Tag Archives: puzzles

02

Jul. ’12

Math Cats, Frieze Music, and Numbers

Welcome to this week’s Math Munch!

I just ran across a website that’s chock full of cool math applets, links, and craft ideas – and perfect for fulfilling those summer math cravings!  Math Cats was created by teacher and parent Wendy Petti to, as she says on her site, “promote open-ended and playful explorations of important math concepts.”

Math Cats has a number of pages of interesting mathematical things to do, but my favorite is the Math Cats Explore the World page.  Here you’ll find links to cool math games and explorations made by Wendy, such as…

… the Crossing the River puzzle!  In this puzzle, you have to get a goat, a cabbage, and a wolf across a river without any of your passengers eating each other!  And…

… the Encyclogram!  Make beautiful images called harmonograms, spirographs, and lissajous figures with this cool applet.  Wendy explains some of the mathematics behind these images, too. And, one of my favorites…

… Scaredy Cats!  If you’ve ever played the game NIM, this game will be very familiar.  Here you play against the computer to chase cats away – but don’t be left with the last cat, or you’ll lose!

These are only a few of the fun activities to try on Math Cats.  If you happen to be a teacher or parent, I recommend that you look at Wendy’s Idea Bank.  Here Wendy has put together a very comprehensive and impressive list of mathematics lessons, activities, and links contributed by many teachers.

Next, Vi Hart has a new video that showcases one of my favorite things in mathematics – the frieze.  A frieze is a pattern that repeats infinitely in one direction, like the footsteps of the person walking in a straight line above.  All frieze patterns have translation symmetry – or symmetry that slides or hops – but some friezes have additional symmetries.  The footsteps above also have glide reflection symmetry – a symmetry that flips along a horizontal line and then slides.  Frieze patterns frequently appear in architecture.  You can read more about frieze patterns here.

Reading about frieze patterns is all well and good – but what if you could listen to them?  What would a translation sound like?  A glide reflection?  The inverse of a frieze pattern?  Vi sings the sounds of frieze patterns in this video.

[youtube http://www.youtube.com/watch?v=Av_Us6xHkUc&feature=BFa&list=UUOGeU-1Fig3rrDjhm9Zs_wg]

Do you have your own take on frieze music?  Send us your musical compositions at MathMunchTeam@gmail.com .

Finally, if I were to ask you to name the number directly in the middle of 1 and 9, I bet you’d say 5.  But not everyone would.  What would these strange people say – and why would they also be correct?  Learn about this and some of the history, philosophy, and psychology of numbers – and hear some great stories – in this podcast from Radiolab.  It’s called Numbers.

Bon appetit!

P.S. – Paul made a new Yoshimoto video!  The Mega-Monster Mesh comes alive!  Ack!

[youtube https://www.youtube.com/watch?v=PMpr8pA5lJw&feature=player_embedded]

P.P.S. – Last week – June 28th, to be exact – was Tau Day.  For more information about Tau Day and tau, check out the last bit of this old Math Munch post.  In honor of the occasion, Vi Hart made this new tau video.  And there’s a remix.

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29

May. ’12

Slides and Twists, Life in Life, and Star Art

Welcome to this week’s Math Munch!

I ran across the most wonderful compendium of slidey and twisty puzzles this past week when sharing the famous 15-puzzle with one of my classes.  It’s called Jaap’s Puzzle Page and it’s run by a software engineer from the Netherlands named Jaap Scherphuis. Jaap has been running his Puzzle Page since 1999.


Jaap Scherphuis
and some of his many puzzles

Jaap first encountered hands-on mathematical puzzles when he was given a Rubik’s Cube as a present when he was 8 or 9. He now owns over 700 different puzzles!

Jaap’s catalogue of slidey and twisty puzzles is immense and diverse. Each puzzle is accompanied by a picture, a description, a mathematical analysis, and–SPOILER ALERT–an algorithm that you can use to solve it!

On top of this, all of the puzzles in Jaap’s list with asterisks (*) next to them have playable Java applets on their pages–for instance, you can play Rotascope or Diamond 8-Ball. Something that’s especially neat about Jaap’s applets is that you can sometimes customize their size/difficulty. If you find the 15-puzzle daunting, you can start with the 8-puzzle or even the 3-puzzle instead. The applets also have a built in solver. I really enjoy watching the solver crank through solving a puzzle–it’s so relentless, and sometimes you can see patterns emerge.

Over ten solves, I found that the autosolve for the 15-puzzle averaged 7.1 seconds. How long do you think on average the 63-puzzle would take to solve?

You can read more about Jaap in this interview on speedcubing.com or on his about page.

puzzle

The 15-puzzle

Rotascope

Diamond 8-ball

Next, I recently read about an amazing feat: Brice Due created a copy of Conway’s Game of Life inside of a Game of Life! This video shows you what it’s all about. It starts zoomed in on some activity, following the rules of Life. The it zooms out to show that this activity conspires to make a large unit cell that is “turned on.” This large cell was dubbed a “OTCA metapixel” by its creator, where OTCA stands for Outer Totalistic Cellular Automata.

Finally, the video zooms out even more to show that this cell and others around it interact according to the rules of Life! The activity at the meta-level that is shown at the end exactly corresponds to the activity on the micro-level that we began with.  Check it out!

This metapixel idea has been around since 2006, but the video was created just recently by Philip Bradbury. It was made using Golly, a cellular automata explorer that is one of my favorite mathematical tools.

Last up, some star art! (STart? STARt? st-art?)  It turns out that the Math Munch team members all converged toward doing some StArT this semester as a part of our mathematical art (MArTH) seminar. Here is some of our work, for your viewing pleasure. Bon appetit!

by Anna Weltman

by Anna Weltman

Stars of the Mind’s Sky
by Paul Salomon

Star Ring 24
by Paul Salomon

300 Stars in Orbit
by Paul Salomon

by Justin Lanier

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06

May. ’12

Scott Kim, Puzzles, and Games

Welcome to this week’s Math Munch!

Scott Kim

Meet Scott Kim.  He’s loved puzzles ever since he was a kid, so these days he designs puzzles for a living.  He’s been writing puzzles for Discover Magazine for years in a monthly column called “The Boggler.”  Click that link to look through some of his Boggler archives.  Here’s a cool one he wrote in 2002 about hypercubes and the 4th dimension.

Ambigram

In his 11-minute TED talk, Scott tells the story of his career and shares some of his favorite puzzles, games, and ambigrams.  It’s also completely clear how much he really loves what he does (as do I.)

Knights on Horseback – M.C. Escher

I’ve always loved “figure/ground” images, where the leftover space from one shape creates another recognizable shape.  M.C. Escher created some of the most famous and well-known examples of figure/ground art, but Scott Kim took the idea a step further – making an interactive puzzle game based on the ideas.  Naturally, the game is called “Figure Ground,” and it’s delightfully tricky.  You can even create your own levels.  Scott has a whole page of web games.  Go play!

Symmetrical Alphabet – Ambigram by Scott Kim

Still hungry for more Scott Kim?  He gave a presentation for the Museum of Math‘s lecture series, Math Encounters.  You can watch the full-length video here.  You can also watch an interview he did with Vi Hart by clicking here.

Finally, after you read a Math Munch (or right in the middle) do you ever have a question you wish someone could answer or something you want explained?  Or do you ever wish we could help you find more of something you liked in the post?  Well we can do that!  Just leave a comment on the bottom of the page, and the Math Munch team will be very happy to answer.  We’d love to hear from our readers.

Bon appetit!

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