Monthly Archives: October 2013

Digital Art, Mastermind, and Pythagoras

Welcome to this week’s Math Munch… on (approximately) Math Munch’s second birthday! Hooray!

Check out this video of mathematical art made by artist Nathan Selikoff:

Cool, right? This piece is called “Beautiful Chaos.” The curves on the screen are made from equations (if you’ve ever graphed a line or a parabola you’ll know what I mean). As the viewer waves her hands around, the equations change– and as the equations change, so do the curves! The result is something that might remind you of the images your computer makes when you play music on it or maybe of something you’d make using a spirograph. All in all, a beautiful and interactive piece of mathematical art.

nathanNathan lives and works as a mathematical artist in Orlando, Florida. As he writes on his website, Nathan uses computer code along with other materials to make art that plays with the mathematical ideas of space, motion, and interaction between objects. To see more of how Nathan does this, check out his giant, interactive marionette or this song that explores the first, second, third, and fourth dimensions:

My school is really lucky to be hosting Nathan this week! We didn’t want any of you, dear readers, to miss out on the excitement, though– so Nathan has kindly agreed to answer your interview questions! Got a question for Nathan? Write it in the box below. He’ll answer seven of your best questions in two weeks!

565px-MastermindNext up, who doesn’t love to play Mastermind? It’s a great combination of logic, patterns, and trickery… but I just hate having to use all those tiny pegs. Well, guess what? You can play it online— no pegs (or opponent) necessary!

As I was playing Mastermind, I started wondering about strategy. What’s the best first guess to make? If I were as smart as a computer, is there a number of guesses in which I could guess any Mastermind code? (This kind of question reminds me of God’s Number and the Rubik’s cube…)

Well, it turns out there is a God’s Number for Mastermind – and that number is five. Just five. If you played perfectly and followed the strategy demonstrated by recreational mathematician Toby Nelson on his website, you could guess ANY Mastermind code in five guesses or less. Toby shares many more interesting questions about Mastermind on his website— I suggest you check it out.

What ARE those irrational numbers, so weird that they get their own bubble??

What ARE those irrational numbers, so weird that they get their own bubble??

Finally, sometime in your mathematical past you may have heard of irrational numbers. These are numbers like the square-root of 2 or pi or e that can’t be written as a fraction– or so people claim. When you start thinking about this claim, however, it may seem strange. There are A LOT of fractions– and none of them equal the square-root of 2? Really? What kind of number is that? It seems like only an irrational person would believe that, at least without proof.

Vi Hart to the rescue! Irrational numbers were encountered long, long ago by the ancient Greek mathematician (and cult leader) Pythagoras– and he didn’t like them much. In this great video, Vi tells all about Pythagoras and the controversial discovery of numbers that aren’t fractions.

If you didn’t follow her explanation of why the square-root of 2 is irrational on your first watch, don’t worry– it’s a complicated idea that’s worth a second (or third or fourth) run-through.

Thanks for a great two years of Math Munch! 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!

Tsoro Yematatu, Fano’s Plane, and GIFs

Welcome to this week’s Math Munch!

Board and pieces for tsoro yematatu.

Here’s a little game with a big name: tsoro yematatu. If you enjoyed Paul’s recent post about tic-tac-toe, I think you’ll like tsoro yematatu a lot.

I ran across this game on a website called Behind the Glass. The site is run by the Cincinnati Art Museum. (What is it with me and art museums lately?) The museum uses Behind the Glass to curate many pieces of African art and culture, including four mathematical games that are played in Africa.

The simplest of these is tsoro yematatu. It has its origin in Zimbabwe. Like tic-tac-toe, the goal is to get three of your pieces in a row, but the board is “pinched” and you can move your pieces. Here’s an applet where you can play a modified version of the game against a computer opponent. While the game still feels similar to tic-tac-toe, there are brand-new elements of strategy.

Tsoro yematatu reminds me of one that I played as a kid called Nine Men’s Morris. I learned about it and many other games—including go—from a delightful book called The Book of Classic Board Games. Kat Mangione—a teacher, mom, and game-lover who lives in Tennessee—has compiled a wonderful collection of in-a-row games. And wouldn’t you know, she includes Nine Men’s Morris, tsoro yematatu, tic-tac-toe, and dara—another of the African games from Behind the Glass.

The Fano plane.

The Fano plane.

The board for tsoro yematatu also reminds me of the Fano plane. This mathematical object is very symmetric—even more than meets the eye. Notice that each point is on three lines and that each line passes through three points. The Fano plane is one of many projective planes—mathematical objects that are “pinched” in the sense that they have vanishing points. They are close cousins of perspective drawings, which you can check out in these videos.

Can you invent a game that can be played on the Fano plane?

