Tag Archives: games

Spheres, Gears, and Souvenirs

92GearSphere-20-24-16Welcome to this week’s Math Munch!

Whoa. What is that?

Is that even possible?

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!

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?

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

This Wuzzit is in trouble!

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.

Poster2

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.

Arctic Circle Theorem

The Arctic Circle Theorem

The Warsaw Circle

The Warsaw Circle

Cairo Pentagonal Tiling

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!

Girls’ Angle, Spiral Tilings, and Coins

Welcome to this week’s Math Munch!

GirlsAngleCoverGirls’ 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:

hexspiral

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

Spirals

Circles and spheres

Circles & spheres

Coloring

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?triangleflip

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?

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!

 

Math Awareness Month, Hexapawn, and Plane Puzzles

Welcome to this week’s Math Munch!

April is Mathematics Awareness Month. So happy Mathematics Awareness Month! This year’s theme is “Mathematics, Magic, and Mystery”. It’s inspired by the fact that 2014 would have marked Martin Gardner’s 100th birthday.

MAM

A few of the mathy morsels that await you this month on mathaware.org!

Each day this month a new piece of magical or mysterious math will be revealed on the MAM site. The mathematical offering for today is a card trick that’s based on the Fibonacci numbers. Dipping into this site from time to time would be a great way for you to have a mathy month.

It is white

It is white’s turn to move. Who will win this Hexapawn game?

Speaking of Martin Gardner, I recently ran across a version of Hexapawn made in the programming language Scratch. Hexapawn is a chess mini-game involving—you guessed it—six pawns. Martin invented it and shared it in his Mathematical Games column in 1962. (Here’s the original column.) The object of the game is to get one of your pawns to the other side of the board or to “lock” the position so that your opponent cannot move. The pawns can move by stepping forward one square or capturing diagonally forward. Simple rules, but winning is trickier than you might think!

The program I found was created by a new Scratcher who goes by the handle “puttering”. On the site he explains:

I’m a dad. I was looking for a good way for my daughters to learn programming and I found Scratch. It turns out to be so much fun that I’ve made some projects myself, when I can get the computer…

puttering's Scratch version of Conway's Game of Life

puttering’s Scratch version of Conway’s Game of Life

Something that’s super cool about puttering’s Hexapawn game is that the program learns from its stratetgy errors and gradually becomes a stronger player as you play more! It’s well worth playing a bunch of games just to see this happen. puttering has other Scratch creations on his page, too—like a solver for the Eight Queens puzzle and a Secret Code Machine. Be sure to check those out, too!

Last up, our friend Nalini Joshi recently travelled to a meeting of the Australian Academy of Science, which led to a little number puzzle.

nalini3

What unusual ways of describing a number! Trying to learn about these terms led me to an equally unusual calculator, hosted on the Math Celebrity website. The calculator will show you calculations about the factors of a numbers, as well as lots of categories that your number fits into. Derek Orr of Math Year-Round and I figured out that Nalini’s clues fit with multiple numbers, including 185, 191, and 205. So we needed more clues!

Can you find another number that fits Nalini’s clues? What do you think would be some good additional questions we could ask Nalini? Leave your thoughts in the comments!

unusualcalc

A result from the Number Property Calculator

I hope this post helps you to kick off a great Mathematics Awareness Month. Bon appetit!

2048, 2584, and variations on a theme

Welcome to this week’s Math Munch! It’s a week of mathematical games, including a devilish little game and variations on the theme.

2048

2048

First up, check out this simple little game called 2048. Really, you must go try that game before reading on.

Gabriele Cirulli

Gabriele Cirulli

2048 was created by Gabriele Cirulli, a 20-year old who lives in northern Italy. He was inspired by a couple of very similar games called 1024 and threes, and he wanted to see if he could code a game from scratch. Nice work, Gabriele! (Stay tuned for a Q&A with Gabriele. Coming soon.)

The first time I played, I thought randomly moving the pieces around would work as well as anything, but wow was I wrong. Give it a try and see how far you get. Now watch how this AI (artificial intelligence) computer program plays 2048. You’ll probably notice some patterns that will help you play on your own.

A beautiful chain of powers of two.

A beautiful chain of powers of two.  Can you solve from here?

Did you notice that the smallest tiles are 2’s, and you can only combine matching tiles to create their double? This makes all of the tile values powers of two! (e.g. 2048=2^11) These are the place values for the binary number system! (Did you see our recent post binary?) This has something to do with the long chains that are so useful in solving the game. It’s just like this moment in the marble calculator video.

4, a silly, but interesting little variation

4, a silly, but interesting little variation

If you’re finding 2048 a bit too hard, here’s an easier version.  It’s called 4. It’s a little silly, but it’s also quite interesting. After you make the 4 tile (tying the world record for fewest moves), click “keep going” and see how far you can get. I’ve never been able to get past the 16 tile. Can anyone make the 32? What’s the largest possible tile that can be made in the original 2048 game? Amazingly, someone actually made a 16384 tile!!!

