It is sometimes easy to lose sight of the fact that math contests are meant to be fun and engaging, but not ultimate ends in and of themselves.
The main value of math contests is not to be found in the honors and awards (which are very nice, of course, but somewhat beside the point). The main value of math contests is to be found in wonderful mathematics it can draw students into discovering. Not only do they discover mathematics, but they can also discover the joys of working hard towards shared goals with a community of kindred spirits.
Math contests are a little bit like mountain climbing. The views from the top are nice, but even spectacular views do not justify the extreme efforts it can take to get there. What can justify those mountain-climbing efforts are the joys of the journey, especially if shared along with others, and the beautiful things you can learn about the natural world and the power within yourselves as you help one another to discover hidden potentials and problem solving abilities within you that you did not know existed.
Just as you should "stop and smell the flowers" as you climb the mountain, so too should you remember to "savor the problems" you encounter in math competitions. By construction, those contest problems have already been solved, but reflecting on them after the contests are over may inspire you to create new and fresh ones of your own devising--or you may discover exciting new ways to solve old problems.
A recently revived discussion on the Art of Problem Solving discussion forums reminded me of a remarkable essay written three years ago by a student on his contest experiences, which has resonated with many readers. The essay is very moving, thoughtful, and beautifully written. Both the essay (linked in the first post on the thread) and the discussion which followed it are readings that I would highly recommend to students, parents, and math teachers and coaches.
I would also recommend reading an inspiring speech given by our alumna speaker at Math Prize for Girls last fall, which also addresses these issues, and gives some excellent advice well worth bearing in mind for making the most of your math contest adventures.
You may also want to explore the links to the authors' blogs, which discuss some of the wonderful mathematics they have been inspired to learn and share with others.
There is more recommended reading on these topics here. Some of our older alumni may also find useful advice and insights in a similar vein here.
Sunday, April 29, 2012
Monday, April 23, 2012
Drawing new circles
The folks at the First Unitarian Society of Schenectady celebrate this poem:
Many people feel "shut out" of the mathematical community. They see math as a superpower that others have but that they hopelessly lack. They can't imagine math as a joyful and empowering activity, as hard yet rewarding work. They see the world as divided into non-intersecting circles of "People who can do math," and "People who can't do math." They place themselves squarely in the latter and can't imagine that they could ever find joy and empowerment in visiting the other circle. They may even be inclined to disparage or make fun of others who claim to enjoy math.
Can we find a way to draw math circles in a way that draws those folks in? I think we can. That is where my concept of "Guerrilla Math Circles" come in.
To be continued ...
He drew a circle that shut me out—
Heretic, rebel, a thing to flout.
But Love and I had the wit to win:
We drew a circle that took him in!
This poem resonates with me in many ways, but today I want to focus on the way it connects to my vision for guerrilla math circles.— "Outwitted" — Edwin Markham
Many people feel "shut out" of the mathematical community. They see math as a superpower that others have but that they hopelessly lack. They can't imagine math as a joyful and empowering activity, as hard yet rewarding work. They see the world as divided into non-intersecting circles of "People who can do math," and "People who can't do math." They place themselves squarely in the latter and can't imagine that they could ever find joy and empowerment in visiting the other circle. They may even be inclined to disparage or make fun of others who claim to enjoy math.
Can we find a way to draw math circles in a way that draws those folks in? I think we can. That is where my concept of "Guerrilla Math Circles" come in.
To be continued ...
Guerrilla Math Circles, Math Super Powers, Math as Performance Art, and Math for the 99%?
I love the dual messages encoded in the T-shirt logo design above:
- Math is a superpower!
- Share with it with everybody!
(∀ is a mathematical symbol that means "for all.")
If you also love this logo (designed by mathematician Cindy Traub of the always awesome St. Mary's College math department), please go here to vote for it. If the design wins the contest, then mathematicians from all over the country attending this summer's Math Fest will get t-shirts with this logo, and I think it would be outstanding to have this message spread far and wide.
