Haven’t got the Math Awareness Month bug yet? Here are three teasers to get you started:
1. What read the same right side up and upside down, and combine mathematics, art, and language? Ambigrams, of course, like this one:
2. How can a 5 by 13 rectangle be cut into 4 pieces which can be rearranged to form an 8 by 8 square? (Could doing it in reverse be used to turn an 8 by 8 bar of chocolate into one that is 5 by 13?) Follow the linked image below and you’ll have your answer:
3. Into how many pieces can you slice a bagel (or doughnut) with one straight cut? Clearly two. What about with two such cuts? It turns out to be six, if you are careful to maximize the impact. It’s much trickier if three planar cuts allowed: that challenged was posed by renowned math popularizer Martin Gardner (1914-2010) early on in his career as Scientific American columnist. The surprising answer is revealed by the image below (more generally, is there a formula for the biggest possible number of bagel (or doughnut) pieces that can result if n cuts are allowed?):
Are you feeling engaged yet? Gardner would hope so. As he wrote in the introduction to Mathematical Carnival (Knopf, 1975), one of the fifteen compendiums of his of Scientific American columns, “The best way, it has always seemed to me, to make mathematics interesting to students and laymen is to approach it in a spirit of play.”
That’s certainly the approach of this year’s Mathematics Awareness Month (MAM) whose theme is Mathematics, Magic, and Mystery—echoing the title of an early and influential book by Gardner. MAM was created to increase public understanding of and appreciation for mathematics.
In a sense, MAM is bringing to a digital-age audience many updated multimedia versions of fun topics that Gardner introduced to earlier generations, both in Mathematics, Magic and Mystery (Dover, 1956) and in his 300-odd legendary Scientific American columns.
Those topics range from magic squares, knots and braids, geometrical vanishes, mathematical card tricks, hexaflexagons and polyominoes, to paradoxes and optical illusions, Penrose tiles and the Mandelbrot set. This year marks the centennial of the birth of this prolific writer—he also left us over 100 books—who for many decades inspired enthusiasts of all ages to engage deeply with mathematics. As Jordan Ellenberg, a mathematician at the University of Wisconsin at Madison, wrote in his recent review of Gardner’s memoirs for The Wall Street Journal:
“Gardner understood instinctively that mathematics was bigger, richer and more fun than what was allowed in class. And he did more than anyone else in the 20th century to get that message to the public.”
To underscore that point, MAM launched on Tuesday April 1 with a brace of videos from rising math stand-up star Ethan Brown—a mere 9th grader—on customized magic square construction:
As hinted by that kickoff—the first of 30 activities being revealed one per day this month—it is hoped that MAM will reach high school students and their teachers, as well as those in colleges and universities. Each topic is introduced with a catchy video, followed by discussion (and more video) that deepens in complexity. There will always be something to look forward to, as well as something for everybody from middle school to middle age and beyond! As co-organizers Bruce and Eve Torrence of Randolph-Macon College put it:
“We've worked hard to put together a compelling collection of magic and mystery, all based on mathematical ideas that are equally amazing. We've corralled top-shelf magicians and mathematicians to present the material. In short, we're really excited to share this resource with the world, and celebrate the spirit of discovery that comes with every mathematical ‘Aha!’ moment.”
MAM is making extensive use of social media, too. There’s a Twitter handle, @MathAware, a Facebook page and several scheduled Connected Classrooms events powered by Google Hangouts on Air. The hangouts are archived on YouTube, and the daily activity webpages will remain active so that students and teachers can go back and explore them at any time in the future.
Day 2 sees Mathematical Association of America (MAA) Mathematician in Residence James Tanton unravel a counterintuitive strands of knots and braids, showing that contrary to first impressions, the two-slit image below on the left can be transformed into the knotted one on the right without any cheating:
Day 7 highlights a series of musings on hexaflexagons—Gardner’s first big hit at Scientific American—by mathematical video veteran Vi Hart, whose quartet of innovative productions on this topic has already attracted over 8 million views.
Hart’s father, SUNY Stony Brook mathematician and sculptor George Hart, has a different twist on bagel slicing on Day 11, showing how to cut such a shape into two interlocking bagelettes, or alternatively into a pretzel-like knotted structure:
Day 13 shows how to create your own ambigrams. Several days explore the magic and mystery of Fibonacci numbers, from their use in a baffling card trick (Day 3, highlighting my creation) to area preservation paradoxes—which, as already hinted, suggests a method for producing free squares of chocolate—to the fads and fallacies of the Golden Ratio featuring Keith Devlin of Stanford, a.k.a. NPR’s “Math Guy.”
Mid-month, mathematician Doug Ensley discusses a clever work-around for people keen to explore card tricks based on the notoriously difficult perfect Faro shuffle. Later in the month, expect a patriotic spin-off of a classic card trick from the 1970s—developed by mathematician Martin Kruskal—that was first brought to the public’s attention by Gardner. This time, it serves as the jumping off point for a demonstration that the Declaration of Independence guarantees “happiness”—a fine contribution from A Mathematician Reads the Newspaper author John Allen Paulos of Temple University.
Some of the topics explored in MAM are purely mathematical, others are magical with mathematical underpinnings, and many are both. What they all share is a sense of mystery. While traditional magic and illusion-spinning depend of sleight-of-hand and misdirection, with the aid of mathematics, it turns out that anyone can perform mysterious card tricks or entertain with made-to-order magic squares. Then there are complex, mysterious activities that also draw heavily on mathematics—which many of us can only marvel at as spectators—such as juggling and lightning-fast mental calculation.
Towards the end of the month, Princeton mathematician John H. Conway appears in person to explain his famous Doomsday algorithm for calculating the day of the week for any date in history (one you’ve mastered that, you will be able to work out that October 21 1914, the day on which Martin Gardner was born, was a Wednesday).
Participants in MAM of all ages are encouraged to share their planned and completed experiences here. Also, visit the MAM news page for MAA resources and this page for American Mathematical Society resources.
The MAM activities aim to engage a new generation with mathematics, leading people of all ages to their own magical and mysterious Aha! moments. As Gardner himself also wrote in that Mathematical Carnival introduction, “Surely the best way to wake up a student is to present him with an intriguing mathematical game, puzzle, magic trick, joke, paradox, model, limerick, or any of a score of other things that dull teachers tend to avoid because they seem frivolous….” To which we say, Amen!
Images courtesy of Math Awareness Month and www.martin-gardner.org.