Back-to-School Blogs, 2017 Edition

Today, I’m taking my chances with traffic and driving up to Idaho to try to get in the path of eclipse totality. (Fun fact: according to my back-of-the-envelope calculations, if everyone in the country went to the path of totality, its population density would be a bit higher than that of Salt Lake City. Get more eclipse math from Bedtime Math and NASA. Don’t have eclipse glasses? Ask the past how to behold it safely.) While I’m on my sojourn, my spouse and millions of other teachers and students will be getting back to the classroom. Last year around this time, I posted a roundup of some of my favorite math education blogs. It’s a good list, and you should check it out.

Welcome back to school! Image: US Department of Education.

This year, I want to add a few more recommendations for good math resources for parents, teachers, and students.

Not awful and boring ideas for teaching statistics I’m a sucker for this blog name. Are you teaching statistics? Do you want some ideas that aren’t awful and boring? Here you go! The author, Jessica Hartnett, is a professor in the psychology department at Gannon University. She finds interesting data in the news and other places and gives her advice about how to use it in a statistics class. She’s recently posted about a Harry Potter sorting quiz, unpopular wedding songs, and the statistics that indicate that ride-sharing apps might curb drunk driving. (I’ve idly wondered about this myself while waiting for a ride after pub trivia night.)

Math Hombre John Golden is a math teacher and math education professor at Grand Valley State University. Follow his blog for posts about teaching teachers and making cool math art, games, and designs.

On This Day in Math Pat Ballew rounds up math-adjacent births, deaths, and events every day. I also enjoy the quotes he includes at the top of each post.

MatthewMaddux Education University of Saskatchewan math education professor Egan Chernoff compiles this feed of articles and blog posts related to math and math education. It helps me keep up with the latest news, especially in mainstream media outlets I don’t normally check for math stories.

inclusion/exclusion My co-blogger Anna Haensch wrote about this new AMS blog here a few months ago. It’s not specifically about teaching, but as math professors make up a lot of the intended audience, it discusses aspects of teaching that intersect with its mission of addressing diversity and inclusion in mathematics. In light of the heartbreaking, infuriating events in Charlottesville earlier this month, Brian Katz wrote about how to discuss justice on the first day of class.

Have other suggestions? Feel free to add them in the comments. If you’re a teacher, parent, or student, have a great beginning of the school year!

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With Profound Sadness

It was an incredible day in 2014 when Maryam Mirzakhani became the first woman to win the Fields Medal. I remember feeling absolutely overwhelmed with emotion and thinking to myself, alright, beginning today winning the Fields Medal is officially something that women do. It went from something impossible to something possible, just like that. It felt like the breaking of such a monumental glass ceiling, and like the opening up of this entire alternate universe of possibility. It was huge. At the time, Erica Klarreich wrote about Mirzakhani’s early life and work for Quanta Magazine.

It was with almost unbearable sadness that news of her passing broke last month. Mirzakhani died at only 40 years old, and the world became dimmer one shining star.

The weeks since have seen an outpouring of writing celebrating the life and work of Mirzakhani and mourning her death. Fellow Fields Medalist Terrence Tao shared a post the day after Mirzakhani’s passing, highlighting her contributions to the field and his own experiences in meeting her. Blogger and mathematician John Baez also wrote a thoughtful piece about Mirzakhani’s life and mathematics. The blog Mathsbyagirl featured a tribute post to Mirzakhani with links to expository articles on the key areas of Mirzakhani’s research.

Our own inclusion/exclusion blog posted about how Mirzakhani shone in all of her various roles, as mathematician, mother, trailblazer, and role model, with reflections on her life by various notable women in math. From Tai-Danae Bradley, “While reading through the many beautifully written tributes to Maryam, I am especially touched by one theme that pervades them all: her character. Words like persistent, determined, and resolute appear time and time again. And her humility and modesty seem to have garnered as much attention as her mathematical accomplishments.”

Mirzakhani working through ideas on a large sheet of butcher paper. This image is a screen shot from the Simon’s Foundation short film on Mirzakhani.

RAGE of the Blackboard, a blog exploring the bridge between scientists and artists, featured an illustration of Mirzakhani and spoke about the importance of drawing in her work. Mirzakhani often said in interviews that she enjoyed doing math by writing, drawing, and doodling on large pieces of butcher paper.