Mathbabe has been blogging about how math is a superpower, and that wonderfully evocative and inspirational phrase has been reverberating around in my head ever since I encountered it for the first time on her blog.
Those messages especially reverberated in my head last weekend, when I was in Washington, DC helping out at the Julia Robinson Math Festival held at the Smithsonian during the Math Circles on the Road event.
You will have to forgive me--my head is truly exploding with all the inspiration and ideas I brought away from that experience, so this post (and most likely my next few posts) will be rambling all over the place as I share them. A giant brainstorm hit me at the end of the weekend, a new concept I will call "Guerrilla Math Circles," which I will explain in a later post. I will get there...I promise.
The event at the Smithsonian was really wonderful, with 60 enthusiastic math circle leaders from all over the country (including Elizabeth Parizh and myself from Albany Area Math Circle) helping to run free and fun public math circle demonstration activities for hundreds of enthusiastic participants.
There was a strong theme of "math as performance art" running through many of the sessions, including the one that Elizabeth and I helped Anna Burago to run, along with Ashley Reiter Ahlin, Berhrooz Parmani, Yulin Qing, and Jack Reynolds. We engaged a group of students (around 8 or 9 years old) in acting out mathematical logic problems set on an island inhabited by Knights (who always tell the truth), Liars (who always lie), and Tourists (who can go either way.) Simple props (leis for the tourists and pennants bearing a K or an L for the knights and liars) added greatly to the engagement of the event. (The kids loved waving the pennants--and it was a really nice way to get their heads inside the logic exercise. We had originally intended to use paper hats bearing K or L, but after consideration of hygiene/sanitation issues that could arise from switching hats around, we decided to go with pennants instead. Props to my ever-resourceful 80-year-old problem-solving mother for suggesting that chopsticks leftover from takeout orders work much better than drinking straws for constructing inexpensive pennants! The kids loved waving the pennants around so much that drinking straws would have quickly drooped.)
Another great session I had the chance to observe also involved math as performance art. Blake Thornton of the Washington University in St. Louis Math Circle adapted an idea from Terry Tao's blog into a great session on the Island of the Blue and Brown Eyes. (Again, it was fascinating to see what a difference the use of a simple but concrete prop made with the young children, who were acting out the roles of islanders trying to reason through a problem of inferring their own "eye-colors" based on their observations of the "eye-colors" they observed on the other islanders along with a remark made by a clueless tourist who did not understand the island's taboo against discussing eye color. In the first run of this activity, Blake and his assistants gave each child an index card that told the child how many of the other islanders in the room had blue eyes, and how many of the other islanders had brown eyes. In the second run of the activity, Blake and his assistants asked all the children to close their eyes as they placed a colored sticker on each child's forehead to designate that child's "eye color". When the children were told to open their eyes, they could then immediately observe the eye color of all the other islanders. This very simple expedient worked *much* better for the students involved, and I was really impressed at the way the children were then able to reason through the problems presented to them.)
I also got the chance to observe a Math Wrangle organized by Tatiana Shubin from the San Jose Math Circle. The wrangle involved six very impressive young members of the Fairfax Math Circle, who wore awesome t-shirts bearing a translation of a famous quote from Georg Cantor as they wrangled in front of an adult audience awed by their poise in presenting their mathematical reasoning.
Their shirts said: "The essence of mathematics of mathematics is its freedom."
Freedom...yes, freedom and free were more words that reverberated in my head last weekend. Math is free--you can do it with scratching in the sand or dirt (as Archimedes did) or even just in your head (as prisoners of war have done in order to maintain their sanity) or with the simplest of materials such as stones or paper and string or colored sidewalk chalk.
And I was troubled by that message.
Why?
Turnout at the free math festival and at the nearby free Math Alive! exhibit at the Smithsonian was excellent. It was a beautiful spring day with a Cherry Blossom Parade that had brought huge crowds downtown.
There were thousands of children eagerly passing through the Math Alive! exhibits, with hundreds of them checking into our math festival and staying to participate in an activity or game with us.
