# George Washington, Tessellation Kit, and Langton’s Ant

Welcome to this week’s Math Munch!

What will you do with your math notebook at the end of the school year? Keep it as a reference for the future? Save it as a keepsake? Toss it out? Turn it into confetti? Find your favorite math bits and doodles and make a collage?

Lucky for us, our first president kept his math notebooks from when he was a young teenager. And though it’s passed through many hands over the years—including those of Chief Justice John Marshall and the State Department—it has survived to this day. That’s right. You can check out math problems and definitions copied out by George Washington over 250 years ago. They’re all available online at the Library of Congress website.

Or at least most of them. They seem to be out of order, with a few pages missing!

That’s what mathematician and math history detective Fred Rickey has figured out. Fred has long been a fan of math history. Since he retired from the US Military Academy in 2011, Fred has been able to pursue his historical interests more actively. Fred is currently studying the Washington cypher books to help prepare a biography about Washington’s boyhood years. You can see two papers that Fred has co-authored about Washington’s mathematics here.

Fred writes:

Washington valued his cyphering books and kept them as a ready source of reference for the rest of his life. This would seem to be particularly true of his surveying studies.

Surveying played a big role in Washington’s career, and math is important for today’s surveyors, too.

Do you have a question for Fred about the math that George Washington learned? Send it to us and we’ll try to include it in our upcoming Q&A with Fred!

A tessellation, by me!

Next up, check out this Tessellation Kit. It was made by Nico Disseldorp, who also made the geometry construction game we featured recently. The kit is a lot of fun to play with!

One thing I like about this Tessellation Kit is how it’s discrete—it deals with large chunks of the screen at a time. This restriction make me want to explore, because it give me the feeling that there are only so many possible combinations.

I’m also curious about the URL for this applet—the web address for it. Notice how it changes whenever you make a change in your tessellation? What happens when you change some of those letters and numbers—like bababaaaa to bababcccc? Interesting…

For another fun applet, check out this doodling ant:

Langton’s Ant.

Langton’s Ant is following a simple set of rules. In a white square? Turn right. In a black square? Turn left. And switch the color of the square that you leave. This ant is an example of a cellular automaton, and we’ve seen several of these here on Math Munch before. This one is different from others because it changes just one square at a time, and not the whole screen at once.

Breaking out of chaos.

There’s a lot that is unknown about Langton’s ant, and it has some mysterious behavior. For example, after thousands of steps of seeming randomness, the ant goes into a steady pattern, paving a highway out to infinity. What gives? Well, you can try out some patterns of your own in the applets on the Serendip website. (previously). And you can read some amusing tales—ant-ecdotes?—about Langton’s ant in this lovely article.

I learned about Langton’s Ant from Richard Evan Schwartz in our new Q&A. In the interview, Rich shares his thoughts about computers, art, what to pursue in life, and of course: Really Big Numbers.

Check it out, and bon appetit!

# Numenko, Turning Square, and Toilet Paper

Welcome to this week’s Math Munch!

Have you ever played Scrabble or Bananagrams? Can you imagine versions of these games that would use numbers instead of letters?

Meet Tom Lennett, who imagined them and then made them!

Tom playing Numenko with his grandkids.

Numemko is a crossnumber game. Players build up number sentences, like 4×3+8=20, that cross each other like in a crossword puzzle. There is both a board game version of Numenko (like Scrabble) and a bag game version (like Banagrams). Tom invented the board game years ago to help his daughter get over her fear of math. He more recently invented the bag game for his grandkids because they wanted a game to play where they didn’t have to wait their turn!

The Multichoice tile.

One important feature of Numenko is the Multichoice tile. Can you see how it can represent addition, subtraction, multiplication, division, or equality?

How would you like to have a Numenko set of your own? Well, guess what—Tom holds weekly Numenko puzzle competitions with prizes! You can see the current puzzle on this page, as well as the rules. Here’s the puzzle at the time of this post—the week of November 3, 2013.

Challenge: replace the Multichoice tiles to create a true number sentence.

