Monthly Archives: October 2012

Pentago, Geometry Daily, and The OEIS

Welcome to this week’s Math Munch!

Pentago Board

Hurricane Sandy is currently slamming the East coast, but the Math Munch Team is safe and sound, so the math must go on.  First up, if you’ve visited our games page lately, you may have noticed a recent addition.  Pentago is a 2-player strategy with simple rules and an enticing twist.

  • Rules: Take turns playing stones.  The first person to get 5 in a row wins.  (5 is the “pent” part.)
  • Twist: After you place a stone you must spin one of the 4 blocks.  This makes things very interesting.

Why don’t you play a few games before you read on?  You can play the computer on their website, play with a friend by email, or download the Pentago iPhone app.  But if you’re ready, let’s dig into some Pentago strategy and analysis.

Mindtwister CEO, Monica Lucas

Mindtwister (the company that sells Pentago) put out a free strategy guide that names 4 different kinds of winning lines and rates their relative strengths.  The weakest strategy is called Monica’s Five, and it’s named after Mindtwister CEO and Pentago lover, Monica Lucas.  You can read our Q&A for more expert game strategies and insights.  We also had a chance to speak with Tomas Floden, the inventor of Pentago, so it’s a double Q&A week.

As you play, you start to build your own strategy guide, so let me share three basic rules from mine.  I call them the first 3 Pentago Theorems.  (A theorem is a proven math fact.)

  1. If you have a move to win, take it!  This one is obvious, but you’ll see why I include it.
  2. If your opponent is only missing one stone from a line of 5 you must play there.  It seems like you could play somewhere else and spin the line apart, but your opponent can play the stone and spin back!  The only exception to this rule is rule 1.  If you can win, just do that!
  3. 4 in a row, with both ends open will (almost always) win.  This is a classic double trap.  Either end will finish the winning line, so by rule 2 both must be filled, but this is impossible.  The exceptions of course will come when your opponent is able to win right away, so you still have to pay close attention.

Up next, check out the beautiful math art of Geometry Daily.

#288 Fundamental

#132 Eight Squares

#259 Dudeney’s Dissection

#296 Downpour

#236 Nova

#124 Cuboctahedron

#136 Tesseract

#26 Pentaflower

#92 Circular Spring

The site is the playground for the geometrical ideas of Tilman Zitzmann, a German designer and teacher, who’s been creating a new image every day for almost a year now!  He also took some time to write about his creative process, so if you’re interested, have a read.  Visit the Geometry Daily archives to view all the images.

Finally, an amazing resource – the On-Line Encyclopedia of Integer Sequences.  What’s the pattern here?  1, 3, 6, 10, 15, 21, …  Any idea?  Do you know what the 50th number would be?  Well if you type this sequence into the OEIS, it’ll tell you every known sequence that matches.  Here’s what you get in this case.  These are the “triangular numbers,” also the number of edges in a complete graph.  It also tells you formula for the sequence:

  • a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+…+n.

If you make n=1, then you get 1.  If n=2, then you get 3.  If n=5, you get the 5th number, so to get the 50th number in the sequence, we just make n=50 in the formula.  n(n+1)/2 becomes 50(50+1)/2 = 1275.  Nifty.  Who’s got a pattern that needs investigating?

Have a great week, and bon appetit!

Pixel Art, Gothic Circle Patterns, and First Past the Post

Welcome to this week’s Math Munch!

Guess what? Today is Math Munch’s one-year anniversary!

We’re so grateful to everyone who has made this year so much fun: our students and readers; everyone who has spread the word about Math Munch; and especially all the people who do and make the cool mathy things that we so love to find and share.

Speaking of which…

Mathematicians have studied the popular puzzle called Sudoku in numerous ways. They’ve counted the number of solutions. They’ve investigated how few given numbers are required to force a unique solution. But Tiffany C. Inglis came at this puzzle craze from another angle—as a way to encode pixel art!

Tiffany studies computer graphics at the University of Waterloo in Ontario, Canada. She’s a PhD candidate at the Computer Graphics Lab (which seems like an amazing place to work and study—would you check out these mazes!?)

Tiffany C. Inglis, hoisting a buckyball

Tiffany tried to find shading schemes for Sudoku puzzles so that pictures would emerge—like the classic mushroom pictured above. Sudoku puzzles are a pretty restrictive structure, but Tiffany and her collaborators had some success—and even more when they loosened the rules a bit. You can read about (and see!) some of their results on this rad poster and in their paper.

Thinking about making pictures with Sudoku puzzles got Tiffany interested in pixel art more generally. “I did some research on how to create pixel art from generic images such as photographs and realized that it’s an unexplored area of research, which was very exciting!” Soon she started building computer programs—algorithms—to automatically convert smooth line art into blockier pixel art without losing the flavor of the original. You can read more about Tiffany’s pixelization research on this page of her website. You should definitely check out another incredible poster Tiffany made about this research!