So why was I troubled?
Because, among the hundreds of students that I personally observed passing through the festival and the museum that day I did not see a single African-American child visit our festival--and this in a city where the overwhelming majority of public school students are African-American and where the black-white educational gap is the greatest in the country. (I did hear a report from other attendees that they did see a few African-American students attending, but there was general agreement that they were very few in number.)
Because, among the hundreds of students that I personally observed passing through the festival and the museum that day I did not see a single African-American child visit our festival--and this in a city where the overwhelming majority of public school students are African-American and where the black-white educational gap is the greatest in the country. (I did hear a report from other attendees that they did see a few African-American students attending, but there was general agreement that they were very few in number.)
We were at the Smithsonian in a FREE math festival, held in a FREE museum, on a national mall surrounded by monuments and memorials in the capital of a country that cherishes FREEDOM. Our society is far from perfect, and yet it represents a beacon of freedom and opportunity to the entire world. The free museums of the Smithsonian and our free-to-the-public math festival were emblematic of that freedom.
The newest memorial celebrating freedom near the national mall just opened last fall, the Martin Luther King Memorial. For me, it brought back many memories of my childhood growing up in Washington, DC in the 1960s. I spent much of the summer of 1963 at my grandmother's apartment, where she was dying of cancer, and tenderly cared for by a much-beloved African-American woman. I still remember us sitting together in the living room as we watched the black-and-white television in awe of the vast crowds assembled on the mall downtown and heard Martin Luther King's powerful words reverberate: "Free at last! Free at last! Thank God Almighty, we are free at last!"
I too have a dream. And it began to take concrete shape last weekend as I contemplated all these memories that reverberated in my head last weekend in Washington, DC.
To be continued....
Monday, March 26, 2012
"Dumb" questions and STEM bullies
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I hasten to point out that the folks in these pictures are most definitely NOT bullies!
They are the Stanford professors who have been teaching some wonderful on-line classes that I have been taking this year. In the fall, I took Introduction to Artificial Intelligence with Professors Peter Norvig and Sebastian Thrun. (They were amazing. Among other things, Prof. Thrun headed the Stanford team that designed and built the driverless car that won the DARPA desert challenge. He has an inspiring Ted Talk that I highly recommend.) Now I am taking Probabilistic Graphical Models with Professor Daphne Koller. (Her photo above is from wikipedia. This NYT article tells a bit more about the cool work she does.)
There are tens of thousands of students all over the world taking these classes along with me, and students helping one another on the course discussion boards has been an essential and exciting part of the learning process.
There is absolutely no way that Professors Thrun, Norvig, and Koller or the few official teaching assistants who help them could answer all our questions. There are so many unanticipated sources of confusion and technical difficulties (for example, some students live in countries where they use commas instead of periods to denote decimal points, people are using many different operating systems on their computers, for many students English is a second language, etc.) I am once again struck by the spirit of generosity among my classmates. While observing the rules of the Stanford Honor Code (which prohibit giving help on the substantive content of graded homework assignments), my classmates have generously provided assistance in dealing with various technical issues that have arisen with downloading and installing and running the required software. This has been very helpful to many of us.
However, very occasionally there is an obnoxious comment posted on the discussion boards making a snide remark such as "Anyone who asks such a dumb question clearly does not belong in this class."
I cringe when I read remarks like these. I think of the people who make such posts as STEM bullies.
My feeling is that the askers of the questions DO belong in the course. The ones who do NOT belong are those who put others down for asking "dumb questions".
I feel the same way about our math circle as I do about the on-line classes I am taking.