I can assure you that it’s possible to win Tom’s competitions, because one of my students and I won Competition 3! I played my first games of Numenko today and really enjoyed them. I also tried making some Numenko puzzles of my own; see the sheet at the bottom of this post to see some of them.

Tom in 1972.

In emailing with Tom I’ve found that he’s had a really interesting life. He grew up in Scotland and left school before he turned 15. He’s been a football-stitcher, a barber, a soldier, a distribution manager, a paintball site operator, a horticulturist, a property developer, and more. And, of course, also a game developer!

Do you have a question you’d like to ask Tom? Send it in through the form below, and we’ll try to include it in our upcoming Q&A!

The level editor.

Say, do you like Bloxorz? I sure do—it’s one of my favorite games! So imagine my delight when I discovered that a fan of the game—who goes by the handle Jz Pan—created an extension of it where you can make your own levels. Awesome, right? It’s called Turning Square, and you can download it here.

(You’ll need to uncompress the file after downloading, then open TurningSquare.exe. This is a little more involved than what’s usual here on Math Munch, but I promise it’s worth it! Also, Turning Square has only been developed for PC. Sorry, Mac fans.)

But wait, there’s more! Turning Square also introduces new elements to Bloxorz, like slippery ice and pyramids you can trip over. It has a random level generator that can challenge you with different levels of difficulty. Finally, Turning Square includes a level solver—it can determine whether a level that you create is possible or not and how many steps it takes to complete.

Jz Pan is from China and is now a graduate student at the Chinese Academy of Sciences, majoring in mathematics and studying number theory. Jz Pan made Turning Square in high school, back in 2008.

Jz Pan has agreed to answer some of your questions! Use the form below to send us some.

If you make a level in Turning Square that you really like, email us the .box file and we can share it with everyone through our new Readers’ Gallery! Here is my level from above, if you want to try it out.

Jz Pan has also worked on an even more ambitious extension of Bloxorz called Turning Polyhedron. The goal is the same, but like the game Dublox, the shape that you maneuver around is different. Turning Polyhderon features several different shapes. Check out this video of it being played with a u-polyhedron!

And if you think that’s wild, check out this video with multiple moving blocks!

Last up this week, have you ever heard that it’s impossible to fold a piece of paper in half more than eight times? Or maybe it’s seven…? Either way, it’s a “fact” that seems to be common knowledge, and it sure seems like it’s true when you try to fold up a standard sheet of paper—or even a jumbo sheet of paper. The stack sure gets thick quickly!

Britney and her 11th fold.

Well, here’s a great story about a teenager who decided to debunk this “fact” with the help of some math and some VERY big rolls of toilet paper. Her name is Britney Gallivan. Back in 2001, when she was a junior in high school, Britney figured out a formula for how much paper she’d need in order to fold it in half twelve times. Then she got that amount of paper and actually did it!

Due to her work, Britney has a citation in MathWorld’s article on folding and even her own Wikipedia article. After high school, Britney went on to UC Berkeley where she majored in Environmental Science. I’m trying to get in touch with Britney for an interview—if you have a question for her, hold onto it, and I’ll keep you posted!

EDIT: I got in touch with Britney, and she’s going to do an interview!

A diagram that illustrates how Britney derived her equation.

The best place to read more about Britney’s story in this article at pomonahistorical.org—the historical website of Britney’s hometown. Britney’s story shows that even when everyone else says that something’s impossible, that doesn’t mean you can’t be the one to do it. Awesome.

I hope you enjoy trying some Numenko puzzles, tinkering with Turning Square, and reading about Britney’s toilet paper adventure.

Bon appetit!

PS Want to see a video of some toilet-paper folding? Check out the very first “family math” video by Mike Lawler and his kids.

Reflection Sheet – Numenko, Turning Square, and Toilet Paper

# 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:

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?

Finally, 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!

# Prime Gaps, Mad Maths, and Castles

Welcome to this week’s Math Munch!

It has been a thrilling last month in the world of mathematics. Several new proofs about number patterns have been announced. Just to get a flavor for what it’s all about, here are some examples.