To read more of my interview with Tiffany, you can click here.

Cartoon Tiffany explains what makes a good pixelization. Check out the full comic!

I met Tiffany this past summer at Bridges, where she both exhibited her artwork and gave an awesome talk about circle patterns in Gothic architecture. You may be familiar with Apollonian gaskets; Gothic circle patterns have a similar circle-packing feel to them, but they have some different restrictions. Circles don’t just squeeze in one at a time, but come in rings. It’s especially nice when all of the tangencies—the places where the circles touch—coincide throughout the different layers of the pattern. Tiffany worked on the problem of when this happens and discovered that only a small family has this property. Even so, the less regular circle patterns can still produce pleasing effects. She wrote about this and more in her paper on Gothic circle patterns.

I’m really inspired by how Tiffany finds new ideas in so many place, and how she pursues them and then shares them in amazing ways. I hope you’re inspired, too!

A rose window at the Milan Cathedral, with circle designs highlighted.

A mathematical model similar to the window, which Tiffany created.

An original design by Tiffany. All of these images are from her paper.

Here’s another of Tiffany’s designs. Now try making one of your own!

Using the Mathematica code that Tiffany wrote to build her diagrams, I made an applet where you can try making some circle designs of your own. Check it out! If you make one you really like—and maybe color it in—we’d love to see it! You can send it to us at

(You’ll may have to download a plug-in to view the applet; it’s the same plug-in required to use the Wolfram Demonstrations Project.)

Finally, with Election Day right around the corner, how about a dose of the mathematics of voting?

I’m a fan of this series of videos about voting theory by C.G.P. Grey. Who could resist the charm of learning about the alternative vote from a wallaby, or about gerrymandering from a weasel? Below you’ll find the first video in his series, entitled “The Problems with First Past the Post Voting Explained.” Majority rule isn’t as simple of a concept as you might think, and math can help to explain why. As can jungle animals, of course.

Thanks again for being a part of our Math Munch fun this past year. Here’s to a great second course! Bon appetit!

PS I linked to a bunch of papers in this post. After all, that’s the traditional first anniversary gift!

Harmonious Sum, Continuous Life, and Pumpkins

Welcome to this week’s Math Munch!

We’ve posted a lot about pi on Math Munch – because it’s such a mathematically fascinating little number.  But here’s something remarkable about pi that we haven’t yet talked about. Did you know that pi is equal to four times this? Yup.  If you were to add and subtract fractions like this, for ever and ever, you’d get pi divided by 4.  This remarkable fact was uncovered by the great mathematician Gottfried Wilhelm Leibniz, who is most famous for developing the calculus.  Check out this interactive demonstration from the Wolfram Demonstrations Project to see how adding more and more terms moves the sum closer to pi divided by four.  (We’ve written about Wolfram before.)

I think this is amazing for a couple of reasons.  First of all, how can an infinite number of numbers add together to make something that isn’t infinite???  Infinitely long sums, or series, that add to a finite number have a special name in mathematics: convergent series.  Another famous convergent series is this one:

The second reason why I think this sum is amazing is that it adds to pi divided by four.  Pi is an irrational number – meaning it cannot be written as a fraction, with whole numbers in the numerator and denominator.  And yet, it’s the sum of an infinite number of rational numbers.

In this video, mathematician Keith Devlin talks about this amazing series and a group of mathematical musicians (or mathemusicians) puts the mathematics to music.

This video is part of a larger work called Harmonious Equations written by Keith and the vocal group Zambra.  Watch the rest of them, if you have the chance – they’re both interesting and beautiful.

Next up, Conway’s Game of Life is a cellular automaton created by mathematician John Conway.  (It’s pretty fun: check out this to download the game, and this Munch where we introduce it.)  It’s discrete – each little unit of life is represented by a tiny square.  What if the rules that determine whether a new cell is formed or the cell dies were applied to a continuous domain?  Then, it would look like this:

Looks like a bunch of cells under a microscope, doesn’t it?  Well, it’s also a cellular automaton, devised by mathematician Stephan Rafler from Nurnberg, Germany.  In this paper, Stephan describes the mathematics behind the model.  If you’re curious about how it works, check out these slides that compare the new continuous version to Conway’s model.

Finally, I just got a pumpkin.  What should I carve in it?  I spent some time browsing the web for great mathematical pumpkin carvings.  Here’s what I found.

A pumpkin carved with a portion of Escher’s Circle Limit.

A pumpkin tiled with a portion of Penrose tiling.

A dodecapumpkin from Vi Hart.

I’d love to hear any suggestions you have for how I should make my own mathematical pumpkin carving!  And, if you carve a pumpkin in a cool math-y way, send a picture over to!

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!

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 [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!