Tag Archives: chess

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!

Virtual Hyenas, Markov Chains, and Random Knights

Welcome to this week’s Math Munch!

It’s amazing how a small step can lead to a chain reaction of adventure.

Arend Hintze

Arend Hintze

Recently a reader named Nico left a comment on the Math Munch post where I shared the game Loops of Zen. He asked why the game has that name. Curious, I looked up Dr. Arend Hintze, whose name appears on the game’s title page. This led me to Arend’s page at the Adami Lab at Michigan State University. Arend studies how complex systems—especially biological systems—evolve over time.

Here is a video of one of Arend’s simulations. The black and white square is a zebra. The yellow ones are lions, the red ones are hyenas, and guess who’s hungry?

Arend’s description of the simulation is here. The cooperative behavior in the video—two hyenas working together to scare away a lion—wasn’t programmed into the simulation. It emerged out of many iterations of systems called Markov Brains—developed by Arend—that are based upon mathematical structures called Markov chains. More on those in a bit.

You can read more about how Arend thinks about his multidisciplinary work on biological systems here. Also, it turns out that Arend has made many more games besides Loops of Zen. Here’s Blobs of Zen, and Ink of Zen is coming out this month! Another that caught my eye is Curve, which reminds me of some of my favorite puzzle games. Curve is still in development; here’s hoping we’ll be able to play it soon.

Arend has agreed to do an interview with Math Munch, so share your questions about his work, his games, and his life below!

Eric Czekner

Eric Czekner

Arend’s simulations rely on Markov chains to model animal behavior. So what’s a Markov chain? It’s closely related to the idea of a random walk. Check out this video by digital artist, musician, and Pure Data enthusiast Eric Czekner. In the video, Eric gives an overview of what Markov chains are all about and shows how he uses them to create pieces of music.

On this page, Eric describes how he got started using Markov chains to make music, along with several of his compositions. It’s fascinating how he captures the feel of a song by creating a mathematical system that “generates new patterns based on existing probabilities.”

Now there’s a big idea: exploring something randomly can capture structures that might be hard to perceive otherwise. Here’s one last variation on the Markov chain theme that involves a pure math question. This blog post ponders the question: what happens when a knight takes a random walk—or random trot?—on a chessboard? It includes some colorful images of chessboards along the way.

How likely it is that a knight lands on each square after five moves, starting from b1.

How likely it is that a knight lands on each square after five moves, starting from b1.

The probabili

How likely it is that a knight lands on each square after 200 moves, starting from b1.

The blogger—Leonid Kovalev—shows in his analysis what happens in the long run: the number of times a knight will visit a square will be proportional to the number of moves that lead to that square. For instance, since only two knight moves can reach a corner square while eight knight moves can reach a central square, it’s four times as likely that a knight will finish on a central square after a long, long journey than on a corner square. This idea works because moving a knight around a chessboard is a “reversible Markov chain”—any path that a knight can trace can also be untraced. The author also wrote a follow-up post about random queens.

It’s amazing the things you can find by chaining together ideas or by taking a random walk. Thanks for the inspiration for this post, Nico. Keep those comments and questions coming, everyone—we love hearing from you.

Bon appetit!

Solitons, Contours, and Thinking Sdrawkcab

Welcome to this week’s Math Munch!

Meet Nalini Joshi, a mathematician at the University of Sydney in Australia. I’ll let her introduce herself to you.

Nalini has an amazing story and amazing passion. What does her video make you think? To hear more from Nalini, you can watch this talk she gave last month at the Women in Mathematics conference at the Isaac Newton Institute in Cambridge, England. Her talk is called “Mathematics and life: a personal journey.” You might also enjoy reading this interview or others on her media page.

Nalini Joshi lecturing about solitons.

Nalini Joshi lecturing about solitons.

I’d like to share three clumps of ideas that might give you a flavor for the math that Nalini enjoys doing. Most of it is way over my head, but I’m reaching for it! You can, too, if you try.

Here’s clump number one. Two of the main objects that Nalini studies are dynamical systems and differential equations. You can think of a dynamical system as some objects that interact with each other and evolve over time. Think of the stars that Nalini described in the video, heading toward each other and tugging on each other. Differential equations are one way of describing these interactions in a mathematically precise way. They capture how tiny changes in one amount affect tiny changes in another amount.

Vlasov billiards.

Vlasov billiards.

To play around with some simple dynamical systems that can still produce some complex behaviors, check out dynamical-systems.org. Vlasov billiards was new to me. I think it’s really cool. The three-body problem is one of the oldest and most famous dynamical systems, and you can tinker around with examples of it here and here. There’s even a three-body problem game you can try playing. I’m not too crazy about it, but maybe you’ll enjoy it. It certainly gives you a sense for how chaotic the a three-body system can be!