I can make 15 by adding together three prime numbers: 3+5+7. I can do this with 49, too: 7+11+31. Can all odd numbers be written as three prime numbers added together? The Weak Goldbach Conjecture says that they can, as long as they’re bigger than five. (video)

11 and 13 are primes that are only two apart. So are 107 and 109. Can we find infinitely many such prime pairs? That’s called the Twin Prime Conjecture. And if we can’t, are there infinitely many prime pairs that are at most, say, 100 apart? (video, with a song!)

 Harald Helfgott Yitang “Tom” Zhang

People have been wondering about these questions for hundreds of years. Last month, Harald Helfgott showed that the Weak Goldbach Conjecture is true! And Yitang “Tom” Zhang showed that there are infinitely many prime pairs that are at most 70,000,000 apart! You can find lots of details about these discoveries and links to even more in this roundup by Evelyn Lamb.

What’s been particularly fabulous about Tom’s result about gaps between primes is that other mathematicians have started to work together to make it even better. Tom originally showed that there are an infinite number of prime pairs that are at most 70,000,000 apart. Not nearly as cute as being just two apart—but as has been remarked, 70,000,000 is a lot closer to two than it is to infinity! That gap of 70,000,000 has slowly been getting smaller as mathematicians have made improvements to Tom’s argument. You can see the results of their efforts on the polymath project. As of this writing, they’ve got the gap size narrowed down to 12,006—you can track the decreasing values down the page in the H column. So there are infinitely many pairs of primes that are at most 12,006 apart! What amazing progress!

Two names that you’ll see in the list of contributors to the effort are Andrew Sutherland and Scott Morrison. Andrew is a computational number theorist at MIT and Scott has done research in knot theory and is at the Australian National University. They’ve improved arguments and sharpened figures to lower the prime gap value H. They’ve contributed by doing things like using a hybrid Schinzel/greedy (or “greedy-greedy”) sieve. Well, I know what a sieve is and what a greedy algorithm is, but believe me, this is very complicated stuff that’s way over my head. Even so, I love getting to watch the way that these mathematicians bounce ideas off each other, like on this thread.

 Andrew Sutherland Andrew. Click this! Scott Morrison

Andrew and Scott have agreed to answer some of your questions about their involvement in this research about prime gaps and their lives as mathematicians. I know I have some questions I’m curious about! You can submit your questions in the form below:

I can think of only two times in my life where I was so captivated by mathematics in the making as I am by this prime gaps adventure. Andrew Wiles’s proof of Fermat’s Last Theorem was on the fringe of my awareness when it came out in 1993—its twentieth anniversary of his proof just happened, in fact. The result still felt very new and exciting when I read Fermat’s Enigma a couple of years later. Grigori Perelman’s proof of the Poincare Conjecture made headlines just after I moved to New York City seven years ago. I still remember reading a big article about it in the New York Times, complete with a picture of a rabbit with a grid on it.

This work on prime gaps is even more exciting to me than those, I think. Maybe it’s partly because I have more mathematical experience now, but I think it’s mostly because lots of people are helping the story to unfold and we can watch it happen!

Next up, I ran across a great site the other week when I was researching the idea of a “cut and slide” process. The site is called Mad Maths and the page I landed on was all about beautiful dissections of simple shapes, like circles and squares. I’ve picked out one that I find especially charming to feature here, but you might enjoy seeing them all. The site also contains all kinds of neat puzzles and problems to try out. I’m always a fan of congruent pieces problems, and these paper-folding puzzles are really tricky and original. (Or maybe, origaminal!) You’ll might especially like them if you liked Folds.

Christian’s applet displaying the original four-room castle.

Finally, we previously posted about Matt Parker’s great video problem about a princess hiding in a castle. Well, Christian Perfect of The Aperiodical has created an applet that will allow you to explore this problem—plus, it’ll let you build and try out other castles for the princess to hide in. Super cool! Will I ever be able to find the princess in this crazy star castle I designed?!

My crazy star castle!

