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

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

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 sphere-like degenerate torus.

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.

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…

Posted by Justin Lanier. Categories: Math Munch. Tags: art, dots-and-boxes, Fibonacci, fractals, games, interview, irrational, Martin Gardner, numbers, pi, psi, spirals, video, waves. 3 Comments

Mar. ’13

Collaborative Math, Petals, and Theseus

Welcome to this week’s Math Munch!

Let’s start with a great new blog – a place for you to do math – Collaborative Mathematics. It’s the pet project of mathematician, teacher, and juggler, Jason Ermer. The idea is simple. Jason posts videos about a little mathematical idea, and he offers up a challenge question for viewers to solve. In fact, he has lots of ideas for how you can do some mathematical research of your own. After that, you make a response video explaining what you’ve come up with. That’s Collaborative Mathematics.

His first video was about ERMER numbers, like 12312 or 94794. Core Challenge: How many ERMER numbers are even? To learn all about it and get involved, check out this video.

On his site, Jason says, “when possible, students should work with a team of problem solving peers. Our ideas are formed and refined as we communicate our thoughts to others and as we hear a diversity of ways of interpreting the same concepts.” So don’t feel like you have to do it all alone. It is collaborative after all!

And don’t worry if things are tough! “Struggling in mathematics is not a bad thing! We expect sore muscles when we exercise and try to improve at, say, basketball. Why would we expect mathematical growth to be painless? We must exert ourselves to grow. There is glory in the struggle! “

And if you liked that. Here’s the second video challenge.

* * *

Up next is a simple little site I found called the Petals Challenge.

The secret of the game is in the name of the game: Petals around a rose
How many petals?

It’s a kind of riddle, because there aren’t really instructions. The only way to make sense of it is to give it a try. Good luck, and never tell anyone the secret of the game!

* * *

Lastly, here’s a great game called Theseus and the Minoataur. You’re Theseus, and you must exit a labyrinth while a minotaur chases you. The Minotaur is faster than you are, though, so you’ll have to be clever!

Unfortunately, this is a java game, which some computers won’t be able to play, so as a bonus, watch this beautiful animation from Numberphile showing the creation of the Dragon Fractal.

Bon appetit!

Posted by Paul Salomon. Categories: Math Munch. Leave a comment

Feb. ’13

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.

Marjorie 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.

When 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.

There 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.

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

$wild about math logo$ Next 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!

Finally, 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, 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!

Posted by Anna Weltman. Categories: Math Munch. Tags: art, geometry, graphs, history, interview, Martin Gardner, open problem, podcast, tessellation. 5 Comments

Feb. ’13

Folds, GIMPS, and More Billiards

Welcome to this week’s Math Munch!

First up, we’ve often featured mathematical constructions made of origami. (Here are some of those posts.) Origami has a careful and peaceful feel to it—a far cry from, say, the quick reflexes often associated with video games. I mean, can you imagine an origami video game?

One of Fold’s many origami puzzles.

Well, guess what—you don’t have to, because Folds is just that! Folds is the creation of Bryce Summer, a 21-year-old game designer from California. It’s so cool. The goal of each level of its levels is simple: to take a square piece of paper and fold it into a given shape. The catch is that you’re only allowed a limited number of folds, so you have to be creative and plan ahead so that there aren’t any loose ends sticking out. As I’ve noted before, my favorite games often require a combo of visual intuition and careful thinking, and Folds certainly fits the bill. Give it a go!

Once you’re hooked, you can find out more about Bryce and how he came to make Folds in this awesome Q&A. Thanks so much, Bryce!

Next up, did you know that a new largest prime number was discovered less than a month ago? It’s very large—over 17 million digits long! (How many pages would that take to print or write out?) That makes it way larger than the previous record holder, which was “only” about 13 million digits long. Here is an article published on the GIMPS website about the new prime number and about the GIMPS project in general.

What’s GIMPS you ask? GIMPS—the Great Internet Mersenne Primes Search—is an example of what’s called “distributed computing”. Testing whether a number is prime is a simple task that any computer can do, but to check many or large numbers can take a lot of computing time. Even a supercomputer would be overwhelmed by the task all on its own, and that’s if you could even get dedicated time on it. Distributed computing is the idea that a lot of processing can be accomplished by having a lot of computers each do a small amount of work. You can even sign up to help with the project on your own computer. What other tasks might distributed computing be useful for? Searching for aliens, perhaps?