The AMS has collected a full list of tributes and obituaries for Mirzakhani, including words from AMS president Ken Ribet and a short video from the Simon’s Foundation.

This month has also seen the passing of several other notable women in math. Cathleen Morawetz, who did pioneering work related to airflow at supersonic speeds, died last week at the ago of 94. Earlier in July, Marina Ratner, who found acclaim later in life, died at the age of 78. Mathematician Amie Wilkinson wrote for the NYTimes about the shared aspects of Ratner’s and Mirzakhani’s works, despite very divergent lives. This month also saw the passing of Marjorie Rice at the age of 94. Rice was an amateur mathematician who made a big discoveries in the study of pentagonal tiling.

It is with tremendous sadness that we bid farewell to these mathematicians who inspired us and gave us so much. The impact of their work will surely continue to resound for many decades to come.

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Conversations with Women of Color in STEM

I online-met Williams College mathematician Pamela Harris last year through Lathisms, a Hispanic Heritage Month project that highlights Latinx and Hispanic mathematicians. She was one of the organizers of the effort, and I spoke with her and another organizer, Gabriel Sosa, for a post about it. It wasn’t until a few weeks ago that I learned more about her background via an interview for the Vanguard STEM blog. Unusually for a professional mathematician, she started college in intermediate algebra and worked up to a Ph.D. from there. It got me thinking about how mathematicians could help more students excel in math after coming to it relatively late in their academic careers. I also enjoyed reading about her childhood encounters with infinity and one of her hobbies, jiu jitsu.

Just a week later, Vanguard STEM published an interview with Alicia Prieto Langarica, a mathematical biologist at Youngstown State University in Ohio, who was one of my colleagues in the Project NExT class of 2013. (Go Brown 13s!)

Seeing two math friends in a row featured on the site convinced me that I really needed to add Vanguard STEM to my blogroll. Vanguard STEM was launched a few years ago by Jedidah Isler, an astrophysicist at Vanderbilt and TED Fellow. She created it as a place to facilitate conversations between established and up-and-coming women of color in STEM (science, technology, engineering, and math) fields, particularly African American and Hispanic/Latina women. The site has monthly Google Hangouts in which women of color discuss different aspects of their lives and careers in STEM in addition to articles with advice for people considering or working in STEM careers. But my favorite is the weekly #WCWinSTEM series where I read the interviews with Harris and Prieto Langarica. WCW stands for “women crush Wednesday,” and #WCWinSTEM is a great series of interviews and articles about STEM professionals in many different jobs and stages of their careers.

Vanguard STEM has featured mathematics and mathematicians several times on the site. Last winter when Hidden Figures fever was gripping the world, the site had several articles about the black women who worked for NASA as “human computers.” It was great to see Katherine Johnson, Dorothy Vaughn, and Mary Jackson portrayed by movie stars on the big screen, but there’s something very special about seeing their real faces, too, even if they weren’t quite as glamorous as Taraji P. Henson, Octavia Spencer, and Janelle Monáe. I especially enjoyed the Vanguard STEM article about Christine Darden, one of the heroes of the Hidden Figures book whose story was not included in the movie.

In the past, the Blog on Math Blogs has featured several other websites that provide information about and/or resources for women and people of color in math and other STEM careers. Check out our recommendations. (Warning: there’s a lot of celebrating going on.)

Celebrating the Grandmothers of STEM
Celebrating our Sisters in STEM
Beyond Banneker: Resources for Learning about Black Mathematicians
Celebrating Latin@s and Hispanics in Mathematics
Celebrating Black Mathematicians
They Answered the Call of Numbers
Diversify Your Blogfolio
Adding to the Faces of Mathematics on Wikipedia

You can also subscribe to the AMS inclusion/exclusion blog, which covers diversity and inclusion in mathematics.

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Searching For Einstein

No, not Einstein. We’re searching for einstein. Literally “ein Stein,” which translated from German means “one stone.” The one stone we’re looking for is a very special type of tile which, when repeated, can cover an infinite floor without leaving any gaps and without admitting any sort of pattern.

