Here’s a puzzle for you: You have 12 weights, 11 of which weigh the same amount and 1 of which is different. Luckily you also have a balance, but you’re only allowed to use it three times. Can you figure out which weight is the different weight?

You certainly can! I won’t tell you how, but you can figure it out for yourself while playing this interactive weight game. This puzzle is tricky, but definitely fun. If one weight puzzle isn’t enough for you, you’re in luck– there are many, many variations! Check out this site to try a similar puzzle with nine weights, ten weights, and 27 weights.

My solution to the Circle Pack 2 challenge. Can you do it in only 5 moves?

Next up, if you like drawing challenges, this is the game for you. Check out this crazy geometry game, in which you have to draw different shapes (like perfect equilateral triangles, squares, pentagons, and groups of circles of particular sizes) using only circles and straight lines! Here’s my solution to one of the challenges, the Circle Pack 2. See the two smaller circles inside of the larger middle circle? That’s what I wanted to draw– but I had to make all of those other circles and lines to get there! I did the Circle Pack 2 challenge in 8 moves, but apparently there’s a way to do it in only 5…

Finally, it’s pumpkin season again! Every year I scour the internet for new math-y ways to carve pumpkins. We’re all in luck this year– because I found great instructions for how to carve pumpkin polyhedra from Math Craft! Check out this site to learn how to carve all the basics– tetrahedra, cubes, octahedra, dodecahedra, and (my favorite) icosahedra– and a bonus polyhedron, the truncated icosahedron (also know as the soccer ball).

Pumpkin Platonic polyhedra!

Don’t forget to make pi with the leftover pumpkin! Oh, and, bon appetit!

This gear sphere and many others are the creations of Paul Nylander. There are 92 gears in this gear sphere. Can you figure out how many there are of each color? Do you notice any familiar shapes in the gears’ layout?

What’s especially neat are the sizes of the gears—how many teeth each gear color has. You can see the ratios in the upper left corner. Paul describes some of the steps he took to find gears sizes that would work together. He wrote a computer program to do some searching. Then he did some precise calculating and some trial and error. And finally he made some choices about which possibilities he liked best. Sounds like doing math to me!

Along the way Paul figured out that the blue gears must have a number of teeth that is a multiple of five, while the yellow ones must have a multiple of three. I think that makes sense, looking at the number of red gears around each one. So much swirly symmetry!

Spiral shadows!

Be sure to check out some of Paul’s other math art while you’re on his site. Plus, you can read about a related gear sphere in this post by mathematician John Baez.

I figured there had to be a good math game that involves gears. I didn’t find quite what I expected, but I did find something I like. It’s a game that’s called—surprise, surprise—Gears! It isn’t an online game, but it’s easy to download.

Can you find the moves to make all the gears point downwards?

Wuzzit Trouble!

And if you’re in the mood for some more gear gaming and you have access to a tablet or smartphone, you should check out Wuzzit Trouble. It’s another free download game, brought to you by “The Math Guy” Keith Devlin. Keith discusses the math ideas behind Wuzzit Trouble in this article on his blog and in this video.

Last up this week, I’d like to share with you some souvenirs. If you went on a math vacation or a math tour, where would you go? One of the great things about math is that you can do it anywhere at all. Still, there are some mathy places in the world that would be especially neat to visit. And I don’t mean a place like the Hilbert Hotel (previously)—although you can get a t-shirt or coffee mug from there if you’d like! The mathematician David Hilbert actually spent much of his career in Göttingen, a town and university in Germany. It’s a place I’d love to visit one day. Carl Gauss lived in Göttingen, and so did Felix Klein and Emmy Noether—and lots more, too. A real math destination!

Lots of math has been inspired by or associated with particular places around the world. Just check out this fascinating list on Wikipedia.