Thus, I was heartened to read this powerful post on the subject of "dumb questions" by Professor Thrun--it captures my own beliefs so well that I wanted to share it--I will be reading this aloud at this Friday's math circle:
I really hope that this new digital medium makes it easier to ask "stupid" questions. Let me report on myself. I work with a 200+ people team at Google (reporting into me), I co-founded Udacity, I am an authority in my area of research. I ask many many "stupid" questions. I have learned that asking questions is power. The problem is if others respond to such questions with "you should have known." People rarely do this to me, but they do this to my students. I really dislike this, and I usually confront them. We should remember that there is NO learning without asking questions. In this class, there are people with many different levels of knowledge and skills. What brings us together at this point is that we are all 100% dedicated to make this class. be kind. Reach out to people asking questions whose answer appears trivial to you. Be a friend. And make a friend. remember the question that seems obvious to you once was non-obvious to you. You find that people respect you for being kind. Being kind is one of the highest levels of achievement. I will respect you for it, and so will the people around you. There will come the day when you are asking the stupid question - and you will appreciate the kindness of others.
Saturday, March 3, 2012
Young student honors
The American Mathematics Competition has a special national public honor list for students who score high on a contest designed for older students.
Congratulations to the following students who made those national honor lists this year:
AMC12A:
(Students in tenth grade or below with scores over 90)
Cecilia Holodak (Niskayuna HS) 99
AMC12B:
(Students in tenth grade or below with scores over 90)
Matt Gu (Guilderland HS) 93
AMC10A:
(Students in eighth grade or below with scores over 90)
AMC10B:
(Students in eighth grade or below with scores over 90)
Students with light blue backgrounds behind their names are members of Albany Area Math Circle and/or our affiliated middle school outreach programs. (If you know any of the others--or any other local students who might enjoy our math circle activities, please invite them to subscribe to our email lists by sending an email to AlbanyAreaMathCircle-subscribe@yahoogroups.com for high school students and their parents or middleschoolmathcircle-subscribe@yahoogroups.com for parents of middle school students.)
Congratulations to the following students who made those national honor lists this year:
AMC12A:
(Students in tenth grade or below with scores over 90)
Cecilia Holodak (Niskayuna HS) 99
AMC12B:
(Students in tenth grade or below with scores over 90)
Matt Gu (Guilderland HS) 93
AMC10A:
(Students in eighth grade or below with scores over 90)
|
AMC10B:
(Students in eighth grade or below with scores over 90)
| Alex | Wei | Van Antwerp MS | 135 |
| Ziqing | Dong | Farnsworth MS | 126 |
| William | Wang | Farnsworth MS | 124.5 |
| Jason | Tang | Van Antwerp MS | 118.5 |
| Andrei | Akhmetov | Van Antwerp MS | 108 |
| Alex | Cao | Shaker JHS | 108 |
| Patrick | Chi | Iroquois MS | 106.5 |
| Liam | McGrinder | Van Antwerp MS | 103.5 |
| Vladimir | Malcevik | Van Antwerp MS | 99 |
| Gwenda | Law | O'Rourke MS | 97.5 |
Students with light blue backgrounds behind their names are members of Albany Area Math Circle and/or our affiliated middle school outreach programs. (If you know any of the others--or any other local students who might enjoy our math circle activities, please invite them to subscribe to our email lists by sending an email to AlbanyAreaMathCircle-subscribe@yahoogroups.com for high school students and their parents or middleschoolmathcircle-subscribe@yahoogroups.com for parents of middle school students.)