We call an arrangement of tiles that covers the plane without any gaps or overlaps a tiling, and a tiling is called non-periodic if it has no translational symmetry. That means, if I pick the tiling up and move it in any direction, I won’t be able to fit it back down on itself. A nice example of a non-periodic tiling where we allow two types of tiles is the Penrose tiling. And if we loosened our restrictions slightly to allow tiles which are not connected, Socolar and Taylor found such a tiling in 2010. So more formally, the search for einstein is the search for a single connected tile that tiles only non-periodically. Recently, in the quest for einstein, some interesting progress has been made.

First some basics. Let’s think about tilings that only use a single convex polygon, that is, a polygon whose angles all bulge out instead of in. If we allow patterns and periodicity, then it’s easy to imagine how you could achieve a non-overlapping gap-free tiling that with a square, triangular, or even hexagonal tile. Even though sometimes they can be in disguise.

We now know that these are the only three possible regular hexagonal tilings. They were first discovered by Karl Reinhardt in 1918. Image via Wikimedia Commons.

Things get a bit more interesting when we consider pentagons. In the early 1900’s Karl Reinhardt found five examples of families of pentagonal tilings. Several more were found by various people over the years, including 4 families which were found by housewife and mathematical enthusiast Marjorie Rice, who recently passed away. And just last year, as reported on this blog by Evelyn Lamb, another pentagonal tiling was found, bringing the total number of known families of pentagonal tilings to 15. For some fun teachable moments involving pentagonal tilings, check out mikesmathpage.

These are the 15 families of tilings with convex pentagons. Here colors indicate the number of edges touching each pentagon. Image courtesy of Wikimedia Commons.

In breaking news, mathematician Michaël Rao of France’s CNRS proved that these are precisely all of the convex tilings of the plane. There are just the 15 known families of pentagonal tilings, 3 hexagonal tilings, and all triangular and quadrilateral tilings. Of note, is that Rao’s work involves a computer assisted proof, which allowed him first to establish some bounds via theoretical methods and then do an exhaustive search. Rao’s conclusion: there are no convex polygons that admit only non-periodic tilings, that means, the einstein tile must not be a convex polygon.

This means, if we want to find einstein, we need to start looking at concave tiles.

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Resources for People Who Wanna Present Stuff Good and Do Other Stuff Good Too

Presentations are hard. You’ve been thinking about something for a long time, and you can get tunnel vision. What do you mean, everyone looking at your poster or going to your talk doesn’t already know why you care about the components of the representation space of π1(M) into PSL(2,R) with extremal Euler characteristic??? Luckily, if you want to up your presentation game, you’re not entirely on your own.

One good way to improve your posters and presentations, of course, is to go to lots of poster sessions and talks and keep an eye on what’s working and isn’t for the presenter and their audience. But you can also get advice from around the math and science blogsophere.

For presentations, Dan Meyer has a good post about how to prepare for a talk. It boils down to “testify and practice,” but he gives a lot of specific advice on the nitty gritty details of how exactly he does that. He’s writing specifically for math teachers, but his presentation tips will be applicable to math research talks and other professional topics. The comments also have some feedback, and he wrote a follow-up post of some advice from 14 of his favorite math education speakers.

Zen Faulkes, an invertebrate neuroethologist (I had to look it up too) at the University of Texas Rio Grande Valley has a free e-book (pdf) of presentation tips from his blog, NeuroDojo. He also has a blog devoted to helping people make better posters. I especially appreciate the constructive critiques of real posters, like this one about two posters by mathematical biology graduate student Chris Miles.

When I asked for suggestions for this post on Twitter, astronomers stepped up. Thanks, astronomers! Meredith Rawls, an astronomy postdoc at the University of Washington, wrote a blog post about how she made an award-winning poster for a conference. She and other astronomers also pointed to tips from Kimberly Cartier and Jason Wright and this list from a blog called Astrobetter whose goal is “to provide information and tips about streamlining all the things we need to do Astronomy well.” People also suggested Edward Tufte, particularly for presentations that have a lot of data visualization.