The Arctic Circle Theorem

The Warsaw Circle

The Cairo Pentagonal Tiling

Did you know that our word souvenir comes from the French word for “memory”? One thing that I like about math is that I don’t have to memorize very much—I can just work things out! But every once in a while, there is something totally arbitrary that I just have to remember. Here’s one memory-helper that has stuck with me for a long time.

May you, like our alligator friend, find some good math to munch on. Bon appetit!

I’ve been really into squares lately. Maybe it’s because I recently ran across a new puzzle involving squares– something called Mrs. Perkin’s quilt.

69 by 69 Mrs. Perkin’s quilt.

The original version of the puzzle was published way back in 1907, and it went like this: “For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches.”

18 by 18 Mrs. Perkin’s quilt

Said in another way, if you have a 13 by 13 square, how can you divide it up into the smallest number of smaller squares? Don’t worry, you get to solve it yourself– I’m not including a picture of the solution to that version of the puzzle because there are so many beautiful pictures of solutions to the puzzle when you start with larger and smaller squares. Some are definitely more interesting than others. If you want to start simple, try the 4 by 4 version. I particularly like the look of the solution to the 18 by 18 version.

152 by 152 Mrs. Perkin’s quilt

Maybe you’re wondering where I got all these great pictures of Mrs. Perkin’s quits. And– wait a second– is that the solution to the 152 by 152 version? It sure is– and I got it from one of my favorite math websites, the Wolfram Demonstrations Project. The site is full of awesome visualizations of all kinds of things, from math problems to scans of the human brain. The Mrs. Perkin’s quilts demonstration solves the puzzle for up to a 1,098 by 1,098 square!

Next up, we here at Math Munch are big fans of unusual calculators. Marble calculators, domino calculators… what will we turn up next? Well, here for your strange calculator enjoyment is a water calculator! Check out this video to see how it works:

I might not want to rely on this calculator to do my homework, but it certainly is interesting!

Finally, meet Snap the Turtle! This cute little guy is here to teach you how to make beautiful math art stars using computer programming.

On the website Tynker, Snap can show you how to design a program to make intricate line drawings– and learn something about computer programming at the same time. Tynker’s goal is to teach kids to be programming “literate.” Combine computer programming with a little math and art (and a turtle)– what could be better?

I hope something grabbed your interest this week! Bon appetit!

Up first, a follow up to our post about the World Cup a while back. We received an email from Marc Chamberland linking us to a nice little video (below) about World Cup Balls and their various properties. You may remember seeing Marc’s mathematical art in this post. Below you can see another nice piece that was included in the mathematical art exhibit at the 2013 Joint Mathematics Meetings. Click for a nice description of the math puzzle it solves. (in short: What’s the area of the red square?)

“Inner Square” by Marc Chamberland

Marc is a math professor at Grinnell College. In March of 2014 (3-14?) he began working on Tipping Point Math, a youtube channel full of videos showing “math as you never imagined.” I encourage you to find something nice there. For now, here’s that video about World Cup Balls I promised you.

A font based on glass bending

Up next are some nifty, fun fonts based on mathematics. Erik Demaine is no stranger to Math Munch readers, and it’s no wonder why. His stuff is clever and downright intriguing. He and his father Martin published a very interesting paper last April about a series of mathematical typefaces they’ve created over the course of their last decade of research and play.

A conveyor belt typeface by Erik and Martin Demaine

Their paper was published to the arXiv (pronounced “archive”) where it is publicly available. You can read it here. Or, if you like something slightly more plain-language, here’s a nice review over on medium.com.

The 4051 Tektronix Graphics Terminal

Finally, I want to share a sleepy little video called “Congruent Triangles.” I like to think of it as a slice of mathematical cultural history. This film was made in 1977 on an early computer called a Tektronix 4051 Graphics Terminal. It was made by Bruce and Katherine Cornwell as part of a series of mathematical videos. The way the shapes move and deform to present the ideas and connect the pieces together is so very cool. I also love the choice of music. It tells you something about what math was like for people then. I’d say sort of “groovy.”