2584, the Fibonacci variant of 2048

2584, the Fibonacci version of 2048

Silly versions aside, there are lots and lots of ways you could alter 2048 to make an interesting game. I wondered about a version where three tiles combined instead of two, but I couldn’t quite figure out how it would work. Can you? (See below.) When I thought about different types of numbers that could combine, I thought of the perfect thing. The Fibonacci numbers!!! 1, 1, 2, 3, 5, 8, 13, 21, … The great thing is that someone else had the same idea, and the game already exists! Take some time now to play 2584, the Fibonacci version of 2048.

2048 and 2584 might seem like very similar games at first, (they’re only 536 apart), but there are some really sneaky and important differences. In the Fibonacci version, a tile doesn’t combine with itself. It has two different kinds of tiles it can match with. I think this makes the game a little easier, but the website says 2584 is more difficult than the original. What do you think?

I have a few more 2048 variations to share with you, as if you didn’t have enough already. These are my favorites:

I hope you dig into some of these games this week. Really think and analyze. If you come up with clever strategies or methods to solve these puzzles, please let us know in the comments. Have a great week, and bon appetit!

Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!

thomas-edison

Edison with his light bulb.

On this day in 1880, Thomas Edison was given a patent for his most famous bright idea—the light bulb.

Edison once said, “Genius is one per cent inspiration, ninety-nine per cent perspiration”—a good reminder that putting in some work is important both in math and in life. He also said, “We don’t know a millionth of one percent about anything.” A humbling thought. Also, based on that quote, it sounds like Edison might have had a use for permilles or even permyraids in addition to percents!

Mike's octahedron.

Mike’s octahedron-in-a-light-buld.

In celebration of this illustrious anniversary, I’d like to share some light mathematical fare relating to, well, light bulbs. For starters, J. Mike Rollins of North Carolina has created each of the Platonic solids inside of light bulbs, ship-in-a-bottle style. Getting just the cube to work took him the better part of twelve hours! Talk about perspiration. Mike has also made a number of lovely Escher-inspired woodcuts. Check ‘em out!

Evelyn's Schwartz lantern.

Evelyn’s Schwartz lantern.

Next up is a far-out example from calculus that’s also a good idea for an art project. It’s called the Schwartz lantern. I found out about this amazing object last fall when Evelyn Lamb tweeted and blogged about it.

The big idea of calculus is that we can find exact answers to tough problems by setting up a pattern of approximations that get better and better and then—zoop! take the process to its logical conclusion at infinity. But there’s a catch: you have to be careful about how you set up your pattern!

A "nicely" triangulated cylinder.

A “nicely” triangulated cylinder.

For example, if you take a cylinder and approximate its surface with a bunch of triangles carefully, you’ll end up with a surface that matches the cylinder in shape and size. But if you go about the process in a different way, you can end up with a surface that stays right near the cylinder but that has infinite area. That’s the Schwartz lantern, first proposed by Karl Hermann Amandus Schwarz of Cauchy-Schwartz fame. The infinite area happens because of all the crinkles that this devilish pattern creates. For some delightful technical details about the lantern’s construction, check out Evelyn’s post and this article by Conan Wu.

Maybe you’ll try folding a Schwartz lantern of your own. There’s a template and instructions on Conan’s blog to get you started. You’ll be glowing when you finish it up—especially if you submit a photo of it to our Readers’ Gallery. Even better, how about a video? You could make the internet’s first Schwartz lantern short film!

Robert Torrence and his Lights Out puzzle.

Robert and his Lights Out puzzle.

At the MOVES Conference last fall, Bruce Torrence of Randolf-Macon College gave a talk about the math of Lights Out. Lights Out is a puzzle—a close relative of Ray Ray—that’s played on a square grid. When you push one of the buttons in the grid it switches on or off, and its neighbors do, too. Bruce and his son Robert created an extension of this puzzle to some non-grid graphs. Here’s an article about their work and here’s an applet on the New York Times website where you can play Lights Out on the Peterson graph, among others. You can even create a Lights Out puzzle of your own! If it’s more your style, you can try a version of the original game called All Out on Miniclip.

The original Lights Out handheld game from 1995.

The original Lights Out handheld game from 1995.

There’s a huge collection of Lights Out resources on Jaap’s Puzzle Page (previously), including solution strategies, variations, and some great counting problems. Lights Out and Ray Ray are both examples of what’s called a “sigma-plus game” in the mathematical literature. Just as a bonus, there’s this totally other game called Light Up. I haven’t solved a single puzzle yet, but my limitations shouldn’t stop you from trying. Perspiration!

All this great math work might make you hungry, so…bon appetit!

Virtual Hyenas, Markov Chains, and Random Knights

Welcome to this week’s Math Munch!

It’s amazing how a small step can lead to a chain reaction of adventure.

Arend Hintze

Arend Hintze

Recently a reader named Nico left a comment on the Math Munch post where I shared the game Loops of Zen. He asked why the game has that name. Curious, I looked up Dr. Arend Hintze, whose name appears on the game’s title page. This led me to Arend’s page at the Adami Lab at Michigan State University. Arend studies how complex systems—especially biological systems—evolve over time.

