Author Archives: Anna Weltman

Combinatorial Games, Redistricting Game, and Graph Music

Welcome to this week’s Math Munch!

Have you ever played tic-tac-toe? If so, maybe you’ve noticed that unless you or your opponent makes a bad move, the game always ends in a tie! (Oops– spoiler alert!) Why is that? And what makes tic-tac-toe different from other games that have unpredictable outcomes, like Monopoly or the card game War?


We wrote about tic-tac-toe in this post! Click to learn more.

Tic-tac-toe is similar to other kinds of game that mathematicians call combinatorial games— or games where there is no chance involved in the outcome and neither player has information that the other one doesn’t. This means that depending on who starts, where they go, and where each player decides to go next, the outcome is completely predictable and everyone playing could know what it is before it even happens. No surprises!

Now, this might also sound like NO FUN to you (why play the game at all if everyone knows what’s going to happen?) but I think it introduces a new kind of fun– figuring out what the outcomes could be! One of my favorite combinatorial games is the game NIM.


Here’s an example of a starting NIM board. If you go first, can you win? (Assuming your opponent never makes a mistake.)

NIM is a two-player game. You start with several piles or rows of objects (here they’re matches). On each turn, a player removes some objects from a pile– any number they want. BUT the player who’s forced to remove the last match loses!


There’s no chance in NIM– no dice determining how many matches you can remove, for example. Also both players know the rules and how many matches are in the piles at all times. That means that if you thought about it for a while, you could figure out who should win or lose any game of NIM. Maybe playing the game NIM isn’t super fun– but thinking about it like a puzzle is!

More versions of online NIM can be found here and here. And to read about combinatorial games we’ve written about in the past, check out this interview with mathematician Elwyn Berlekamp!

Next up, it’s presidential election time here again in the U.S.! Did you know that there’s a lot of mathematics behind what makes elections work? Four years ago, before the last presidential election, we shared a great series of YouTube videos about the math of elections.

redistricting game

A map in the redistricting game.

A big way that math gets involved in elections is through how politicians decide to draw districts, or regions of states that get to elect their own representative to the House of Representatives and elector to the Electoral College. The math behind drawing districts ranges from simple arithmetic to graph theory, or the field of math that deals with how parts of a shape or diagram are connected. To learn more about drawing election districts and the math behind it, check out the Re-Districting Game! In this game, you play the part of a map maker who works with the Congress, governor of your state, and courts to make a district map that meets everyone’s needs.

Finally, I recently ran across a series of graph music videos! What’s that? Videos in which a graph (made on Desmos) dances along to music, much like people would in a regular music video. Here’s one of my favorites:

The equations on the left-hand side of the screen create the images you see and the rhythm of the animation. Want to make your own graph music video? Share it with us!

Bon appetit!

Halving-Fun, Self-Tiling Tile Sets, and Doodal

This post comes to us all the way from June of 2014! Enjoy this blast from the past!

Welcome to this week’s Math Munch!

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Our first bit of fun comes from a blog called Futility Closet (previously featured). It’s a neat little cut-and-fold puzzle. The shape to the right can be folded up to make a solid with 5 sides. Two of them can be combined to make a solid with only 4 sides, the regular tetrahedron. If you’d like, you can use our printable version, which has two copies on one sheet.

What do you know, I also found our second item on Futility Closet! Check out the cool family of tiles below. What do you notice?

A family of self-tiling tiles

A family of self-tiling tiles

Did you notice that the four shapes in the middle are the same as the four larger shapes on the outside? The four tiles in the middle can combine to create larger versions of themselves! They can make any and all of the original four!!

Lee Sallows

Recreational Mathematician, Lee Sallows

Naturally, I was reminded of the geomagic squares we featured a while back (more at, and then I came to realize they were designed by the same person, the incredible Lee Sallows! (For another amazing one of Lee Sallows creations, give this incredible sentence a read.) You can also visit his website,


A family of 6 self-tiling tiles

For more self-tiling tiles (and there are many more amazing sets) click here. I have to point out one more in particular. It’s like a geomagic square, but not quite. It’s just wonderful. Maybe it ought to be called a “self-tiling latin square.”