Nalini doesn’t study just any old dynamical systems. She’s particularly interested in ones where the chaotic parts of the system cancel each other out. Remember in the video how she described the stars that go past each other and don’t destroy each other, that are “transparent to each other”? Places where this happens in dynamical systems are called soliton solutions. They’re like steady waves that can pass through each other. Check out these four videos on solitons, each of which gives a different perspective on them. If you’re feeling adventurous, you could try reading this article called What is a Soliton?


Making a water wave soliton in the Netherlands.


A computer animation of interacting solitons.


Japanese artist Takashi Suzuki tests a soliton to be used in a piece of performance art.


Students studying and building solitons in South Africa.

Level curves that are generalized Cassini curves. Also, kind of looks like a four-body problem. (click for video)

Level curves that are generalized Cassini curves.
Also, it kind of looks like a four-body problem.
(click for video)

The second idea that Nalini uses that I’d like to share is level curves, or contours. Instead of studying complicated differential equations directly, it’s possible to get at them geometrically by studying families of curves—contours—that are produced by related algebraic equations. They’re just like the lines on a topographic map that mark off areas of equal elevation.

Here’s a blog post by our friend Tim Chartier about colorful contour lines that arise from the differential equation governing heat flow. The temperature maps by Zachary Forest Johnson from a few weeks ago also used contour lines. And I found some great pieces of art that take contours as their inspiration. Click to check these out!

level_curves Utopia-70 Visual_Topography_of_a_Generation_Gap_Brooklyn_2

The last idea clump I’ll share involves integrable systems. In an integrable system, it’s possible to uniquely “undo” what has happened—the rules are such that there’s only one possible past that could lead to the present. Most systems don’t work this way—you can’t tell what was in your refrigerator a week ago by looking at it now! Nalini mentions on her research page that “ideas on integrable differential equations also extend to difference equations, and even to extended versions of cellular automata.” I enjoyed reading this article about reversible cellular automata, especially the section about Critters.

What move did Black just play? A puzzle by Raymond Smullyan.

What move did Black just play?
A puzzle by Raymond Smullyan.

And this made me think of a really nifty kind of chess puzzle called retrograde analysis—a fancy way of saying “thinking backwards”. Instead of trying to find the best chess move to play next, you instead have to figure out what move was made to get to the position in the puzzle. Most chess positions could be arrived at through multiple moves, but the positions in these puzzles are specially designed so that only one move will work. There’s a huge index of this kind of problem at The Retrograde Analysis Corner, and there are some great starter problems on this page.

Maurice Ashley

Maurice Ashley

And perhaps you’d like to hear a little bit about thinking backwards from one of the greatest teachers of chess, Grandmaster Maurice Ashley. Check out his TED video here.

I hope you’ve enjoyed finding out about Nalini Joshi and the mathematics that she loves. I asked Nalini if she would do a Q&A with us, and she said yes! Do you have a question you’d like to ask her? Send it to us below and we’ll include it in the interview, which I send to Nalini in about a week.

UPDATE: We’re no longer accepting questions for Nalini, because the interview has happened! Check it out!

Bon appetit!

A Closet Full of Puzzles, Sphereland, and Math Doodles

Welcome to this week’s Math Munch!

After a few weeks off, we’re back with some exciting things to share.  First up is Futility Closet, a blog featuring “an idler’s miscellany of compendious amusements.”  The blog is full of big-worded phrases like that, but I most love the puzzles they often post – everything from chess to numbers, codes, and devilish word play.  I also love that the name of the person who wrote each puzzle accompanies it.  Take a look at the few I’ve posted below and click here for the full list of puzzles.

Here’s a puzzle called Swine Wave, by Lewis Carroll. The puzzle: Lace 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one. Want to know the solution? Click on the image above to visit Futility Closet.
This puzzle is called Project Management, by Paul Vaderlind. The question: If a blacksmith requires five minutes to put on a horseshoe, can eight blacksmiths shoe 10 horses in less than half an hour? The catch: A horse can stand on three legs, but not on two. Click on the image to visit Futility Closet for the solution!