GIMPS searches only for a special kind of prime called Mersenne primes. These primes are one less than a power of two. For instance, 7 is a Mersenne prime, because it’s one less that 8, which is the third power of 2. For more on Mersenne primes, check out this video by Numberphile.

Finally, we’ve previously shared some resources about the math of billiards on Math Munch. Below you’ll find another take on bouncing paths as Michael Moschen combines the math of billiards with the art of juggling.

So lovely. For more on this theme, here’s a second video to check out.

Bon appetit!

Posted by Justin Lanier. Categories: Math Munch. Tags: big numbers, billiards, computers, games, geometry, interactive puzzle, interview, juggling, numbers, origami, primes, video. 1 Comment

Feb. ’13

The Sierpinski Valentine, Cardioids, and Möbius Hearts

Welcome to this week’s Math Munch!

With Valentine’s Day this Thursday, let’s take a look at some mathy Valentine stuff. If you’re searching for the perfect card design for your valentine, search no more. Math Munch has you covered!

Sierpinski Valentine

xkcd creator Randall Munroe

Above you can see a clever twist on the classic Sierpinski Triangle, which I found on xkcd, a wonderfully mathematical webcomic. You can read more about xkcd creator Randall Munroe in this interview from the Sept. 2012 issue of Math Horizons. (pdf version)

Ron Doerfler designed another math-insprired Valentine’s Day card, which you can check out here. The image to the left is only part of it. Don’t get it? Well it’s a reference to a mathematical curve called the cardioid (from the Greek word for “heart”). Look what happens if you follow a point on one circle as it rolls around another. You’ll have to imagine it tipped the other way so it’s oriented like a typical heart, but that curve is a cardioid. The second animation was created by the amazing and previously featured Matt Henderson. If you have a compass, then you can make the second one at home.

A cardioid generated by one circle rolling around another

A completely different way to generate a cardioid

Pop-up Sierpinski Heart Card

Really though, nothing says “I Love you” like a Möbius strip. Am I right? Here’s a quick little project you can do to make a pair of linked Möbius hearts. You can find directions here on a blog called 360, or you can watch the video below. Oh, and as if that wasn’t enough great stuff, here’s one more project from the 360 blog, a pop-up version of the Sierpinski Heart!

Happy Valentine’s Day, and bon appetit!

Posted by Paul Salomon. Categories: Math Munch. Tags: cardioid, hearts, Mobius strip, paper projects, sierpinski triangle, valentine s day, xkcd. 2 Comments

Feb. ’13

Mathematical Impressions, Modular Origami, and the Tenth Dimension

Welcome to this week’s Math Munch!

First up, check out the latest video in George Hart‘s series called “Mathematical Impressions.” George has been making videos for “Mathematical Impressions,” which is sponsored by the Simons Foundation, since summer, when he made his video debut – so there are many videos to watch! Here’s his newest video, called, “Attesting to Atoms,” about how the geometric structure of crystals gives clues to the existence of atoms. (Click on the picture below to watch the video.)

I love how this video shows a real way in which knowledge of mathematics – which can seem very abstract at times – can help us to understand the structure of the world, which is very concrete. In this second video, one of my favorites, George talks about the reverse of that – allowing our knowledge of something concrete to help us understand abstract mathematics. This video is called, “Knot Possible.” (Again, click on the picture to watch the video!)

I could have used these words of wisdom from George when I was thinking about the problem he poses in this video: “Don’t let your knowledge of mathematics artificially limit what you think is physically possible. Quite to the contrary! Mathematics is a tool which can empower us to do amazing things that no one has ever done before.” Well said, George!

Speaking of using mathematics to do and make amazing things, check out this website of modular origami models and patterns!

This site was put together by Michal Kosmulski, who lives in Poland and works in information technology. In addition, however, he folds these amazing modular origami polyhedra, fractals, and other awesome mathematical objects! Michal’s site is full of pictures of his modular origami creations and links to patterns for how to make them yourself as well as information about the mathematics behind the objects. He has also included some useful tips on how to make the more challenging shapes.

One of my favorites is the object to the left, “Five Intersecting Tetrahedra.” I think that this structure is both beautiful and very interesting. It can be made by intersecting five tetrahedra, or triangular-based pyramids, as shown, or by making a stellation of an icosahedron. What does that mean? Well, an icosahedron is a polyhedron with twenty equilateral triangular faces. To stellate a polyhedron, you extend some element of the polyhedron – such as the faces or edges – in a symmetric way until they meet to form a new polyhedron. There are 59 possible stellations of the icosahedron! Michal has models of several of them, including the Five Intersecting Tetrahedra and the great stellated icosahedron shown below on the left. The figure on the right is called “Cube.”