And for a final item this week, we have a powerful drawing tool. It’s a website that reminds me a lot of recursive drawing, but it’s got a different feel and some excellent features. It’s called Doodal. Basically, whatever you draw inside of the big orange frame will be copied into the blue frames.  So if there’s a blue frame inside of an orange frame, that blue frame gets copied inside of itself… and then that copy gets copied… and then that copy…!!!

To start, why don’t you check out this amazing video showing off some examples of what you can create. They go fast, so it’s not really a tutorial, but it made me want to figure more things out about the program.

I like to use the “delete frame” button to start off with just one frame. It’s easier for me to understand if its simpler. You can also find instructions on the bottom. Oh, and try using the shift key when you move the blue frames. If you make something you like, save it, email it to us, and we’ll add it to our readers’ gallery.

Start doodaling!

Make something you love. Bon appetit!

A fractal Math Munch Doodal

A fractal Math Munch Doodal

Forest Fires, Scrubbing Calculator, and Bongard Problems

Welcome to this week’s Math Munch!

Last summer where I live, in California, there were a lot of forest fires. We’re having a big drought, and that made fires started for lots accidental of reasons– lightning, downed power lines, things like that– get much bigger than usual. I thought I’d learn a little about forest fires so that I can be a more responsible resident of my state.

And I found this great website with an awesome computer simulation that you can manipulate to experiment with the factors that lead to forest fires!

Forest fire.png

My forest fire, just starting to burn. Click on this picture to play with the simulation! (Note: It doesn’t work as well in certain browsers. I recommend Firefox.)


This site was built by Nicky Case, who studies things that mathematicians and scientists call complex systems. Basically, a complex system is some phenomenon that has a simple set of causes but unpredictable results. An environment with forest fires is a good example– a simple lightning strike in a drought-ridden forest can cause a wildfire to spread in patterns that firefighters struggle to predict. It turns out that simulations are perfect for modeling complex systems. With just a few simple program rules we can create a huge number of situations to study. Even better, we can change the rules to see what will happen if, say, the weather changes or people are more careful about where and when they set fires.

Forest fire what if

What do you think? Click the picture to see Nicky’s simulation.

You can use Nicky’s program to change the probability that trees grow, a burning tree sets its neighbors on fire, and many other factors. You can even invent your own and model them with emojis. What if there were two kinds of trees and one was more flammable than the other? What if trees grew quickly but lightning was common? As Nicky shows, simulations are useful for exploring what-if questions in complex systems. Use your imagination and explore!

Next up, have you ever heard of a scrubbing calculator? No, it’s not a calculator that doubles as a sponge. It’s a calculator that helps you solve for unknowns in equations by “scrubbing,” or approximating, the answer until you find a number that works.

Scrubbing calculator equation

Here’s how it works: Say you’re trying to solve for an unknown, like the x in the equation above (maybe for some practical reason or just because you’re doing your homework). You could do some pretty complicated algebraic manipulations so that the x alone equals some number. But what if you could make a guess and change it until the equation worked?

Scrubbing calculator

Click on this image to see a scrubbing calculator in action!

If your guess was too big, you’d know because the expression wouldn’t equal 768 anymore– it would equal something larger. And if you had a calculator that instantly told you the solution based on your guess, you could do this guessing and checking pretty quickly.

Well, lucky for you I found a scrubbing calculator that you can use online! It’s very simple– just type in your equation and your guess, and click on the number you want to change (most likely the guess) to make it larger or smaller. It’s useful for solving equations, like I said. But I actually find it most interesting to watch how the whole expression changes as you change one of the numbers in it. For instance, check out the Pythagorean triple calculator I built. What do you notice as you gradually change one of the numbers in the expression?

Finally, I’m excited to share with you one of my favorite kinds of puzzles– Bongard problems!
Bongard 1A Bongard problem has two sets of pictures, with six pictures in each set. All of the pictures on the left have something in common that the pictures on the right do not. The challenge is to figure out what distinguishes the two groups of pictures.

