next year? How about Christmas, 2009?

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

    • Problem #8 from your handout

  • 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
    • On my last birthday 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. 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 check a few samples...
    • Life Cereal
    • Keebler Club Crackers
    • Post Raison Bran Cereal
    • Kroger Brand Crackers