And as summer gets into full swing, the other kind of castle that’s on my mind is the sandcastle. Take a peek at these photos of geometric sandcastles by Calvin Seibert. What shapes can you find? Maybe Calvin’s creations will inspire your next beach creation!

Bon appetit!

# 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!

# Sandpiles, Prime Pages, and Six Dimensions of Color

Welcome to this week’s Math Munch!

Four million grains of sand dropped onto an infinite grid. The colors represent how many grains are at each vertex. From this gallery.

We got our first snowfall of the year this past week, but my most recent mathematical find makes me think of summertime instead. The picture to the right is of a sandpile—or, more formally, an Abelian sandpile model.

If you pour a bucket of sand into a pile a little at a time, it’ll build up for a while. But if it gets too tall, an avalanche will happen and some of the sand will tumble away from the peak. You can check out an applet that models this kind of sand action here.

A mathematical sandpile formalizes this idea. First, take any graph—a small one, a medium sided one, or an infinite grid. Grains of sand will go at each vertex, but we’ll set a maximum amount that each one can contain—the number of edges that connect to the vertex. (Notice that this is four for every vertex of an infinite square grid). If too many grains end up on a given vertex, then one grain avalanches down each edge to a neighboring vertex. This might be the end of the story, but it’s possible that a chain reaction will occur—that the extra grain at a neighboring vertex might cause it to spill over, and so on. For many more technical details, you might check out this article from the AMS Notices.

This video walks through the steps of a sandpile slowly, and it shows with numbers how many grains are in each spot.

A sandpile I made with Sergei’s applet

You can make some really cool images—both still and animated—by tinkering around with sandpiles. Sergei Maslov, who works at Brookhaven National Laboratory in New York, has a great applet on his website where you can make sandpiles of your own.

David Perkinson, a professor at Reed College, maintains a whole website about sandpiles. It contains a gallery of sandpile images and a more advanced sandpile applet.

Hexplode is a game based on sandpiles.

I have a feeling that you might also enjoy playing the sandpile-inspired game Hexplode!

Next up: we’ve shared links about Fibonnaci numbers and prime numbers before—they’re some of our favorite numbers! Here’s an amazing fact that I just found out this week. Some Fibonnaci numbers are prime—like 3, 5, and 13—but no one knows if there are infinitely many Fibonnaci primes, or only finitely many.

A great place to find out more amazing and fun facts like this one is at The Prime Pages. It has a list of the largest known prime numbers, as well as information about the continuing search for bigger ones—and how you can help out! It also has a short list of open questions about prime numbers, including Goldbach’s conjecture.

Be sure to peek at the “Prime Curios” page. It contains intriguing facts about prime numbers both large and small. For instance, did you know that 773 is both the only three-digit iccanobiF prime and the largest three-digit unholey prime? I sure didn’t.

Last but not least, I ran across this article about how a software company has come up with a new solution for mixing colors on a computer screen by using six dimensions rather than the usual three.

The arithmetic of colors!

Well, there are actually several ways that computers store colors. Each of them encodes colors using three numbers. For instance, one method builds colors by giving one number each to the primary colors yellow, red, and blue. Another systems assigns a number to each of hue, saturation, and brightness. More on these systems here. In any of these systems, you can picture a given color as sitting within a three-dimensional color cube, based on its three numbers.

A color cube, based on the RGB (red, green, blue) system.

If you numerically average two colors in these systems, you don’t actually end up with the color that you’d get by mixing paint of those two colors. Now, both scientists and artists think about combining colors in two ways—combining colored lights and combining colored pigments, or paints. These are called additive and subtractive color models—more on that here. The breakthrough that the folks at the software company FiftyThree made was to assign six numbers to each color—that is, to use both additive and subtractive ideas at the same time. The six numbers assigned to a given number can be thought of as plotting a point in a six-dimensional space—or inside of a hyper-hyper-hypercube.

I think it’s amazing that using math in this creative way helps to solve a nagging artistic problem. To get a feel for why mixing colors using the usual three-coordinate system is such a problem, you might try your hand at this color matching game. For even more info about the math of color, there’s some interesting stuff on this webpage.

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.

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!