Tag Archives: Martin Gardner

Scary-o-graphic Projection, Thinky the Dragon, and Martin Gardner

Welcome to this week’s Math Munch!

Halloween is quickly approaching, which is why last week, Anna shared some pumpkin polyhedra. It just so happens that Justin did some pumpkin-y math of his own last year. He created a must-watch video called “Scary-o’-graphic Projection,” which was shown in the 2014 Bridges Short Film Festival. Enjoy, but don’t get too scared.

A stereographic projection sculpture by Henry Segerman.

A stereographic projection sculpture by Henry Segerman.

To learn some more about stereographic projection, watch one of Henry Segerman’s videos. You’ll also get to see some of his 3D printed sculptures.  (1 2)

In other news, Oct. 21 marked the 100th anniversary of the birthday of Martin Gardner!! (previously featured here, here, and here, among others) Around this time every year people get together to do math in his honor as part of Celebration of Mind.

This year we’re featuring one of Gardner’s optical illusions. Let’s begin with a video. Meet Thinky the Dragon.

You can find printable make-your-own templates here. (There are other colors as well.) Thinky is an example of a “hollow face” illusion, many more of which can be found on mathaware.org. There you can also find this video explaining the geometry behind this illusion.

And look at this AMAZING movie trailer that one of our readers made.  Thank you Lily!!!

Can you fold this strip of 7 squares into a cube?

Can you fold this strip of 7 squares into a cube?

Whats special about this square?  More than you think!

Whats special about this square? More than you think!

Thank you to Colm Mulcahy for his recent post on the BBC website, where Colm put together a list of 10 really wonderful problems from the hundreds that Gardner wrote about and popularized during his career. Gardner helped show the world that thinking about problems and mathematics was a really fun way to spend time. Watch the video below to learn more about Celebration of Mind events, and click here to see if there’s an event near you.  Note: You can even host an event of your own.

BONUS: I just have to mention MoSAIC for any math art enthusiasts in our audience. Around the country, small mathematical art conferences and exhibitions will go on this year. Click to learn more or find an event near you.

Munch in honor of Martin Gardner. Bon appetit!

Picture from the MoSAIC website.

Picture from the MoSAIC website.

Math Awareness Month, Hexapawn, and Plane Puzzles

Welcome to this week’s Math Munch!

April is Mathematics Awareness Month. So happy Mathematics Awareness Month! This year’s theme is “Mathematics, Magic, and Mystery”. It’s inspired by the fact that 2014 would have marked Martin Gardner’s 100th birthday.


A few of the mathy morsels that await you this month on mathaware.org!

Each day this month a new piece of magical or mysterious math will be revealed on the MAM site. The mathematical offering for today is a card trick that’s based on the Fibonacci numbers. Dipping into this site from time to time would be a great way for you to have a mathy month.

It is white

It is white’s turn to move. Who will win this Hexapawn game?

Speaking of Martin Gardner, I recently ran across a version of Hexapawn made in the programming language Scratch. Hexapawn is a chess mini-game involving—you guessed it—six pawns. Martin invented it and shared it in his Mathematical Games column in 1962. (Here’s the original column.) The object of the game is to get one of your pawns to the other side of the board or to “lock” the position so that your opponent cannot move. The pawns can move by stepping forward one square or capturing diagonally forward. Simple rules, but winning is trickier than you might think!

The program I found was created by a new Scratcher who goes by the handle “puttering”. On the site he explains:

I’m a dad. I was looking for a good way for my daughters to learn programming and I found Scratch. It turns out to be so much fun that I’ve made some projects myself, when I can get the computer…

puttering's Scratch version of Conway's Game of Life

puttering’s Scratch version of Conway’s Game of Life

Something that’s super cool about puttering’s Hexapawn game is that the program learns from its stratetgy errors and gradually becomes a stronger player as you play more! It’s well worth playing a bunch of games just to see this happen. puttering has other Scratch creations on his page, too—like a solver for the Eight Queens puzzle and a Secret Code Machine. Be sure to check those out, too!