Congratulations to our American Invitational Math Exam qualifiers
| Congratulations to all the Albany area students who embraced the "extreme math" challenge of this year's AMC10 and AMC12 contests. Here are the criteria for invitation to the AIME along with the honor lists: American Invitational Math Exam (AIME) qualification: 115.5 or above on AMC10A 120 or above on AMC10B 94.5 or above on AMC12A 99 or above on AMC12B Congratulations and best wishes to the following students from the Albany area who have qualified to take the AIME, the next step in a series of progressively more challenging mathematics exams leading to the International Mathematics Olympiad. American Invitational Math Exam (AIME) AMC12B qualifiers: |
||||
| Matthew Babbitt (heeg) 117 | ||||
| Wyatt Smith (heeg) 114 | ||||
| Elizabeth Parizh (Niskayuna HS) 99 | ||||
| American Invitational Math Exam (AIME) AMC10B qualifiers: |
||||
| Alex Wei (Van Antwerp MS) 135 Ziqing Dong (Farnsworth MS) 126 William Wang (Farnsworth MS) 124.5 Aniket Tolpadi (Niskayuna HS) 123 |
||||
| American Invitational Math Exam (AIME) AMC12A qualifiers: |
||||
| Matthew Babbitt (heeg) 130.5 Zubin Mukerjee (Guilderland HS) 102 Sherry He (Emma Willard School) 101 Cecilia Holodak (Niskayuna HS) 99 Wyatt Smith (heeg) 99 J Chung (Emma Willard School) 95 N Xie (Albany Academies) 95 | ||||
| American Invitational Math Exam (AIME) AMC10A qualifiers Alex Wei (Van Antwerp MS) 127.5 Philip Sun (Shenendahoah HS East) 126 William Wang (Farnsworth MS) 123 Patrick Chi (Iroquois MS) 120 Vineet Velandula (Niskayuna HS) 120 Ziqing Dong (Farnsworth MS) 117 Gili Rusak (Shaker HS) 117 |
||||
| Students with light blue backgrounds behind their names are members of Albany Area Math Circle and/or our affiliated middle school outreach programs. (If you know any of the others--or any other local students who might enjoy our math circle activities, please invite them to subscribe to our email lists by sending an email to AlbanyAreaMathCircle-subscribe@yahoogroups.com for high school students and their parents or middleschoolmathcircle-subscribe@yahoogroups.com for parents of middle school students.) Please report any errors or omissions by sending email to mathcircle at gmail. | ||||
Wednesday, December 28, 2011
Winter holiday mathematics
Albany Area Math Circle is taking a brief break from our regular Friday evening meetings, but that doesn't mean we aren't surrounded by wonderful mathematics, so here are some treasures to share with your friends and family members. I would love to hear from other math circle members about interesting mathematical aspects of their cultures. I am sure there are many that I do not know about.
Last night was the last of the eight nights of Hanukkah. There is all sorts of fun math in Hanukkah, such as figuring out how many candles you need in a box to cover all eight nights of Hanukkah, which will be one less than a triangular number. There are many fun combinatorics problems to construct and explore as well. How many different color combinations can be constructed if you have n different colors of candles available?
Today's Albany Times Union had a great mathematics of Hanukkah story, A one-in-trillions dreidel game, about a remarkable string of luck by a first-time dreidel player, who managed to spin the four-sided top 68 consecutive times without ever seeing the losing side come up and winning the entire pot 56 of those spins. This inspired his great-nephew, a Princeton sophomore studying operations research and financial engineering, to pull out his calculator to compute the astronomically large odds against such a long streak of good luck.
There are several ways to model this probability calculation and it would be interesting to discuss the pros and cons of each approach. One simple possibility is to compute the likelihood of never losing in 68 spins, which would be 1 - (3/4)68, which works out to 1 in 22.5 trillion. You can make the odds even more astronomical if you also consider the likelihood that he actually wins the pot on 56 of those 68 non-losing spins, because the remaining three sides are equally likely to come up, and only one of those spins yields the entire pot to the spinner.
Of course, this kind of analysis gives rise to interesting philosophical discussions of the sort that physicist Richard Feynman raised in his book, The Meaning of It All: Reflections of a Citizen Scientist, when he talked about the probability of seeing a particular license plate in a parking lot as well as the work of Stanford mathematician Persi Diaconis on the mathematics of coincidences.
Moving onto Christmas, we are currently in the midst of the fabled "Twelve Days of Christmas," as memorialized in the song that starts with one gift given on the first day (December 25, "a partridge in a pear tree"), three gifts given on the second day (December 26, "two turtle doves" plus another "partridge in a pear tree"), and so on through the twelfth day (January 6, which happens to be our next math circle meeting date!)
John Cook at the Endeavor blog has a great post on the mathematics of the Twelve Days of Christmas. He observes that the total number of gifts received each day is a triangular number and also notes that the cumulative number of gifts received through the end of each day is a tetrahedral number. He has written up several nice proofs demonstrating that this is true in general, and he also includes a wonderful link and illustration from the Math is Fun blog.
