Tag Archives: algebra

Sphericon, National Curve Bank, and Cardioid String Art

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

Wooden sphericonIt was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:How to make a sphericon

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

Sphericon pattern

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

ncbmastertitleI found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

epicycloidaThere are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String cardioid

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

Weights, Crazy Geometry Game, and Pumpkin Polyhedra

Welcome to this week’s Math Munch!

Weighing puzzleHere’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.

Circle two pack

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…

Truncated icosahedron pumpkinFinally, 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 polyhedra

Pumpkin Platonic polyhedra!


Don’t forget to make pi with the leftover pumpkin! Oh, 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.

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.

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.

Can you replace the Multichoice tiles to create a true number sentence?

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.

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.)

The level!

The level I made!

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 Gallivan and her 11th fold.

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.

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