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 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.
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!
Some more sliding puzzle resources you might like:
Thanks, Brett! Those look great!
Can you point us to directions for making the art?
Yes! For my StArT, I made hyperbolic paraboloids. Here’s a pattern: https://www.math.lsu.edu/~verrill/origami/parabola/ (I do the folds in a different order, but that’s up to you.) I assembled the polyhedra with some guidance from this awesome paper by Erik Demaine: http://erikdemaine.org/papers/BRIDGES99/paper.pdf I made the polyhedra shown in this post myself, but I’ve also built some in class with my students.
For most of my animations, I used Cinderella (http://www.cinderella.de). The basic element is made of a line segment, another segment coming off the first’s endpoint, and then a large number of segments that follow in “geometric sequence”. As the first is to the second–in terms of size and angle–so is the second to the third, the third to the fourth, and so on. You can produce an enormous variety of shapes using this simple element. The art of it is finding ones you like!
Here are two Cinderella files that you can tinker with: http://bit.ly/O9LFOe and http://bit.ly/PPhpFx
Each contains the basic element, preconstructed. To make your own, use the Modes>Transformation>Similarity tool. Enjoy!
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At first, it wasn’t making much sense until on of the last shots. I saw that the inner most shot was the same as the outer most shot. Technically the big shot is made up of lots of the same little ones. I was wondering what all of the moving figures were on the outside of the big squares on the 3rd or 4th viewing because they eventually disappeared, and didn’t play a big role in the final shot. I was also wondering, is this in an actual cell, and is that how they move?
You’ve watched this video with a careful eye. Great observations.
First, these are not the cells that make up plants and animals. They are just cells on a mathematical grid. They do sort of behave like living cells, though, and that’s why this program is called the Game of Life.
About the moving figures on the outside of the big squares—I think they are there to make the whole thing “work”. Kind of like scaffolding. Or another way to think about it is that the big squares need to communicate with each other, to know when to turn on and off. Those extra bits probably help to “wire” the big squares together.
I hope you find more great math to dig into on the site!
Thanks for the information to help make it more clear!
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Reblogged this on Math Munch and commented:
Happy second Thursday, and get your engines star-ted! We hope you’ll enjoy this throwback post from May 2012. Bon appetit!