Last up, our friend Nalini Joshi recently travelled to a meeting of the Australian Academy of Science, which led to a little number puzzle.


What unusual ways of describing a number! Trying to learn about these terms led me to an equally unusual calculator, hosted on the Math Celebrity website. The calculator will show you calculations about the factors of a numbers, as well as lots of categories that your number fits into. Derek Orr of Math Year-Round and I figured out that Nalini’s clues fit with multiple numbers, including 185, 191, and 205. So we needed more clues!

Can you find another number that fits Nalini’s clues? What do you think would be some good additional questions we could ask Nalini? Leave your thoughts in the comments!


A result from the Number Property Calculator

I hope this post helps you to kick off a great Mathematics Awareness Month. Bon appetit!

Celebration of Mind, Cutouts, and the Problem of the Week

Welcome to this week’s Math Munch!  We’re going to revisit the work of Martin Gardner, look at some beautiful mathematical art, and see if we can dig into a college’s “problem of the week” program.

Martin Gardner

Martin Gardner

Last October, I wrote about Martin Gardner. He is one of the great popularizers of mathematics, known for his puzzles, columns in Scientific American, and over 100 books. Around the time of his birthday, October 21st, each year, people around the world participate in a global “Gathering4Gardner” — a so-called Celebration of Mind.

Two Sipirals

One of Martin Gardner’s many puzzles

These are gatherings of two or more people taking time to dig into the kinds of mathematics that Martin Gardner loved so much. Below you can find lots of ways to participate and share with family, friends, or strangers.

First, If you want to learn more about Gardner himself, here’s a very detailed interview. You can also try solving some of Gardner’s great puzzles. We featured both of these last year, but I recently found a whole new page of resources and activities for the Celebration of Mind.

In the video on the left you can see a geometric vanish like those we’ve previously featured (Get off the Earth, and Chocolate). The second is a surprising play on the Möbius Strip which we’ve also featured before (Art and Videos + Möbius Hearts). I hope you’ll find some time this week to celebrate Martin Gardner’s love of math and help grow your own. (Though, I guess if you’re reading this, you already are!)

Up next, check out the work of artist Elena Mir. This video shows a series of artworks she created over the last four years. They feature stacks of cut paper to form geometric shapes, and they make me wonder what I could make out of cut paper. If you make something, please let us now.

It reminds me of the work of Matt Shlian that we featured in our very first post.  You can watch Matt’s TED Talk or visit his website to see all sorts of cutouts and other paper sculptures, plus incredible videos like the one below. It might be my favorite video I’ve ever posted on Math Munch.

Finally, Macalester College in St. Paul, Minnesota has a weekly problem that they offer to their students, and the problem archives can be found online. These are for college students, so some of them are advanced or phrased in technical language, but I think we can find some that all of us can dig in to. Give these a try:

Have a mathematical week, and let us know if you do anything for the Celebration of Mind. Bon appetit!

TED, Bridges, and Silk

Welcome to this week’s Math Munch!


The Math Munch team at TEDxNYED

Marjorie Rice

Marjorie Rice | click to watch her interview video

On Saturday, the Math Munch team gave a 16-minute presentation at TEDxNYED about Math Munch!  (Eventually there will be a video, and we’ll be sure to share it with you right away, but you’ll have to wait a month, maybe.)

We started with the story of Marjorie Rice, and in searching for a good picture of her, we came across this wonderful interview in a documentary about Martin Gardner.  It’s so neat to hear her speak about her discoveries.  You can see how proud she is and how much she truly loves math.  Feel free to watch the whole documentary if you like.  I haven’t gotten a chance yet, but I know it’s full of incredible stuff.

In the spirit of TED, I decided to share a few mathematical TED talks.  This one is absolutely fascinating.  In it, mathematician Ron Eglash describes how fractals underly the african designs.  You know how we love fractals.

If you’re hungry for another TED talk, here’s one about connections between music, mathematics, and sonar.