Bongard 2

This problem was made by Douglas Hofstadter, who introduced Bongard problems to the U.S. I haven’t figured it out yet. Can you?

I got the Bongard problems shown above from a collection of problems put together by cognitive scientist Harry Foundalis. He has almost 300 of them, some made by Mikhail Bongard himself, who developed these problems while studying how to train computers to recognize patterns.

Harry also has guidelines for how to develop your own Bongard problems. He encourages people to send their problems to him and says he might even put them up on his site!

Bon appetit!

Mobiles, Mathematical Objects, and Math Magazine

Welcome to this week’s Math Munch!

Before we start, a little business. You may have noticed that posts have been few and far between lately. Those of you who know us, the members of the Math Munch Team, know that we’ve all made a lot of life changes in the past year or two. We started out together teaching in the same school in New York– but now we live on far corners of the country and spend our time doing very different things. In case you’re curious, here are some pictures of the things we’ve been up to!


Justin’s genus 19, rotationally symmetric surface

But even though we’ve moved apart physically, we’ve decided that we really want to keep the Math Munch Team together. We LOVE sharing our love of math with you– and we love hearing from you about the amazing things you make and do with math, too.

So, we’ve decided to revamp our posting process and came up with a schedule for when you can expect posts. There will be a new post every Thursday. (Though if Anna is posting from the West Coast, it might come out in the wee hours of Friday morning for some of you!) And here’s the monthly schedule of Thursday posts:

  • The first Thursday of the month will be a post from Justin
  • The second Thursday of the month will be a rerun!! Did you know we have over 150 posts on this site?? And we’ve been posting for almost five years??
  • The third Thursday of the month will be a post from Anna
  • The last Thursday of the month will be a post from Paul

And for those mysterious months with five Thursdays (ooh, when will that be, I wonder?)… There will be a surprise!

And now… for some math!

Screen Shot 2016-04-22 at 1.44.50 AMFirst up is a little game called SolveMe Mobiles! This game is full of little puzzles in which you have to figure out what each of the different shapes in a mobile weighs. You’re given different clues in different puzzles. So, for instance, in the puzzle to the left, you’re given the weight of the red circle and you have to figure out how much a blue triangle is. But you’re not given the weight of the whole mobile… Hmmm…

Screen Shot 2016-04-22 at 1.53.04 AMAnd this one, to the right, gives you the weight of one of the shapes and of the whole mobile– but now there are three shapes! Tricky!

Even better, you can build your own mobile puzzle for others to solve! I made this one, shown below– like my use of a mobile within a mobile?

Screen Shot 2016-04-22 at 2.04.48 AM

Next up, I found a beautiful Tumblr account that I’d like to share with you full of pictures of found mathematical objects. It’s called… Mathematical Objects! (How clever.) The author of the site writes that the aim of the blog is to “show that mathematics, aside from its practicality, is also culturally significant. In other words, mathematics not only makes the trains run on time but also fundamentally influences the way we view the world.


“Counting to One Hundred with my Four-Color Pen”

tumblr_n07pvisVvm1rnwsgbo2_1280Some of the images are mathematical art, like the one above; others are more “practical,” such as plans for buildings or images drawn from science.

Do you ever see an interesting mathematical object in the wild and feel the urge to take a picture of it? If so, go ahead and send it to us! We’d love to see what you find.

I’m very excited to share this last find with you all. It was sent to me by a wonderful math teacher, Mark Dittmer, and his math students. This year, they were inspired by Math Munch to make their own fun online math sites! I think what they made is super awesome– and I want to share it with you. I’ll be featuring some of their work in my next few posts, one thing at a time to give each its own day in the sun. First up is this adorable adventure story about the residents of Number Land. I hope you enjoy it!

Screen Shot 2016-04-22 at 9.46.00 AM

Bon appétit!



Mathy Clocks, Spirolaterals, and Mandalas

Welcome to this week’s Math Munch!

Screen Shot 2016-03-31 at 10.33.37 AM

Hermann’s Abacus Clock. What time was I working on this post?