I got great suggestions from friends on Facebook as well:

  • Georgia Tech mathematician Dan Margalit has a page of talk tips, which includes articles and blog posts from Paul Halmos, Jordan Ellenberg, Bryna Kra, and other mathematicians.
  • The LaTeX package tikzposter was designed specifically for conference posters.
  • Technically Speaking, a page by Lewis D. Ludwig of Denison University, has videos illustrating common presentation pitfalls and how to avoid them.
  • University of Waterloo mathematician Chris Godsil has a webpage of math presentation tips. I particularly appreciated his pointing out that giving a research talk or presentation is not the same as lecturing or teaching. Of course there are skill overlaps, but the goal of a research talk is usually not for an attendee to reproduce your proof later. 
  • Evolutionary biologist Colin Purrington’s website has “geeky tips for scientists,” including a page on designing conference posters.
  • Stephanie Evergreen is a data reporting and visualization expert with a blog of presentation design and visualization suggestions.
  • The NYU library has a good page of poster design tips.

If you have other presentation or poster design tips, please share in the comments!

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Take The Math Less Traveled

Mathlesstraveled is a blog “dedicated to exploring beautiful mathematics.” The blog is written by Brent Yorgey, an assistant professor in the department of math and computer science at Hendrix College, who lives closer to the computer science end of mathematics. As such his posts are often somewhat computational in nature. He has a whole zoo of good looking graphics and everything is easily digestible by anyone interested in learning a bit of math.

Factorization diagrams for the numbers 1 through 100, courtesy of Brent Yorgey.

One ongoing series featured in the blog are Posts Without Words. These are just graphics depicting some mathematical idea, sometimes it’s easy to see what the pictures are describing and sometimes it’s more difficult. I really like Post Without Words #5. The explanation is a doozy as it involves Hilbert space filling curves and the Thue-Morse sequence. Although I’ll bet you can come up with a simpler explanation!

As an aside, if you like mathematical ideas in graphic form, you should check out Mathematics in the Eye of the Beholder.

Yorgey also has several other series of posts, including those in which he discusses the irrationality of pi, and more recently, the curious powers of 1+sqrt 2. The later series aims to answer the following question posted on Mathstadon: What is the 99th digit to the right of the decimal point in the decimal expansion of (1 + sqrt 2)500? After stating the problem, Yorgey comes up with a reasonable conjecture (motivated by some computational examples), states a clever solution, and then goes on to explain an alternative approach.

Another great offering on the blog are Yorgey’s factorization diagrams. These are pretty pictures generated using diagrams in Haskell, that give a visual representation of prime factorization. These reminded me of a clever little book my brother gave me recently, You Can Count on Monsters. Yorgey sells the factorization diagrams as a deck of cards along with some fun game ideas for teaching factorization. I’m into it.

Yorgey maintains a second blog, aimed (as he says) at his peers. These posts tend to be slightly less accessible but would probably be of interest to anyone studying type systems, category theory, or combinatorics.

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Mathematics in the Eye of the Beholder

If you’re like me, you might get as excited about the intricate patterns in a museum’s parquet floor as in the art hanging on the wall. I love seeing the world through a mathematical lens and celebrating the patterns built into everyday life. We’ve shared image-based math blogs here in the past: complex functions, math gifs, and beautiful geometric designs. Today I’d like to share some of my favorite math photo blogs.

A parquet floor in the palace at Versailles. I guess the stuff on the tables and walls was nice too. Image: Evelyn Lamb

The Mathematical Tourist is a daily blog of math-adjacent photographs by math writer Ivars Peterson. He shares photos of architecture, art with mathematical themes, and occasional not-so-mathy stuff.

I also enjoy Turismo Matemático, a Spanish (both in language and in country of origin, I believe) math picture blog. They tend to share examples of mathematics in art and architecture, such as the many mathematical details of the Sagrada Familia cathedral in Barcelona and the many paintings containing compasses and astrolabes. When I was living in Paris earlier this year, I was excited to see posts about things I had seen, particularly the climate change equations in the Gare du Nord train station. Turismo Matemático has a short article to accompany each photo, too. It’s in Spanish, so an added bonus is that you can learn a few Spanish words or practice your Spanish while you enjoy some math pictures. (Or take your chances with your browser’s translation if your Spanish isn’t quite up to it.)

The MAA used to maintain a page of Found Math images updated every week. It seems to have gone dormant in 2015, but there’s a lot of good stuff in the archives on their website and on their Flickr page.

