# Ghost Diagrams, Three New Games, and Scrabble Tiles

Welcome to this week’s Math Munch!

A ghost diagram composed of two different tiles.

An organism is more than the sum of its organs. When the organs are fitted together, the organism becomes something more. This surprising something more we call “spirit” or “ghost”. Ghost Diagrams finds the ghosts implicit in simple sets of tiles.

So writes Paul Harrison, creator of the amazing Ghost Diagram applet. Paul creates all kinds of free software and has his Ph.D. in Computer Science. I found his Ghost Diagram applet through this huge list of links about generative art.

A ‘111-‘ tile connected to a ‘1aA1’ tile.

Given a collection of tile types, the applet tries to find a way to connect them so that no tile has any loose ends. A tile type is specified through a string of letters, numbers, and dashes. Each of these specifies an edge. You can think of a four-character tile as being a modified square and a six-character tile as being a modified hexagon. Two tiles can connect if they have edges that match. Number edges match with themselves—1 matches with 1—while letter edges match with the same letter with opposite capitalization—a matches with A.

It’s amazing the variety of patterns that can emerge out of a few simple tiles. Here are a couple of ghost diagrams that I created. You can click them to see live versions in the applet. There are many other nice ghost diagrams that Paul has compiled on the site. Also, be sure to check out the random button—it’s a great way to get started on making a pattern of your own. I hope you enjoy tinkering with the ghost diagram applet as much as I have.

And now for some more fun: three new games! When I ran across Loops of Zen, I had ghost diagrams on my mind. I think they have a similar feel to them. The goal in each level of Loops of Zen is to orient the paths and loops so that they connect up without any loose edges. I feel like this game—like good mathematics—requires both a big-picture, intuitive grasp of the playing field and detailed, logical thinking. Put another way, you need both global strategy  and local tactics. Also, if you like playing Entanglement, then I bet you’ll like Loops of Zen, too.

Last week we wrote about Flatland. This book and the movies it inspired describe what it might be like if creatures of different dimensionality were to meet each other. The game Z-Rox puts you in the shoes of a Flatlander. Mystery shapes pass through your field of vision a slice at a time, and it’s up to you to identify what they are. It’s a tricky task that requires a good imagination.

Hat tip to Casual Girl Gamer for both of these great mathy games.

Steppin’ Stones

Steppin’ Stones is a fun little spatial puzzle game I recently came across. You should definitely check it out. It also provides a nice segue to our last mathy item for the week, because a Steppin’ Stones board looks a lot like a Scrabble board. Scrabble, of course, is a word game. Aside from the arithmetic of keeping score, there isn’t much mathematics involved in playing it. In addition, the universe of Scrabble—the English dictionary—is not particularly elegant from a math standpoint. However, it’s the amazing truth that even in arenas that don’t seem very mathematical, math can often be applied in useful ways.

From a comic about Prime Scrabble on Spiked Math.

In Re-evaluating the values of the tiles in Scrabble™, the author—who goes by DTC and is a physics graduate student at Cornell—wonders whether the point values assigned to letters in Scrabble are correctly balanced. The basic premise is that the harder a letter is to play, the more it should be worth. DTC does what any good mathematician does—lays out assumptions clearly, reasons from them to make a model, critiques the arguments of others, and of course makes lots of useful calculations. One tool DTC uses is the Monte Carlo method. In the end, DTC finds that the current Scrabble point values are very close to what the model would assign.

I really enjoyed the article, and I hope you will, too. And since Scrabble is a “crossword game”, I think I’ll leave you with a couple of “crossnumber” puzzles. Here are some straightforward ones, while these require a little more thinking.

Have a great week, and bon appetit!

P.S. I can’t resist sharing this video as a bonus: a cellular automaton of rock-paper-scissors! Blue beats green, green beats red, and red beats blue. Hooray for non-transitive swirls!

# A Closet Full of Puzzles, Sphereland, and Math Doodles

Welcome to this week’s Math Munch!

After a few weeks off, we’re back with some exciting things to share.  First up is Futility Closet, a blog featuring “an idler’s miscellany of compendious amusements.”  The blog is full of big-worded phrases like that, but I most love the puzzles they often post – everything from chess to numbers, codes, and devilish word play.  I also love that the name of the person who wrote each puzzle accompanies it.  Take a look at the few I’ve posted below and click here for the full list of puzzles.

 Here’s a puzzle called Swine Wave, by Lewis Carroll. The puzzle: Lace 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one. Want to know the solution? Click on the image above to visit Futility Closet. This puzzle is called Project Management, by Paul Vaderlind. The question: If a blacksmith requires five minutes to put on a horseshoe, can eight blacksmiths shoe 10 horses in less than half an hour? The catch: A horse can stand on three legs, but not on two. Click on the image to visit Futility Closet for the solution!

Next, have you ever wondered what it would be like to visit another dimension?   In 1884, Edwin A. Abbott wrote about life in the second dimension, in a nice little book called Flatland: A Romance of Many Dimesnions.  (Fun fact: the “A” in Edwin’s name stands for Abbott.  So his name is Edwin Abbott Abbott.)  Click on that link and you can read the whole book, if you like.  The book is about a world of flat beings who have no idea that the third dimension exists.  In the book, the main character, A Square, is visited by a sphere from the unknown world “above” him.  Kind of makes me wonder whether we’re just like the characters in Flatland, three-dimensional creatures ignorant of the fourth dimension that exists “above” us…

Well, the recently released movie Flatland 2: Sphereland deals with precisely that issue.  The Math Munch team had the opportunity to preview this movie, and we loved it.  In Sphereland, the granddaughter of the Square from Flatland, Hex, and her friend Puncto try to understand some mysterious triangles that Puncto thinks will cause the disastrous end of a space exploration mission and go on an adventure to help their three-dimensional friend Spherius with a problem he brought back from the fourth dimension.

Higher dimensions can be very difficult to wrap your head around.  This movie does a great job of helping the movie-watcher to understand how higher and lower dimensions relate to each other through the plot twists and challenges that the characters face.  You can really learn a lot about dimensions and the shape of space by watching this movie.  Plus, the characters are engaging and the images are fun.  Sphereland features the voices of a number of really great actors, including Kristen Bell, Danny Pudi, Michael York, and Danica McKellar.