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HPR - HPR1143: The N Days of Christmas? Intro to Recreational Math

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Hacker Public Radio: 206 203 5729

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:

* 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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