World Tessellation Day in June (the 17th, to coincide with M.C. Escher’s birthday) was a fun celebration of tiling patterns in buildings and on sidewalks. I wrote about the holiday, invented by Emily Grosvenor, here last year and on my blog Roots of Unity this year. The Twitter hashtag #worldtessellationday was a lot of fun to watch that day. I learned about a church in Spain that features a marble Penrose tiling (article in Spanish). It’s enough to make you want to plan a trip to Mahón!

Do you like to take mathy pictures? Share them on Twitter or Instagram using the hashtag #mathphoto or #foundmath.

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Math In The Capital

Visiting with Catherine Paolucci at Senator Franken’s office, where this well known friend of science displays his pride in Minnesota higher ed.

A few weeks ago I had occasion to visit the capital — and the Capitol — and as is the custom in Washington D.C., I had coffee with interesting people. We’ve blogged quite a bit here about the expressionless face emoji state of science and truth in our current world, but I was happy to catch up with the people on the ground kicking butt in the name of the AMS.

Karen Saxe, who is the director of the Washington D.C. office of the AMS, keeps us all in the loop with her blog Capital Currents. Saxe began her position in the D.C. office in January, and her role in that office is to be a link between the mathematicians who are part of the AMS and the lawmakers and legislators of Capitol Hill. She blogs about current legislation of interest to mathematicians, most recently about a bill promoting diversity in STEM fields. Also, particularly relevant to anyone who is seeking NSF funding, Saxe has blogged about the congressional budget approval process and what that means for us.

Saxe isn’t the only AMS insider with access to the legislative branch, Catherine Paolucci is this year’s AMS Congressional Fellow. Paolucci, who is also an assistant professor at SUNY New Paltz, is working in Senator Al Franken’s office and is in a unique position to work with the AMS and its members while also advising and informing policy decisions. Together with Saxe, she creates a pipeline for communicating AMS priorities to Congress and the AMS Committee on Science Policy.

As a fellow, Paolucci also helps AMS members organize successful hill visits and initiate successful grassroots efforts. It is important to have mathematicians in the political space, she says “strength and power comes from people on the ground.” Paolucci also stresses “the power of state level advocacy and importance in constituents engaging with their own senators.”

So, you know how people are always on you to call your congresspeople and tell them what you want?

It’s really important that you do that.

One thing that Paolucci has been interested in is math as a tool for social justice. She’s been studying the role that algebra plays in defining life trajectory and career paths, and she sees the importance of supporting after school programs to make sure students have access to this vital tool. She is surprised to find that many folks in D.C. don’t even realize how important math is, and often its a matter of packing mathematical ideas in a way that makes them resonate with people. Of course we know math is important, but her job as a fellow is to make sure the legislators really know what can (and should!) be done. Believe it or not, very few legislators have a background in math or science.

If you’re interested in applying to be a Congressional Fellow, applications are due each February. In the meantime, don’t forget to call your senators and representatives to remind them how much funding for math and science means to you.

I also had coffee with Senator Bob Casey while I was in the capital. He also told me to keep calling him to tell him what was important to me, and — this part was really exciting to me and should be to you too — he told me to bring my students to D.C. and his staffers would arrange a tour of the Capitol and the White House. So get your Math Club, AMS Grad Student Chapter or AWM Student Chapter pumped up about math and public policy, pack them in a bus, and tell your senator you’re on your way!

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Teaching Math to Incarcerated Students

Last month, Beth Malmskog wrote a post for the AMS blog PhD Plus Epsilon about teaching mathematics at a nearby prison. Malmskog is a math professor at Villanova, and in the post she writes about a course she and her colleague Katie Haymaker taught in the spring at Graterford State Correctional Institution. Malmskog had previously written about giving a one-off lecture at Graterford and the Math Circle she and Haymaker started there the next year. Despite the difficulties of teaching with limited supplies and access to technology, Malmskog describes the course as a great experience.