Next, have you ever wondered what it would be like to visit another dimension?   In 1884, Edwin A. Abbott wrote about life in the second dimension, in a nice little book called Flatland: A Romance of Many Dimesnions.  (Fun fact: the “A” in Edwin’s name stands for Abbott.  So his name is Edwin Abbott Abbott.)  Click on that link and you can read the whole book, if you like.  The book is about a world of flat beings who have no idea that the third dimension exists.  In the book, the main character, A Square, is visited by a sphere from the unknown world “above” him.  Kind of makes me wonder whether we’re just like the characters in Flatland, three-dimensional creatures ignorant of the fourth dimension that exists “above” us…

spherelandWell, the recently released movie Flatland 2: Sphereland deals with precisely that issue.  The Math Munch team had the opportunity to preview this movie, and we loved it.  In Sphereland, the granddaughter of the Square from Flatland, Hex, and her friend Puncto try to understand some mysterious triangles that Puncto thinks will cause the disastrous end of a space exploration mission and go on an adventure to help their three-dimensional friend Spherius with a problem he brought back from the fourth dimension.

portfolio-TorusHigher dimensions can be very difficult to wrap your head around.  This movie does a great job of helping the movie-watcher to understand how higher and lower dimensions relate to each other through the plot twists and challenges that the characters face.  You can really learn a lot about dimensions and the shape of space by watching this movie.  Plus, the characters are engaging and the images are fun.  Sphereland features the voices of a number of really great actors, including Kristen Bell, Danny Pudi, Michael York, and Danica McKellar.

Want to learn more about Sphereland?  Check out the trailer:

And, here’s an interview with Danny Pudi, the voice of Puncto, and Tony Hale, who does a fantastic job as the King of Pointland:

By the way, the makers of Sphereland also made a movie of Flatland!  The Math Munch team loved that one, too.  Here’s a link to the trailer.

tumblr_mgw2ainZDX1s0payeo1_1280Finally, check out this beautiful blog of mathematical doodles by high school math student and artist Chloé Worthington!  Chloé started mathematically doodling a few years ago in… well, in class.  When she doodles in class, Chloé is better able to focus on what’s going on and makes beautiful art.   (We at Math Munch encourage you to pay attention in class while you doodle.)

Chloé does all of her doodles by hand with ink pens.  She does a lot of work with triangles, as shown here.  One of her signature doodles is this nested puzzle piece doodle:


Doodling mathematically is one of the ways that Chloé does math and shares what she loves about it with the world.  She’s a trigonometry student, too.  How do you share what you love about math – or any other subject?

Bon appetit!

Rectangles, Explosions, and Surreals

Welcome to this week’s Math Munch!

What is 3 x 4?   3 x 4 is 12.

Well, yes. That’s true. But something that’s wonderful about mathematics is that seemingly simple objects and problems can contain immense and surprising wonders.

How many squares can you find in this diagram?

As I’ve mentioned before, the part of mathematics that works on counting problems is called combinatorics. Here are a few examples for you to chew on: How many ways can you scramble up the letters of SILENT? (LISTEN?) How many ways can you place two rooks on a chessboard so that they don’t attack each other? And how many squares can you count in a 3×4 grid?

Here’s one combinatorics problem that I ran across a while ago that results in some wonderful images. Instead of asking about squares in a 3×4 grid, a team at the Dubberly Design Office in San Francisco investigated the question: how many of ways can a 3×4 grid can be partitioned—or broken up—into rectangles? Here are a few examples:

How many different ways to do this do you think there are? Here’s the poster that they designed to show the answer that they found! You can also check out this video of their solution.

In their explanation of their project, the team states that “Design tools are becoming more computation-based; designers are working more closely with programmers; and designers are taking up programming.” Designing the layout of a magazine or website requires both structural and creative thinking. It’s useful to have an idea of what all the possible layouts are so that you can pick just the right one—and math can help you to do it!

If you’d like to try creating a few 3×4 rectangle partitions of your own, you can check out www.3x4grid.com. [Sadly, this page no longer works. See an archive of it here. -JL, 10/2016]

Next up, explosions! I could tell you about the math of the game Minesweeper (you can play it here), or about exploding dice. But the kind of explosion I want to share with you today is what’s called a “combinatorial explosion.” Sometimes a problem that appears to be an only slightly harder variation of an easy problem turns out to be way, way harder. Just how BIG and complicated even simple combinatorics problems can get is the subject of this compelling and also somewhat haunting video.

Donald Knuth

Finally, all of this counting got me thinking about big numbers. Previously we’ve linked to Math Cats, and Wendy has a page where you can learn how to say some really big numbers. But thinking about counting also made me remember an experience I had in middle school where I found out just how big numbers could be! I was in seventh grade when I read this article from the December 1995 issue of Discover Magazine. It’s called “Infinity Plus One, and Other Surreal Numbers” and was written by Polly Shulman. I remember my mind being blown by all of the talk of infinitely-spined aliens and up-arrow notation for naming numbers. Here’s an excerpt:

Mathematicians and precocious five-year-olds have long been fascinated by the endlessness of numbers, and they’ve named the endlessness infinity. Infinity isn’t a number like 1, 2, or 3; it’s hard to say what it is, exactly. It’s even harder to imagine what would happen if you tried to manipulate it using the arithmetic operations that work on numbers. For example, what if you divide it in half? What if you multiply it by 2? Is 1 plus infinity greater than, less than, or the same size as infinity plus 1? What happens if you subtract 1 from it?