Here is a video of one of Arend’s simulations. The black and white square is a zebra. The yellow ones are lions, the red ones are hyenas, and guess who’s hungry?

Arend’s description of the simulation is here. The cooperative behavior in the video—two hyenas working together to scare away a lion—wasn’t programmed into the simulation. It emerged out of many iterations of systems called Markov Brains—developed by Arend—that are based upon mathematical structures called Markov chains. More on those in a bit.

You can read more about how Arend thinks about his multidisciplinary work on biological systems here. Also, it turns out that Arend has made many more games besides Loops of Zen. Here’s Blobs of Zen, and Ink of Zen is coming out this month! Another that caught my eye is Curve, which reminds me of some of my favorite puzzle games. Curve is still in development; here’s hoping we’ll be able to play it soon.

Arend has agreed to do an interview with Math Munch, so share your questions about his work, his games, and his life below!

Eric Czekner

Eric Czekner

Arend’s simulations rely on Markov chains to model animal behavior. So what’s a Markov chain? It’s closely related to the idea of a random walk. Check out this video by digital artist, musician, and Pure Data enthusiast Eric Czekner. In the video, Eric gives an overview of what Markov chains are all about and shows how he uses them to create pieces of music.

On this page, Eric describes how he got started using Markov chains to make music, along with several of his compositions. It’s fascinating how he captures the feel of a song by creating a mathematical system that “generates new patterns based on existing probabilities.”

Now there’s a big idea: exploring something randomly can capture structures that might be hard to perceive otherwise. Here’s one last variation on the Markov chain theme that involves a pure math question. This blog post ponders the question: what happens when a knight takes a random walk—or random trot?—on a chessboard? It includes some colorful images of chessboards along the way.

How likely it is that a knight lands on each square after five moves, starting from b1.

How likely it is that a knight lands on each square after five moves, starting from b1.

The probabili

How likely it is that a knight lands on each square after 200 moves, starting from b1.

The blogger—Leonid Kovalev—shows in his analysis what happens in the long run: the number of times a knight will visit a square will be proportional to the number of moves that lead to that square. For instance, since only two knight moves can reach a corner square while eight knight moves can reach a central square, it’s four times as likely that a knight will finish on a central square after a long, long journey than on a corner square. This idea works because moving a knight around a chessboard is a “reversible Markov chain”—any path that a knight can trace can also be untraced. The author also wrote a follow-up post about random queens.

It’s amazing the things you can find by chaining together ideas or by taking a random walk. Thanks for the inspiration for this post, Nico. Keep those comments and questions coming, everyone—we love hearing from you.

Bon appetit!

Math Meets Art, Quarto, and Snow!

Welcome to this week’s Math Munch!

article-0-19F9E81700000578-263_634x286… And, if you happen to write the date in the European way (day/month/year), happy Noughts and Crosses Day! (That’s British English for Tic-Tac-Toe Day.) In Europe, today’s date is 11/12/13– and it’s the last time that the date will be three consecutive numbers in this century! We in America are lucky. Our last Noughts and Crosses Day was November 12, 2013 (11/12/13), and we get another one next year on December 13 (12/13/14). To learn more about Noughts and Crosses Day and find out about an interesting contest, check out this site. And, to our European readers, happy Noughts and Crosses Day!

p3p13Speaking of Noughts and Crosses (or Tic-Tac-Toe), I have a new favorite game– Quarto! It’s a mix of Tic-Tac-Toe and another favorite game of mine, SET, and it was introduced to me by a friend of mine. It’s quite tricky– you’ll need the full power of your brain to tackle it. Luckily, there are levels, since it can take a while to develop a strategy. Give it a try, and let us know if you like it!

BRUCKER-ICS-DARKRYE-SQUARE

Looking to learn about some new mathematical artists? Check out this article, “When Math Meets Art,” from the online magazine Dark Rye. It profiles seven mathematical artists– some of whom we’ve written about (such as Erik and Martin Demaine, of origami fame, and Henry Segerman), and some of whom I’ve never heard of. The work of string art shown above is by artist Adam Brucker, who specializes in making “unexpected” curves from straight line segments.

gauss17_smallAnother of my favorites from this article is the work of Robert Bosch. One of his specialities is making mosaics of faces out of tiles, such as dominoes. The article features his portrait of the mathematician Father Sebastien Truchet made out of the tiles he invented, the Truchet tiles. Clever, right? The mosaic to the left is of the great mathematician Gauss, made out of dominoes. Check out Robert’s website to see more of his awesome art.

Finally, it snowed in New York City yesterday. I love when it snows for the first time in winter… and that got me wanting to make some paper snowflakes to celebrate! Here’s a video by Vi Hart that will teach you to make some of the most beautiful paper snowflakes.

Hang them on your windows, on the walls, or from the ceiling, and have a very happy wintery day! Bon appetit!