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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
- 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: http://en.wikipedia.org/wiki/Doomsday_rule
* 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
Email: catintp@yahoo.com
Charlie + Alpha + Tango + India + November + Tango + Papa.
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