There’s more to the story and many more cool videos to enjoy. You can look forward to seeing more from the Cornwells, but for now, enjoy this one video and do some hunting on your own if you’re interested. That’s called “research.”

As you start a new school year, you might be looking for some new mathy doodle games to play in the margins of your notebooks. Doodling helps me to listen sometimes, and I love making neat patterns. I especially like seeing what new shapes I can make.

This summer I was very happy to run across Zentangle®—”an easy-to-learn, relaxing, and fun way to create beautiful images by drawing structured patterns.” I’ve learned a lot about Zentangle from a blog called Tangle Bucket by Sandy Hunter. She shares how to doodle snircles, snafoozles, and oodles. There’s a whole dictionary of zentangle shapes over at tanglepatterns.com.

My favorite idea in Zentangle is trying to combine two kinds of designs. Sometimes this is described as one pattern “versus” another one. For instance, check out these:

Maybe you’ll pick some tangle patterns to combine with each other. If you try some, maybe you’ll share them in our Readers’ Gallery.

It’s so true that the more I tangle, the more I see the potential in patterns all around me. I catch myself mentally deconstructing them (whether I want to or not) to figure out if they can be broken down into simple steps without too much effort. That’s the trademark of a good tangle pattern.

Try some of Sandy’s weekly challenges, or check out Tiffany Lovering’s time-lapse videos—here’s one with music and one with an interview. Can you learn the names of any of the shapes she creates? I spy a Rick’s Paradox. There are lots of ways to begin zentangling—I hope you enjoy giving it a try.

Squares & dots & crosses, oh my!

If zentangling is too freeform for your doodling tastes, then let me share with you one of my longtime favorite websites. I’ve used it for years to help me to do math and to teach math, and it’s great for math doodling, too. I might even call it a trusty friend, one that I met one day through the simple online search: “free online graph paper”.

Something I love about the site is that it lets you design different aspects of your graph paper. Then you can print it out. First you get to decide what kind of grid you would like: square? triangular? circular? Then you get to tinker with lots of variables, like how big the grid cells are, how dark the lines are, and what color they are. And more!

Free Online Graph Paper was created by Kevin MacLeod, who composes music and shares it for free. That way other people can use it for creative projects. That’s really awesome! I enjoyed listening to Kevin’s “Winner Winner“. It’s always good to be reminded that everything you use or enjoy was almost certainly made by a person—including custom graph paper websites!

A 7/3 star spirocake.

Last up this week is some doodly math that you can really munch on. Everyone knows that breakfast is the most important meal of the day and that the most important food group is roulette curves.

To get your daily recommended allowance of groovy math, look no further than the edible doodles of Nathan Shields and his family over at Saipancakes.

Welcome to this week’s Math Munch! And, if you’re a student or teacher, welcome to a new school year!

One of the most exciting events in the world of math happened this August– the awarding of the Fields Medal! This award honors young mathematicians who have already done awesome mathematical work and who show great promise for the future. It also only happens every four years, at the beginning of an important math conference called the International Congress of Mathematicians, so it’s a very special occasion when it does!

Maryam Mirzakhani, first woman ever to win a Fields Medal

This year’s award was even more special than usual, though. Not only were there four winners (more than the usual two or three), but one of the winners was a woman!

Now, if you’re like me, you probably heard about the Fields Medal and thought, “There’s no way I’ll understand the math that these Field Medalists do.” But this couldn’t be more wrong! Thanks to these great articles from Quanta Magazine, you can learn a lot about the super-interesting math that the Fields Medalists study– and why they study it.

Manjul Bhargava

One thing you’ll immediately notice is that each Fields Medalist has non-math interests that inspire their mathematical work. Take Manjul, for instance. When he was a kid, his grandfather introduced him to Sanskrit poetry. He was fascinated by the patterns in the rhythms of the poems, and the number patterns that he found inspired him to study the mathematics of number patterns– number theory!