The Math is Fun blog's illustration at right is a great way to illustrate the triangular/tetrahedral nature of the 12 Days of Christmas song for your younger friends and relatives. The image shows the situation for the first five days of Christmas, with the top layer representing the number of presents given on the first day (1), the next layer representing the number of presents given on the second day (3), the next layer the presents given on the third day (6), the next layer the presents given on the fourth day (10), the next layer the presents given on the fifth day (15). The cumulative number of presents given on the first through fifth day is the number in the entire five layer tetrahedron, or 1+3+6+10+15 = 35. You can extend this indefinitely, of course. If you stop after 12 days, you will have a 12-layer tetrahedron with a cumulative total of 1+3+6+10+15+21+28+36+45+55+66+78 presents, which turns out to be 364, a very nice number, since it is one less than the number of days in a typical year!
Another fun fact to share with your younger friends and relatives: take the number of cumulative gifts received during the 12 days and multiply it by the number of days in 2012 and you get a nice opportunity to discuss factoring differences of squares since 3652 - 12 = (365-1)(365+1).
There are wonderful mathematical possibilities to explore in every religion and culture--including the modular arithmetic of the various calendar systems generally designed to reconcile discrepancies between lunar and solar numbering systems, which move many holiday observations around relative to one another. It's also interesting to note the frequency with which calendars in so many widely divergent calendars use a 7-day week. This is mathematically very convenient, since the number nearest to 365.24 with a conveniently large number of factors is 364, which is divisible by 7.
The Hebrew calendar has a 19-year cycle with a leap month in seven of those years. The Islamic calendar has a 30-year cycle with a leap day added to the final month in 11 out of those 30 years. The Gregorian calendar commonly used in the west appears to have a four year cycle with a leap day every four years, but it's actually more complicated than that.
New years festivals are observed at many different times in different calendars, starting with the Jewish new year, Rosh Hashanah, which arrives in early fall. The Hindu new year celebration happens on their festival of lights, Divali, in mid-fall. The Chinese New Year generally comes later in winter than the Gregorian new year on January 1. The early Roman calendar (before Caesar came along and reformed it) began its new year in March, had only ten lunar months, and then had a mysterious unlabeled winter period of 61 day that were apparently not considered to belong to any month. Caesar changed the New Year to January 1, but subsequently, Christian rulers moved the New Year to March, before Pope Gregory moved it back to January. The Islamic calendar is a mathematically interesting exception to the general rule that most calendars observe New Year's in the interval between the fall equinox and the spring equinox. The new year in the Islamic calendar rotates through all the seasons over time, so in 2025, the Islamic new year will start in late June very close to the summer solstice. The Islamic calendar advances about 11 days each year, the Islamic new year will fall near the winter solstice about 16 or 17 years later and then will fall near the summer solstice again in 2058.
Last night was the last of the eight nights of Hanukkah. There is all sorts of fun math in Hanukkah, such as figuring out how many candles you need in a box to cover all eight nights of Hanukkah, which will be one less than a triangular number. There are many fun combinatorics problems to construct and explore as well. How many different color combinations can be constructed if you have n different colors of candles available?
Today's Albany Times Union had a great mathematics of Hanukkah story, A one-in-trillions dreidel game, about a remarkable string of luck by a first-time dreidel player, who managed to spin the four-sided top 68 consecutive times without ever seeing the losing side come up and winning the entire pot 56 of those spins. This inspired his great-nephew, a Princeton sophomore studying operations research and financial engineering, to pull out his calculator to compute the astronomically large odds against such a long streak of good luck.
There are several ways to model this probability calculation and it would be interesting to discuss the pros and cons of each approach. One simple possibility is to compute the likelihood of never losing in 68 spins, which would be 1 - (3/4)68, which works out to 1 in 22.5 trillion. You can make the odds even more astronomical if you also consider the likelihood that he actually wins the pot on 56 of those 68 non-losing spins, because the remaining three sides are equally likely to come up, and only one of those spins yields the entire pot to the spinner.