Up next, remember when we wrote about attending last year’s Bridges conference?  Well it happens every year, of course, and this year’s gallery of mathematical art is available online!  Click on one of those images and you get to more of the artists work.  I could easily spend hours staring at this art, trying to understand them, and reading the descriptions and artist statements.  Seriously, there is just way too much cool stuff there, so I’ve picked out a few of my favorites.  Also, I have great news to announce: Chloé Worthington (previously featured) had some of her art accepted to the exhibition!  Congratulations, Chloe!  If you look closely, you’ll see some of my art in there too.  🙂

Bjarne Jespersen

Bjarne Jespersen

Marc Chamberland

Marc Chamberland

Bob Rollings

Bob Rollings

Chloe Worthington

Chloé Worthington

Mehrdad Garousi

Mehrdad Garousi

By the way, if you ever create any mathematical art of your own, we’d love to see it!  Send us an email at mathmunchteam@gmail.com, and maybe we’ll feature your work in an upcoming Math Munch. (Only if you want us too, of course.)

Yuri Vishnevsky

Silk creator Yuri Vishnevsky

Finally, I know many of you like playing around with Symmetry Artist, which can be found on our page of Math Art Tools.  If you like that, then you’ll love Silk!  It’s much the same, but generates a certain kind of whispiness as you draw that looks really cool.  It also lets you spiral your designs toward the center, a feature which Symmetry Artist lacks.  You can download the Silk app for iPad or iPhone, if you like.  Silk was designed by Yuri Vishnevsky, with sound design by Mat Jarvis.  Yuri has agreed to do a Q&A for us, but we haven’t quite finished it just yet.  I’ll upload it as soon as possible, but for now, you can read an interview Mat and Yuri did with a website called Giant Fire Breathing Dragon.

Bon appetit!

Silk1 Silk4 Silk2


Dots-and-Boxes, Choppy Waves, and Psi Day

Welcome to this week’s Math Munch!

And happy Psi Day! But more on that later.


Click to play Dots-and-Boxes!

Recently I got to thinking about the game Dots-and-Boxes. You may already know how to play; when I was growing up, I can only remember tic-tac-toe and hangman as being more common paper and pencil games. If you know how to play, maybe you’d like to try a quick game against a computer opponent? Or maybe you could play a low-tech round with a friend? If you don’t know how to play or need a refresher, here’s a quick video lesson:

In 1946, a first grader in Ohio learned these very same rules. His name was Elwyn Berlekamp, and he went on to become a mathematician and an expert about Dots-and-Boxes. He’s now retired from being a professor at UC Berkeley, but he continues to be very active in mathematical endeavors, as I learned this week when I interviewed him.

Elwyn Berlekamp

Elwyn Berlekamp

In his book The Dots and Boxes Game: Sophisticated Child’s Play, Elwyn shares: “Ever since [I learned Dots-and-Boxes], I have enjoyed recurrent spurts of fascination with this game. During several of these burst of interest, my playing proficiency broke through to a new and higher plateau. This phenomenon seems to be common among humans trying to master any of a wide variety of skills. In Dots-and-Boxes, however, each advance can be associated with a new mathematical insight!”

Elwyn's booklet about Dots-and-Boxes

Elwyn’s booklet about

In his career, Elywen has studied many mathematical games, as well as ideas in coding. He has worked in finance and has been involved in mathematical outreach and community building, including involvement with Gathering for Gardner (previously).

Elywn generously took the time to answer some questions about Dots-and-Boxes and about his career as a mathematician. Thanks, Elywn! Again, you should totally check out our Q&A session. I especially enjoyed hearing about Elwyn’s mathematical heros and his closing recommendations to young people.

As I poked around the web for Dots-and-Boxes resources, I enjoyed listening to the commentary of Phil Carmody (aka “FatPhil”) on this high-level game of Dots-and-Boxes. It was a part of a tournament held on a great games website called Little Golem where mathematical game enthusiasts from around the world can challenge each other in tournaments.

What's the best move?A Sam Loyd Dots-and-Boxes Puzzle

What’s the best move?
A Dots-and-Boxes puzzle by Sam Loyd.

