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 <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: 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: firstname.lastname@example.org Charlie + Alpha + Tango + India + November + Tango + Papa.
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