I had more fun teaching this class than I have ever had teaching. The most striking thing about the course was the amount of energy in the classroom throughout the semester.  The students were engaged and game, willing to dive in to any discussion, to speak up with questions, comments, and occasional complaints, and to try activities for themselves.  Every day when I walked out of class, I felt that I had actually connected with the students.  Along with this gameness, most of the students were fairly mature and serious about learning, while still being ready to make jokes and speak up in class.  I wished I could have brought my on-campus students, as a demonstration of what a classroom can be like.  I love working with my on-campus students, but I feel that self-consciousness and expectations of what a college classroom “should” be can really limit their experience.  What could college be like if students really engaged every minute of class time and saw class as a dialogue?  I have tried to create this classroom atmosphere in many classes, with varying degrees of success.  At Graterford, this atmosphere just happened on its own.

Francis Su spoke about one of his incarcerated students in his MAA Retiring Presidential Address, Mathematics for Human Flourishing, posted at his blog The Mathematical Yawp. I know a few of my friends have taught math in prisons as well. But before reading Malmskog’s posts I hadn’t really thought about how that worked.

It turns out there’s no one way to do it. The Villanova Graterford program is somewhat unusual in that students can earn a degree with in-person classes and professors get the same credit for teaching at Graterford as they do on campus at Villanova. Other schools and states have different programs, including correspondence classes and non-credit classes. The Prison Studies Project website has information about many, though not all, programs around the country. (Thanks to Annie Raymond of the University of Washington for pointing me to the Prison Studies Project.) A blog post from the U.S. Department of Education has information about the Second Chance Pell Program that gives grants to incarcerated students through 67 schools around the country.

Another side to the intersection of mathematicians, education, and prison is the fact that the U.S. has the largest incarcerated population in the world and an incarceration rate far higher than most other countries. The system is rife with racism and inequality. If you’re interested in getting involved in studying and working to fix some of the problems with our criminal justice system, Phil Goff writes on Cathy O’Neil’s blog that Justice Needs Nerds.

Posted in Issues in Higher Education, Math Education | Tagged | 1 Comment

Math Puzzles To Pass The Time

Apparently nothing spoils a mountain vista quite as well as a story about a man with a fox, a goose, and a bag of beans.

There are two types of people in this world: those that can only lie, and those that can only tell the truth.

You might recognize that as the opening clause of so many knights and knaves problems. These are classic logic puzzles that I love to use to torture my siblings while we go hiking every summer. Which brings me to my real point. There are two types of people in this world: those that love math riddles, and those that despise them.

Sadly, my siblings fall into the second camp, but I’m hoping that you fall into the first.

What makes math puzzles so enticing are their pleasant blend of elementary math and logic. There are ones that require some knowledge of math, like maybe some knowledge of geometry or a clever use of arithmetic. And there are others that just require some logic. I always love watching mathematicians solve both types.

I recently had a visitor ask me the Cake Icing Problem. You have an iced birthday cake, you cut a piece of size Θ (that’s the center angle of the piece, the way you would typically cut a cake), flip it upside-down and place it back in the cake. If you continue on in one direction, cutting and flipping pieces of size Θ, will all the icing ever be back on top? Will it ever all be on the bottom? For certain values of Θ the answer is obvious, but can you say something general? Project Euler also poses a slightly trickier modification on the birthday cake puzzle.

A great treasure trove of puzzles is Varsity Math, a series cohosted by the Wall Street Journal Blogs and the National Museum of Math. A new couple of problems show up each week with solutions the following week. Last week there was a particularly fun one about areas of squares inscribed in squares.

Slightly less obvious in their mathematical nature, but no less fun, Popular Mechanics also hosts a Riddle of the Week. This week’s problem was the problem of 7 lit candles arranged in a circle. If you blow on one candle it changes the status of the two neighboring candles (that is lit with become unlit and unlit will become lit). What is the minimum number of moves to extinguish every candle. The solution is quite simple, and doesn’t really require any advanced knowledge of math but it’s a nice one to think about.

Alex Bellos, who posts his Monday Puzzle every two weeks, also wrote about a recent popular internet meme, the math problem for a 5 year old that’s been stumping the web. Bellos writes that these problems are sometimes interesting, but often totally misstated and impossible to solve. He walks us through one particular viral problem from earlier this month.

As you go forth and enjoy these puzzles, one word of advice: If you’re hiking up a mountain with a mixed group and you feel compelled to ask that cake icing problem, I recommend that you stay far back from the edge.

Posted in Recreational Mathematics | Tagged , , , | 3 Comments