And before I move on, here are two Dots-and-Boxes puzzles for you to try out. The first asks you to use the fewest lines to saturate or “max out” a Dots-and-Boxes board without making any boxes. The second is by the famous puzzler Sam Loyd (previously). Can you help find the winning move in The Boxer’s Puzzle?

Next up, check out these fantastic “waves” traced out by “circling” these shapes:

Click the picture to see the animation!

Lucas Vieira—who goes by LucasVB—is 27 years old and is from Brazil. He makes some amazing mathematical illustrations, many of them to illustrate articles on Wikipedia. He’s been sharing them on his Tumblr for just over a month. I’ll let his images and animations speak for themselves—here are a few to get you started!

A colored-by-arc-length Archimedean spiral.

A colored-by-arc-length Archimedean spiral.


A sphere-like degenerate torus.

A Koch cube.

A Koch cube.

There’s a great write-up about Lucas over at The Daily Dot, which includes this choice quote from him: “I think this sort of animated illustration should be mandatory in every math class. Hopefully, they will be some day.” I couldn’t agree more. Also, Lucas mentioned to me that one of his big influences in making mathematical imagery has always been Paul Nylander. More on Paul in a future post!

Psi is the 23rd letter in the Greek alphabet.

Psi is the 23rd letter in the Greek alphabet.

Finally, today—March 11—is Psi Day! Psi is an irrational number that begins 3.35988… And since March is the 3rd month and today is .35988… of the way through it–11 out of 31 days—it’s the perfect day to celebrate this wonderful number!

What’s psi you ask? It’s the Reciprocal Fibonacci Constant. If you take the reciprocals of the Fibonnaci numbers and add them add up—all infinity of them—psi is what you get.


Psi was proven irrational not too long ago—in 1989! The ancient irrational number phi—the golden ratio—is about 1.61, so maybe Phi Day should be January 6. Or perhaps the 8th of May—8/5—for our European readers. And e Day—after Euler’s number—is of course celebrated on February 7.

That seems like a pretty good list at the moment, but maybe you can think of other irrational constants that would be fun to have a “Day” for!

And finally, I’m sure I’m not the only one who’d love to see a psi or Fibonacci-themed “Gangham Style” video. Get it?

Bon appetit!


EDIT (3/14/13): Today is Pi Day! I sure wish I had thought of that when I was making my list of irrational number Days…

Marjorie Rice, Inspired by Math, and Subways

Welcome to this week’s Math Munch!

A few weeks ago, I learned about an amazing woman named Marjorie Rice.  Marjorie is a mathematician – but with a very unusual background.

mrice_picMarjorie had no mathematical education beyond high school.  But, Marjorie was always interested in math.  When her children were all in school, Marjorie began to read about and work on math problems for fun.  Her son had a subscription to Scientific American, and Marjorie enjoyed reading articles by Martin Gardner (of hexaflexagon fame).  One day in 1975, she read an article that Martin Gardner wrote about a new discovery about pentagon tessellations.  Before several years earlier, mathematicians had believed that there were only five different types of pentagons that could tessellate – or cover the entire plane without leaving any gaps.   But, in 1968, three more were discovered, and, in 1975, a fourth was found – which Martin Gardner reported on in his article.

Marjorie's first type of pentagonWhen she read about this, Marjorie became curious about whether she could find her own new type of pentagon that could tile the plane.  So, she got to work.  She came up with her own notation for the relationships between the angles in her pentagons.  Her new notation helped her to see things in ways that professional mathematicians had overlooked.  And, eventually… she found one!  Marjorie wrote to Martin Gardner to tell him about her discovery.  By 1977, Marjorie had discovered three more types of pentagons that tile the plane and her new friend, the mathematician Doris Schattschneider, had published an article about Marjorie’s work  in Mathematics Magazine.

type11There are now fourteen different types of pentagons known to tile the plane… but are there more?  No one knows for sure.  Whether or not there are more types of pentagons that tile the plane is what mathematicians call an open problem.  Maybe you can find a new one – or prove that one can’t be found!

