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hpr1143 :: The N Days of Christmas? Intro to Recreational Math

Intro to Recreational Math Part Zero

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mathematics, calendar counting. (Be the first).
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The N Days of Christmas? Intro to Recreational Math
Part Zero: Calendar Counting

First episode of HPR that contains a direct discussion of a math topic.
 - Episode 479 Ohio Linux Fest, Klaatu interviews DWick about math 
      software for Linux
 - Episode 523 Using Petunia software to teach math

Inspired by a traditional song that is proof that some songs do not
   need to be recorded by William Shatner to be annoying.
 - Repetitive and formulaic
 - Involves a lot of counting, and that's our focus here. 
What is the 12 Days of Christmas?
 - Starts on Christmas Day, runs through the day before the next Season
 - Hint: That's 'Epiphany', which starts January 6.
 - Counting calendar days comes hard, so we tend to use our fingers
 - Turns out that using our fingers is quite mathematical. Here's why.
Finger Counting: How do I count Twelve Days?
 - Let's start easy, with the fingers on one hand. My hands have five.
 - To name the Five Days of New Years is easy: January 1-5 <done>
 - What about the Five Days of Christmas?
        Physical way                   General way
   * Christmas Day gets 1 (thumb)     Dec 25 is one day after Dec 24
   * Dec 26 gets 2 (index)            26 - 24 = 2 days
   * Dec 27 gets 3 (salute finger)    27 - 24 = 3 days
   * Dec 28 gets 4 (ring)             28 - 24 = 4 days
   * Dec 29 gets 5 (pinky)            29 - 24 = 5 days
 - Notice that counting 5 days, starting with Dec 25, is the same
     as numbering the days after Dec 24 (Christmas Eve).
   * In math, we call this "1-1 correspondence with natural numbers"
   * Math can give you the same certainty as using your fingers.  
   * But it handles larger problems, because you don't run out.
 - Example: I'm booked to speak on Day 4 of a 5-day conference
   * Starts on the 25th of the month
   * When do I have to show up?
     - Wrong: Add 4 to first day (25), and arrive a day late.
     - Correct: Add 4 to date of pre-registration cocktail party (24),
          and arrive on time.
 - OK. Back to Twelve Days of Christmas.  
   * The labeling approach tells us that December can hold only the 
       first seven of the Twelve Days of Christmas, 
   * December 31 - December 24 gives me 7 days.
 Partitioning: Adding hands full of additional fingers as needed
  - How do we handle the case where we go into the next month?
  - Key insight: Running out of December days for the Twelve Days is 
       like running out of fingers on one hand when we count to 8.
  - We are so good at counting on our fingers that we don't recognize 
       the act of partitioning the number 8 between our two hands.
    *  Left hand gets 1, 2, 3, 4 and 5.
    *  Right hand picks up 6, 7 and 8 by mapping them to fingers 1,2,3.

  - To count even higher, we could:
      1) keep borrowing other people's hands, or
      2) track the number of times we reuse our two hands as we go
    * First method mirrors calendar math ("Annexing" hands, or months)
    * Second is positional notation ("base 10" and all that)

Back to the Twelve Days
 - I have Twelve Days: 1, 2, ... 12 to assign to dates, even though I 
     may only be interested in the first and last dates right now.
   * Start: How many can I fit into December?
   * December 31st is last. It gets assigned 31 - 24, or 7. 
   * By "finger math", that means I have mapped 7 of the Twelve Days
   * That leaves 12 - 7, or 5 days into January.
 - Who can tell me which days are assigned in January? Anyone?
   * That's right, Ken.  January 1, 2, 3, 4 and 5.
   * So the Twelve Days of Christmas runs 25 December to 5 January 

Question: What if there were 72 Days of Christmas?  When would it end?
 - Note: Don't worry.  This is purely hypothetical.

 - Let's attack this with finger math, with partitioning and annexing
   * December, as we have seen, accounts for 7 days: 25 through 31
   * That leaves 72 - 7, or 65 days
   * January easily picks up 31 days: 1 to 31, leaving 65 - 31 = 34 days
   * February can handle either 28 days, or 29 on a leap year.
   * This leaves us either 5 or 6 days into March

 - Final Answer: 72 Days of Christmas would run from Christmas until the
     following March 5 (leap year), or March 6 (all other years).
   * On Day 73, everyone would enter treatment for Christmas overdose.

Let's check the answer: Day 72 would end ten weeks and 2 days after 
   the opening cocktail party (Monday). So Day 72 should be Wednesday.
   * Next year is not a leap year, so last day is March 6.
   * By the Doomsday perpetual calendar method, Feb 28 is Thursday.
   Doomsday method:
   * So March 7 is Thursday, and March 6 is Wednesday.
   * It worked.
Why should I bother with Calendar Math?
 - I learn to look for ways to partition hard problems into easier ones.
 - I learn the same skills that I'll need to debug "off-by-one" errors
     and other boundary violations, which kill you in C programs.
 - I will never miss a speaking engagement, as long as I count my 
     Conference Days from the cocktail party, not from the Keynote.

Next episode: Part One
  Counting partridges and gold rings with Pascal
  - Warning: There will be two semi-magic formulas at the end.
  - I'll show you an easy way to do running sums in a spreadsheet.
  - You can skip the formulas, and I'll never know.
  - Since this is HPR, not school.  We can look up the formulas.

Contact: Charles in NJ

Charlie + Alpha + Tango + India + November + Tango + Papa.


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