Closely related to the Fano plane is an object called the Klein quartic. They have the same symmetries—168 of them. Felix Klein discovered not only the Klein quartic and the famous Klein bottle, but also the gorgeous Kleinian groups and the Beltrami-Klein model. He’s one of my biggest mathematical heroes.

The Klein quartic.

The Klein quartic.

This article about the Klein quartic by mathematician John Baez contains some wonderful images. The math gets plenty tough as the article goes on, but in a thoughtfully-written article there is something for everyone. One good way to learn about new mathematics is to read as far as you can into a piece of writing and then to do a little research on the part where you get stuck.

If you’ve enjoyed the animation of the Klein quartic, then I bet my last find this week will be up your alley, too. It’s a Tumblr by David Whyte and Brian Fitzpatrick called Bees & Bombs. David and Brian create some fantastic GIFs that can expand your mathematical imagination.

This one is called Pass ‘Em On. I find it entrancing—there’s so much to see. You can follow individual dots, or hexagons, or triangles. What do you see?


This one is called Blue Tiles. It makes me wonder what kind of game could be played on a shape-shifting checkerboard. It also reminds me of parquet deformations.


A few of my other favorites are Spacedots and Dancing Squares. Some of David and Brian’s animations are interactive, like Pointers. They have even made some GIFs that are inspired by Tilman Zitzmann’s work over at Geometry Daily (previously).

I hope you enjoy checking out all of these new variations on some familiar mathematical objects. Bon appetit!

Reflection Sheet – Tsoro Yematatu, Fano’s Plane, and GIFs

God’s Number, Chocolate, and Devil’s Number

Welcome to this week’s Math Munch! This week, I’m sharing with you some math things that make me go, “What?!” Maybe you’ll find them surprising, too.

The first time I heard about this I didn’t believe it. If you’ve never heard it, you probably won’t believe it either.

Ever tried to solve one of these? I’ve only solved a Rubik’s cube once or twice, always with lots of help – but every time I’ve worked on one, it’s taken FOREVER to make any progress. Lots of time, lots of moves…. There are 43,252,003,274,489,856,000 (yes, that’s 43 quintillion) different configurations of a Rubik’s cube, so solving a cube from any one of these states must take a ridiculous number of moves. Right?

Nope. In 2010, some mathematicians and computer scientists proved that every single Rubik’s cube – no matter how it’s mixed up – can be solved in at most 20 moves. Because only an all-knowing being could figure out how to solve any Rubik’s cube in 20 moves or less, the mathematicians called this number God’s Number.

Once you get over the disbelief that any of the 43 quintillion cube configurations can be solved in less than 20 moves, you may start to wonder how someone proved that. Maybe the mathematicians found a really clever way that didn’t involve solving every cube?

Not really – they just used a REALLY POWERFUL computer. Check out this great video from Numberphile about God’s number to learn more:

Screen Shot 2013-10-02 at 2.48.01 PM

Here’s a chart that shows how many Rubik’s cube configurations need different numbers of moves to solve. I think it’s surprising that so few required all 20 moves. Even though every cube can be solved in 20 or less moves, this is very hard to do. I think it’s interesting how in the video, one of the people interviewed points out that solving a cube in very few moves is probably much more impressive than solving a cube in very little time. Just think – it takes so much thought to figure out how to solve a Rubik’s cube at all. If you also tried to solve it efficiently… that would really be a puzzle.

Next, check out this cool video. Its appealing title is, “How to create chocolate out of nothing.”

This type of puzzle, where area seems to magically appear or disappear when it shouldn’t, is called a geometric vanish. We’ve been talking about these a lot at school, and one of the things we’re wondering is whether you can do what the guy in the video did again, to make a second magical square of chocolate. What do you think?

infinityJHFinally, I’ve always found infinity baffling. It’s so hard to think about. Here’s a particularly baffling question: which is bigger, infinity or infinity plus one? Is there something bigger than infinity?

I found this great story that helps me think about different sizes of infinity. It’s based on similar story by mathematician Raymond Smullyan. In the story, you are trapped by the devil until you guess the devil’s number. The story tells you how to guarantee that you’ll guess the devil’s number depending on what sets of numbers the devil chooses from.

Surprisingly, you’ll be able to guess the devil’s number even if he picks from a set of numbers with an infinite number of numbers in it! You’ll guess his number if he picked from the counting numbers larger than zero, positive or negative counting numbers, or all fractions and counting numbers. You’d think that there would be too many fractions for you to guess the devil’s number if he included those in his set. There are infinitely many counting numbers – but aren’t there even more fractions? The story tells you about a great way to organize your guessing that works even with fractions. (And shows that the set of numbers with fractions AND counting numbers is the same size as the set of numbers with just counting numbers… Whoa.)

Is there something mathematical that makes you go, “What?!” How about, “HUH?!” If so, send us an email or leave us a note in the comments. We’d love to hear about it!

Bon appetit!