Marjorie has a website called Intriguing Tessellations on which she’s written about her work and posted some of her tessellation artwork.  Here is one of her pentagon tilings transformed into a tessellation of fish.

fishgrid fishsm

By the way, it was Marjorie’s birthday a few weeks ago.  She just turned 90 years old.  Happy Birthday, Marjorie!

wild about math logoNext up, I just ran across a great blog called Wild About Math!  This blog is written by Sol Lederman, who used to work with computers and LOVES math.  My favorite part about this blog is a series of interviews that Sol calls, “Inspired by Math.”  Sol has interviewed about 23 different mathematicians, including Steven Strogatz (who has written two series of columns for the New York Times about mathematics) and Seth Kaplan and Deno Johnson, the producer and writer/director of the Flatland movies.  You can listen to Sol’s podcasts of these interviews by visiting his blog or iTunes.  They’re free – and very interesting!

subway map 2Finally, what New York City resident or visitor isn’t fascinated by the subway system? And what New York City resident or visitor doesn’t spend a good amount of time thinking about the fastest way to get from point A to point B?  Do you stay on the same train for as long as possible and walk a bit?  Or do you transfer, and hope that you don’t miss your train?

chris and matt

Chris and Matt, on the subway.

Well, in 2009, two mathematicians from New York – Chris Solarz and Matt Ferrisi – used a type of mathematics called graph theory to plan out the fastest route to travel the entire New York City subway system, stopping at every station.  They did the whole trip in less than 24 hours, setting a world record!  Graph theory is the branch of mathematics that studies the connections between points or places.  In their planning, Chris and Matt used graph theory to find a route that had the most continuous travel, minimizing transfers, distance, and back-tracking.  You can listen to their fascinating story in an interview with Chris and Matt done by the American Mathematical Society here.

If you’re interested in how graph theory can be used to improve the efficiency of a subway system, check out this article about the Berlin subway system (the U-bahn).  Students and professors from the Technical University Berlin used graph theory to create a schedule that minimized transfer time between trains.  If only someone would do this in New York…

Bon appetit!

Martin Gardner, G4G, and Many More Flexagons

Welcome to this week’s Math Munch!

Meet the incredible Martin Gardner. If you’re a mathematician then chances are good you already have, but any reader of this blog has certainly felt the ripples of his influence. Nearly everything we share on Math Munch can be traced back to the recreational mathematics of Martin Gardner. For 25 years he wrote the “Mathematical Games” column for Scientific American, in which he shared wonderful puzzles, riddles, and games that have inspired generations of mathematicians. We’re trying to do the same here at Math Munch, so Gardner’s a sort of hero for us.

Maybe the best way to connect with this math legend would be to take on a few of his puzzles. (I’ve linked to some of my favorites below.) Many of the puzzles have a “Print ‘n Play” option you may want to take advantage of. His columns and other writings now live in the more than 100 books he wrote. Pick one up at your local library and dig in!

Martin Gardner’s Puzzles

Heavy Weight

Crazy Cut

Two Sipirals

Cube Dates

Scott Kim’s ambigram logo for G4G5

In this video you can hear Scott Kim (remember his ambigrams?) talk about his own connection to Martin Gardner, and here he talks about his involvement with the Gathering 4 Gardner. These are events that take place across the world to celebrate Gardner’s work and legacy. If you’re inspired by some of this stuff, maybe you’ll get a few friends together and share it with them.

Gardner’s very first article for Scientific American was about the hexaflexagon, so as part of this year’s Gathering 4 Gardner, people are making flexagons of all sorts. Here’s a video of Martin himself talking about them, which is part of full-length video about Gardner called “The Nature of Things.” Justin wrote about flexagons here, and Vi Hart followed suit with a pair of fantastic videos telling the true story of their discovery.

And here’s part 2.

That’s not all! There’s a whole world of flexagons to build and play with. To see another kind, check out the cyclic hexa-tetraflexagon as shown off by James Grime in this video.

A Flexagon Bestiary

Here’s one last flexagon resource with instructions you might prefer.  There are so many videos to watch, puzzles to solve, and flexagons to flex. Hopefully you’ll have a very mathy week in honor of Martin Gardner.

Bon appetit!