Of course, this kind of analysis gives rise to interesting philosophical discussions of the sort that physicist Richard Feynman raised in his book, The Meaning of It All: Reflections of a Citizen Scientist, when he talked about the probability of seeing a particular license plate in a parking lot as well as the work of Stanford mathematician Persi Diaconis on the mathematics of coincidences.
Moving onto Christmas, we are currently in the midst of the fabled "Twelve Days of Christmas," as memorialized in the song that starts with one gift given on the first day (December 25, "a partridge in a pear tree"), three gifts given on the second day (December 26, "two turtle doves" plus another "partridge in a pear tree"), and so on through the twelfth day (January 6, which happens to be our next math circle meeting date!)
John Cook at the Endeavor blog has a great post on the mathematics of the Twelve Days of Christmas. He observes that the total number of gifts received each day is a triangular number and also notes that the cumulative number of gifts received through the end of each day is a tetrahedral number. He has written up several nice proofs demonstrating that this is true in general, and he also includes a wonderful link and illustration from the Math is Fun blog.
The Math is Fun blog's illustration at right is a great way to illustrate the triangular/tetrahedral nature of the 12 Days of Christmas song for your younger friends and relatives. The image shows the situation for the first five days of Christmas, with the top layer representing the number of presents given on the first day (1), the next layer representing the number of presents given on the second day (3), the next layer the presents given on the third day (6), the next layer the presents given on the fourth day (10), the next layer the presents given on the fifth day (15). The cumulative number of presents given on the first through fifth day is the number in the entire five layer tetrahedron, or 1+3+6+10+15 = 35. You can extend this indefinitely, of course. If you stop after 12 days, you will have a 12-layer tetrahedron with a cumulative total of 1+3+6+10+15+21+28+36+45+55+66+78 presents, which turns out to be 364, a very nice number, since it is one less than the number of days in a typical year! Another fun fact to share with your younger friends and relatives: take the number of cumulative gifts received during the 12 days and multiply it by the number of days in 2012 and you get a nice opportunity to discuss factoring differences of squares since 3652 - 12 = (365-1)(365+1).
There are wonderful mathematical possibilities to explore in every religion and culture--including the modular arithmetic of the various calendar systems generally designed to reconcile discrepancies between lunar and solar numbering systems, which move many holiday observations around relative to one another. It's also interesting to note the frequency with which calendars in so many widely divergent calendars use a 7-day week. This is mathematically very convenient, since the number nearest to 365.24 with a conveniently large number of factors is 364, which is divisible by 7.
The Hebrew calendar has a 19-year cycle with a leap month in seven of those years. The Islamic calendar has a 30-year cycle with a leap day added to the final month in 11 out of those 30 years. The Gregorian calendar commonly used in the west appears to have a four year cycle with a leap day every four years, but it's actually more complicated than that.
New years festivals are observed at many different times in different calendars, starting with the Jewish new year, Rosh Hashanah, which arrives in early fall. The Hindu new year celebration happens on their festival of lights, Divali, in mid-fall. The Chinese New Year generally comes later in winter than the Gregorian new year on January 1. The early Roman calendar (before Caesar came along and reformed it) began its new year in March, had only ten lunar months, and then had a mysterious unlabeled winter period of 61 day that were apparently not considered to belong to any month. Caesar changed the New Year to January 1, but subsequently, Christian rulers moved the New Year to March, before Pope Gregory moved it back to January. The Islamic calendar is a mathematically interesting exception to the general rule that most calendars observe New Year's in the interval between the fall equinox and the spring equinox. The new year in the Islamic calendar rotates through all the seasons over time, so in 2025, the Islamic new year will start in late June very close to the summer solstice. The Islamic calendar advances about 11 days each year, the Islamic new year will fall near the winter solstice about 16 or 17 years later and then will fall near the summer solstice again in 2058.
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