A few months ago, the Math Munch team got an email from retired mathematician Hermann Hoch with a lead to his amazing website full of (among other things)… clocks! One of the things Hermann does with his spare time in retirement is make creative math-y clocks using html. He calls them “html5 experiments”– and they really do take math art to the next level!

There are many fascinating clocks on Hermann’s site. (Be careful, or you might spend too much time watching the seconds go by!) One of my favorites is a clock he calls the Mondriaan Clock. The display is inspired by the art of Dutch painter Piet Mondriaan, who was known for his paintings of overlapping squares and rectangles in primary colors. The clock also comes with the exciting prompt– “wait until time creates golden ratios for us”! At what time will one of the rectangles in the image have dimensions that approximate the Golden Ratio? Hermann says that this question isn’t easy– he hasn’t even found all of the times himself! (And I’m sure he’d love to know– post your ideas in the comments below.)

Next up, I’ve been obsessed with Spirolaterals lately. What’s a Spirolateral, you ask? It’s a shape made by drawing segments of different lengths (say, 2, 3, and 4) one after another in a cycle (say, right, up, left, and down) until the shape closes up (or doesn’t, and you know it never will). If you follow those instructions (drawing on grid paper helps), you make this flower-like shape:

Screen Shot 2016-03-31 at 10.55.24 AM


You can make Spirolaterals (or Loop-de-Loops, as they’re also called) with any numbers and using any turning angle. This Spirolateral uses three numbers and a turning angle of 90 degrees. (See the square corners?) But what if you use four numbers? Five numbers? Thirteen numbers? You can try drawing by hand- and then coloring them in, to make a beautiful mathematical creation. The Spirolateral below uses the first 50 digits of pi!

Screen Shot 2016-03-31 at 11.00.37 AM.png

But you also don’t have to draw them by hand. The two Spirolaterals shown here were both drawn using a computer program! My favorite program for drawing Spirolaterals with 90 degree turns is this one, made by Chris Lusto. He gives great instructions and allows you to use as many numbers as you like!

But what if you wanted to make a Spirolateral with a… 109 degree turn? Wouldn’t that be cool! Well, yes, it is cool–

Screen Shot 2016-03-31 at 11.04.41 AM.png

You can make this and other crazy Spirolaterals at this awesome website, brought to us by The Mathenaeum.

Finally, I leave you with this mesmerizing video of Dearing Wang drawing a Mandala. If you thought you’d never use your skills with a straight-edge and compass you worked so hard to develop in Geometry class– think twice. And for you straight-edge and compass nerds, keep an eye out for his pentagon construction! Is it perfect??

Bon appetit!

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How long did it take me to make this post? (Hint: This clock tells time in base 6!)

Fold and Cut, My Favorite Spaces, and Hook

Welcome to this week’s Math Munch!

Before you watch this video, think about this question: Do you think you could fold a piece paper so that you could cut a square out of it using exactly one straight cut? How about a triangle? Hexagon? Christmas tree shape??

Give it a try. Then watch this video:

Pattern for a very angular swan, by Erik Demaine

Surprised? As you may have seen in the video, using the “fold and cut” process you can make any shape with straight sides! Isn’t that crazy? I learned about this a few years ago, and now cutting weird shapes out of paper using just one cut is one of my favorite things to do.

The person who proved this amazing result is one of my favorite mathematicians, Erik Demaine. (You might remember him from our post a few years ago about origami mazes.) I think it’s really interesting that this idea that’s now a mathematical theorem appeared throughout history as a magic trick and a method for cutting out five-pointed stars to make American flags. Check out this website about the fold and cut problem to learn more about the history of the theorem, Demaine’s method for cutting out any straight-edged shape, and other related problems.

Evelyn wearing a Borromean ring cowl. Sweet!

I found out about this video from another favorite mathematician of mine, Evelyn Lamb. Evelyn writes a blog about math for Scientific American called Roots of Unity that’s really fun to read. Check it out if you get the chance!

She has a series of posts called “A Few of My Favorite Spaces” (cue Sound of Music song, “My Favorite Things”). Favorite spaces, you may ask? I’m not familiar with spaces plural. There’s more than just regular old 3D space? Yes, in fact there are! And if you read Evelyn’s blog you’ll learn about how mathematicians like to invent new spaces with bizarre properties– and sometime find out that what they thought was a completely new space actually resembles something very familiar.

