# The World Cup Group Stage, Math at First Sight, and Geokone

Welcome to this week’s Math Munch! We’ve got some World Cup math from a tremendous recreational mathematics blog and a new mathematical art tool. Get ready to dig in!

Brazuca: The 2014 World Cup Ball

I’ve been meaning to share the really fantastic Puzzle Zapper Blog, because it’s so full of cool ideas, but the timing is perfect, because IT’S WORLD CUP TIME!!! and the most recent post is about the math of the world cup group stage! It’s called “World Cup Group Scores, and “Birthday Paradox” Paradoxes,” and I hope you’ll give it a read. (For some background on the Birthday Paradox, watch this Numberphile video called 23 and Football Birthdays.)

The thing that got me interested in the article was actually just this chart. I think it’s really cool, probably because I always find myself two games through the group stage, thinking of all the possible outcomes. If you do nothing else with this article, come to understand this chart. I was kind of surprised how many possible outcomes there are.

All Possible World Cup Group Stage Results

Long story short (though you should read the long story), there’s about a 40% chance that all 8 world cup groups will finish with different scores.

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre’s interactive fiction)

Puzzle Zapper is the recreational mathematics blog of Alexandre Owen Muñiz. You can also find much of his work on his Math at First Sight site. He has a lot of great stuff with polyominoes and other polyforms (see the nifty pics below). Alexandre is also a writer of interactive fiction, which is basically a sort of text-based video game. Click on Alexandre’s picture to learn more.

The complete set of “hinged tetrominoes”

A lovely, symmetric family portrait of the “hinged tetriamonds”

I hope you’ll poke around Alexandre’s site and find something interesting to learn about.

For our last item this week, I’ve decided to share a new mathematical art tool called Geokone. This app is a recursive, parametric drawing tool. It’s recursive, because it is based on a repeating structure, similar to those exhibited by fractals, and it’s parametric, because the tool bar on the right has a number of parameters that you can change to alter the image. The artistic creation is in playing with the parameter values and deciding what is pleasing. Below are some examples I created and exported.

I have to say, Geokone is not the easiest thing in the world to use, but if you spend some time playing AND thinking, you can almost certainly figure some things out! As always, if you make something cool, please email it to us!

##### Now go create something!  Click to go to Geokone.net.

I hope you find something tasty this week. Bon appetit!

# Girls’ Angle, Spiral Tilings, and Coins

Welcome to this week’s Math Munch!

Girls’ Angle is a math club for girls. Since 2007 it has helped girls to grow their love of math through classes, events, mentorship, and a vibrant mathematical community. Girls’ Angle is based in Cambridge, Massachusetts, but its ideas and resources reach around the world through the amazing power of the internet. (And don’t you worry, gentlemen—there’s plenty for you to enjoy on the site as well.)

Amazingly, the site contains an archive of every issue of Girls’ Angle Bulletin, a wonderful bimonthly journal to “foster and nurture girls’ interest in mathematics.” In their most recent issue, you’ll find an interview with mathematician Karen E. Smith, along with several articles and puzzles about balance points of shapes.

There’s so much to dig into at Girls’ Angle! In addition to the Bulletins, there are two pages of mathematical videos. The first page shares a host of videos of women in mathematics sharing a piece of math that excited them when they were young. The most recent one is by Bridget Tenner, who shares about Pick’s Theorem. The second page includes several videos produced by Girls’ Angle, including this one called “Summer Vacation”.

Girls’ Angle can even help you buy a math book that you’d like, if you can’t afford it. For so many reasons, I hope you’ll find some time to explore the Girls’ Angle site over your summer break. (And while you’ve got your explorer’s hat on, maybe you’ll tour around Math Munch, too!)

I did a Google search recently for “regular tilings.” I needed a few quick pictures of the usual triangle, square, and hexagon tilings for a presentation I was making. As I scrolled along, this image jumped out at me:

What is that?! It certainly is a tiling, and all the tiles are the “same”—even if they are different sizes. Neat!

Clicking on the image, I found myself transported to a page all about spiral tilings at the Geometry Junkyard. The site is a whole heap of geometrical odds and ends—and a place that I’ve stumbled across many times over the years. Here are a few places to get started. I’m sure you’ll enjoy poking around the site to find some favorite “junk” of your own.

 Spirals Circles & spheres Coloring

Last up this week, you may have seen this coin puzzle before. Can you make the triangle point downwards by moving just three pennies?

There are lots of variants of this puzzle. You can find some in an online puzzle game called Coins. In the game you have to make arrangements of coins, but the twist is that you can only move a coin to a spot where would it touch at least two other coins. I’m enjoying playing Coins—give it a try!

I solved this Coins puzzle in four moves. Can you? Can you do better?

That’s it for this week’s Math Munch. Bon appetit!

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

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.

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

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!!

Recreational Mathematician, Lee Sallows

Naturally, I was reminded of the geomagic squares we featured a while back (more at geomagicsquares.com), 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, leesallows.com.

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