Finally, all the talking about dimensions that we’ve been doing for the past few weeks reminded me of my favorite video about higher dimensions. It’s called, “Imagining the Tenth Dimension,” and it shows a way of thinking about dimensions, from the zero dimension all the way up to the tenth. I can watch this video again and again and still find it mind-blowing and fascinating.

Bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: applied math, art, dimensions, fractals, George Hart, knots, origami, polyhedra, video. Leave a comment

Jan. ’13

Ghost Diagrams, Three New Games, and Scrabble Tiles

Welcome to this week’s Math Munch!

A ghost diagram composed of two different tiles.

“An organism is more than the sum of its organs. When the organs are fitted together, the organism becomes something more. This surprising something more we call “spirit” or “ghost”. Ghost Diagrams finds the ghosts implicit in simple sets of tiles.“

So writes Paul Harrison, creator of the amazing Ghost Diagram applet. Paul creates all kinds of free software and has his Ph.D. in Computer Science. I found his Ghost Diagram applet through this huge list of links about generative art.

A ’111-’ tile connected to a ’1aA1′ tile.

Given a collection of tile types, the applet tries to find a way to connect them so that no tile has any loose ends. A tile type is specified through a string of letters, numbers, and dashes. Each of these specifies an edge. You can think of a four-character tile as being a modified square and a six-character tile as being a modified hexagon. Two tiles can connect if they have edges that match. Number edges match with themselves—1 matches with 1—while letter edges match with the same letter with opposite capitalization—a matches with A.

It’s amazing the variety of patterns that can emerge out of a few simple tiles. Here are a couple of ghost diagrams that I created. You can click them to see live versions in the applet. There are many other nice ghost diagrams that Paul has compiled on the site. Also, be sure to check out the random button—it’s a great way to get started on making a pattern of your own. I hope you enjoy tinkering with the ghost diagram applet as much as I have.

And now for some more fun: three new games! When I ran across Loops of Zen, I had ghost diagrams on my mind. I think they have a similar feel to them. The goal in each level of Loops of Zen is to orient the paths and loops so that they connect up without any loose edges. I feel like this game—like good mathematics—requires both a big-picture, intuitive grasp of the playing field and detailed, logical thinking. Put another way, you need both global strategy and local tactics. Also, if you like playing Entanglement, then I bet you’ll like Loops of Zen, too.

Last week we wrote about Flatland. This book and the movies it inspired describe what it might be like if creatures of different dimensionality were to meet each other. The game Z-Rox puts you in the shoes of a Flatlander. Mystery shapes pass through your field of vision a slice at a time, and it’s up to you to identify what they are. It’s a tricky task that requires a good imagination.

Hat tip to Casual Girl Gamer for both of these great mathy games.

Steppin’ Stones

Steppin’ Stones is a fun little spatial puzzle game I recently came across. You should definitely check it out. It also provides a nice segue to our last mathy item for the week, because a Steppin’ Stones board looks a lot like a Scrabble board. Scrabble, of course, is a word game. Aside from the arithmetic of keeping score, there isn’t much mathematics involved in playing it. In addition, the universe of Scrabble—the English dictionary—is not particularly elegant from a math standpoint. However, it’s the amazing truth that even in arenas that don’t seem very mathematical, math can often be applied in useful ways.

From a comic about Prime Scrabble on Spiked Math.

In Re-evaluating the values of the tiles in Scrabble™, the author—who goes by DTC and is a physics graduate student at Cornell—wonders whether the point values assigned to letters in Scrabble are correctly balanced. The basic premise is that the harder a letter is to play, the more it should be worth. DTC does what any good mathematician does—lays out assumptions clearly, reasons from them to make a model, critiques the arguments of others, and of course makes lots of useful calculations. One tool DTC uses is the Monte Carlo method. In the end, DTC finds that the current Scrabble point values are very close to what the model would assign.

I really enjoyed the article, and I hope you will, too. And since Scrabble is a “crossword game”, I think I’ll leave you with a couple of “crossnumber” puzzles. Here are some straightforward ones, while these require a little more thinking.

Have a great week, and bon appetit!

P.S. I can’t resist sharing this video as a bonus: a cellular automaton of rock-paper-scissors! Blue beats green, green beats red, and red beats blue. Hooray for non-transitive swirls!

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, applied math, art, board games, cellular automata, comics, dimensions, games, geometry, puzzles, tessellation, video. 6 Comments