But, don’t just take my word for it– you can read all about Manjul and the others in these great articles! And did I mention that they come with videos about each mathematician?

Want to read more about this year’s Fields Medallists? Check out Alex Bellos’s article in The Guardian. Which brings me to…

… What’s your favorite number? Is it 7? If it is, then you’re in good company! Alex polled more than 30,000 people about their favorite number, and the most popular was 7. But why? What’s so special about 7? Here’s why Alex thinks 7 is such a favorite:

Why do you like your favorite number? People gave Alex all kinds of different reasons. One woman said about 3, her favorite number, “3 wishes. On the count of 3. 3 little pigs… great triumvirates!” Alex made these questions the topic of the first chapter of his new book, The Grapes of Math. (Get the reference?) In this book, Alex shares many curious ways that math appears in our world. Did you know that a weird pattern in numbers can be used to catch criminals? Or that the Game of Life, a simple computer program, shares surprisingly many characteristics with real life? These are only a few of the hundreds of topics Alex covers in his book. Whether you’re a math whiz or a newbie, you’ll learn something new on every page.

Alex currently writes about math for The Guardian in a blog called, “Alex’s Adventures in Numberland”– but he also loves and writes about soccer (or futbol, as it’s called in his native Brazil)! He even wrote a few articles for his blog about math and soccer.

Do you have any questions for Alex? (About math, soccer, or their intersection?) Write them here and you might find them featured in our interview with Alex!

Good writing about math is hard to find. If you’ve ever picked up a standard math textbook, you’ll know what I mean. But reading something fascinating, that grabs your interest from the first page and leads you through the most complex ideas like they’re as natural as anything you’ve observed, is a great way to learn. The Grapes of Math and “Alex’s Adventures in Numberland” do just that. Give them a go!

Welcome to this week’s Math Munch! We’ve got a pair of new games for you to play later, but first I want to share something beautiful and impressive.

Ready for some mathematical art? The new issue of Hyperseeing begins with a review of Merete Rasmussen’s ceramic sculpture. Merete is a Danish artist who lives in London, and her recent work features complex and beautiful, smooth two-dimensional surfaces.

Editor Nat Friedman’s writeup begins with this wonderful quote by Rasmussen:

“I want to create a form that you can’t understand until you see the other side. You have to look at it for a while to realize how it is connected.”

Merete Rasmussen at work

A lot of mathematical work is done just trying to describe and understand the ideas or pictures in our head. Merete’s sculpture get us to do math as we try to understand the nature of her sculptural surfaces. How many sides do they have? How many edges? How many holes? I just love that.

The article is very enjoyable, and I encourage you to read the entire text, but what got me hooked, what completely mesmerized and inspired me, was a video about Merete’s work and process that I found referenced at the end of the article. The video is presented in dual screen, which is really fantastic, because just like Merete’s sculptures, you may need to view it a couple of times to catch all that’s going on.

I recommend the full video. I recommend full screen.

You can learn more about Merete Rasmussen and view more of her work at her website, mereterasmussen.com.

“Right from the start I was thinking about creating a simple game, with simple graphics and simple game design.” That’s what 26-yr old Andrey Shevchuk said about his recent creation, “Game About Squares.” You’ll find as you play, however, that these little puzzles can get oh so complicated, despite their simple presentation.

I love imagining how Andrey must have had to think creatively to keep developing his simple idea in new ways, and I love the way that the puzzles get us to think in new ways. All in all, this is just a wonderful game.

Oh, and thinking about the very viral 2048, Andrey had this to say,

“Squares are trendy. Hexagons aren’t even close, let alone triangles.”

That’s short for vaccine, in case you don’t know. The Salathé Group recently released a game about vaccinations and fighting the spread of epidemics (previously). The game is called VAX!, and it’s based on a graph theory representation for the spread of disease. Take the tour and you’ll learn everything you need to play.

There’s also a module that explains herd immunity. That’s where random vaccines are used to isolate the potentially infected from potential carriers of the disease.