House with 2 roomsSuch as… The “house with two rooms.” As I understand it, this a box (“house”) with two floors and two tunnels in it– one punched from the top of the box and another from the bottom. The top tunnel lets you get from the roof of the house to the ground floor; the bottom tunnel lets you get from below the house to the second floor.

If you want to see someone making this crazy house in Minecraft and hear a much better explanation of what the house is like, here’s a video!

Ok, so what’s the point? Well, it turns out you can squish (just squish– no ripping or gluing) this house all the way down to a single point. This means that in topology (the type of math that involves a lot of squishing), the crazy tunnel house space is the same as the really boring space of just one point. I might want to live in a house with all these tunnels– but I definitely don’t want to live in a point. But in topology-world, they’re the same space. Huh.

To learn more about the house with two rooms (aka, point) and other crazy spaces, check out Evelyn’s blog!

Finally, speaking of squishing things down to a point, I want to show you a fun new game I found that involves a lot of squishing– Hook! Here’s a trailer video for the game:

You can find this game online at Kongregate. Enjoy!

Bon appetit!

HYPERNOM, Euclid the Game, and Math Quilts

Welcome to this week’s Math Munch! And, welcome to a new school year! Back to school means back to Math Munch– and we’re super excited to share some great new things that we found over the summer.  The first of which is…


(GIF hoisted from the amazing Aperiodical)

That’s an image from this crazy new game called HYPERNOM, invented by some of our favorite people– Vi Hart, Henry Segerman, and Andrea Hawksley!

Noming through tasty tasty tetrahedra. Mmmm!

Noming through tasty tasty tetrahedra.

In this game, you wiggle around in a projection of 4-dimensional space, eating (or, better put, NOMING– NOM NOM NOM) 4-dimensional objects. Such as the dodecahedra (polyhedron with faces made from regular pentagons) that come together to form the 4-dimensional shape (called a polytope) you’re moving around in.

Playing hypernomThis game is MINDBLOWING. Really. You can play it on your computer– but I got to play it wearing a helmet that plunged me into the fourth dimension and left me feeling very dizzy.

The math behind HYPERNOM is kind of complicated but VERY interesting. If you’d like to learn more about the game and the 4-dimensional math it involves, check out this post from Aperiodical. Or, watch the talk that Vi, Henry, and Andrea gave about HYPERNOM at this year’s Bridges Mathematical Art conference!

Next up, the Math Munch team went back to school a few weeks ago, too– literally! And this member of the Math Munch team is taking a math class! My homework assignment last week was to play a new game called Euclid: The Game.

On my way to constructing an equilateral triangle. What should I do next?

On my way to constructing an equilateral triangle. What should I do next?

The game is pretty much exactly what it sounds like. You get to use just a straight-edge and compass (but a virtual straight-edge and compass, powered by Geogebra, because it’s a computer game!) to make Euclid’s constructions. For instance, the first challenge is to make an equilateral triangle– and all you can do is draw circles and lines! How would you do that?

I love this game for learning geometry because it lets you see how Euclid and his mathematicians peers thought about geometry– but you don’t have to use a real compass! The game saves your constructions so you can use them later– so if you ever want to make an equilateral triangle again, you don’t have to start from scratch. The game also gives you points if you make your construction with the least number of steps or without using any new tools. Give it a try!

Ellison tessellation quiltFinally, I recently ran across the beautiful mathematical quilts of Elaine Ellison. Elaine is a former high school math teacher from Indiana who now creates and gives talks about making mathematical quilts. Her quilts explore some of the most interesting types of mathematics– from tessellations (like the Escher-inspired fish tessellation quilt to the left), to conic sections, to strange geometric spaces.

Ellison poincare quilt

“Poincare Plane”

Ellison parabola quilt


Elaine has a website and a YouTube channel devoted to her gorgeous quilts. Check them out! Here’s a taster:

We hope you’re enjoying your return to school! We definitely are. Bon appetit!