After I read this article, John Conway and Donald Knuth became heros of mine. (In college, I had the amazing fortune to have breakfast with Conway one day when he was visiting to give a lecture!) Knuth has a book about surreals that’s the friendliest introduction to the surreal numbers that I know of, and in this video, Vi Hart briefly touches on surreal numbers in discussing proofs that .9 = 1. Boy, would I love to see a great video or online resource that simply and beautifully lays out the surreal numbers in all their glory!

It was fun for me to remember that Discover article. I hope that you, too, run across some mathematics that leaves a seventeen-year impression on you!

Bon appetit!

Pentomino Puzzles, Knight’s Tours, and Decimal Maxing

Welcome to this week’s Math Munch!

Have a pentomino tiling problem that’s got you stumped?  Then perhaps the Pentominos Puzzle Solver will be right up your alley! Recently I’ve been thinking a lot about using computer programming and search algorithms to solve mathematical problems, and the Pentomino Puzzle Solver is a great example of the power of coding.  Written by David Eck, a professor of math and computer science at Brandeis University, the solver can find tilings of a variety of shapes.  Watch the application in slow-mo to see how it works; put it into high-gear to see the power of doing mathematics with computers!

Next, here’s a wonderful page about knight’s tours maintained by George Jelliss, a retiree from the UK.  He says on his introductory page, “I have been interested in questions related to the geometry of the knight’s move since the early 1970s.” George has investigated “leapers” or “generalized knights”—pieces that move in other L-shapes than the traditional 2×1—and he even published his own chess puzzle magazine for a number of years.  His webpage includes a great section about the history of knights tours, and I’m a fan of the beautiful catalog of “crosspatch” tours. Great stuff!

Multiplication, addition, division: which gives the biggest result?

Last but not “least”, to the left you’ll find a tiny chunk of a very large table that was constructed and colored by Debra Borkovitz, a math professor at Wheelock College.  Debra describes how, “Students often have poor number sense about multiplication and division with numbers less than one.”  She created an investigation where students decide, for any pair of decimals, which is biggest–multiplying them, adding them, or subtracting them.  For 1.0 and 1.0 the answer is easy–you should add them, so that you get 2.  .5 and 1 is trickier–adding yields 1.5, multiplying gives .5, but dividing 1 by .5 makes 2, since there are two halves in 1. Finding the biggest value possible given some restrictions is called “maximization” in mathematics, and it’s a very popular type of problem with many applications.

This investigation about makes me wonder: what other kinds of tables could I try to make?

Debra mentions that she got the inspiration for this problem from a newsletter put out by the Association of Women in Mathematics.  There’s lots to explore on their website, including an essay contest for middle schoolers, high schoolers, and undergraduates.

I hope you found something here to enjoy.  Bon appetit!

Pennies, Knights, and Origami Mazes

Welcome to this week’s Math Munch!

How many pennies do you think this is? Click to find out.

Big numbers are sometimes hard to get a feel for.  A billion is a lot, but so is a million.  The MegaPenny Project is a cool attempt at making the difference between large numbers easier to grasp.  Would 1,000,000 pennies fill a football field or would you need a billion pennies for that?  MegaPenny can help you figure it out.

The first kixote puzzle

Next up, we have kixote, a puzzle in the spirit of Sudoku and Ken-Ken, but involving knight’s moves.  Dan Mackinnon–its creator–has a blog called mathrecreation that he says, “helps me go a little further in my mathematical recreations, helps me understand things better, and sometimes connects me to other people who share similar interests. I hope that it might encourage you to play with math too.”  I’m sure we’ll be linking to more of Dan’s posts in the future!

Finally, since the mazes and paper-folding were so popular last week, we thought that this week we would share some paper-folding mazes! Here is a clip of MIT professor Erik Demaine talking about how he has created origami mazes, preceded by a discussion of origami robots that fold themselves!  The clip is a part of a lecture about origami that Erik gave last spring in New York City for the Math Encounters series put on by the Museum of Mathematics.  You can watch Erik’s entire origami lecture from the beginning by clicking here.

frame from lecture video

Eric Demaine with a sheet of origami cubes

You can also check out Erik’s Maze Folder applet–but if you try it out, take his warning and start with a small maze!

Bon appetit!