# 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!

### 27 responses »

1. Oh WOW!!! Z-Rox is amaaaazzing! It really stretched my brain. Seriously, I could feel it. whoa.

• Sweet! Also, it occurred to me that the name is a play on words: Xerox. Get it?

2. If you like “Loops of Zen” you should check out Grid: http://www.candystand.com/play/grid . Similar concept, but it’s always a connected graph and you can only rotate pieces connected to the starting “power source.”

• Nice! I shared it with a couple of my students and they really enjoyed playing it. I could tell that they were thinking hard. Thanks for sharing!

3. Great post. It would be good to have the Rock, Paper, Scissors as an interactive game too, wouldn’t it.

• Thanks! And that’s a great idea. I’ll look into it.

4. I don’t understand why the colors swirl like that. Is it because they’re interacting and then winning and losing against eachother? Was that a site you found or did you make it yourself because whoever invented that was a genius. Thanks for the post! It really made me feel arsty!

• Hi Lillian,
Yes, that’s it exactly. Each color represents one of paper, rock, and scissors. Since they defeat each other cyclicly, they “chase” each other and this turns into a swirl.
I only found the video—I didn’t make it. The video appears to have been made by Benjamin Grob of Kiwibonga Studios. I have, however, tinkered with some cellular automata of my own using a program called Golly. You can get it on our Software page. There are also cellular automata applets on the web you can try out, like this one:
http://0.s3.envato.com/files/1285887/index.html
I’m glad the post made you feel artsy. That’s a great feeling to have!
Justin

• Here at the International School of Toulouse the kids were recently playing the outdoor game “Poule renard vipère” (Chicken, fox, snake) and it made me think of this.

There are three teams, each with a base or corner. The foxes have to chase and touch the chickens, the chickens have to touch the snakes, and the snakes the foxes. When you’re touched you’re captured and have to go and stand at your captor’s base. You can be released by being touched by one of your own team-mates.
There’s a strange to-ing and fro-ing in this, because, say, the foxes have captured lots of chickens, the snakes no longer have so many predators and threat the foxes so much that they retreat into their base. So the game has sudden swings, and seems to last a lot longer than you’d think it would!

It’s a great game – and I think quite popular in France. I did wonder if there may be a way of creating a board game with a similar negative feedback dynamic, and if I get time I might explore that more…

• Hi Simon,
Thanks for sharing this recess game. It’s helped me to think about the cellular automata in a more concrete way. And it sounds like it would be fun to try out and play!
Cheers,
Justin

5. i found this video very interesting but i got a little confused observing the colorful lines move. But i found it amusing to see that video it was great!! I loved it how the colors kept blending together!!

• Hi Andrea,
I think it’s great that you enjoyed the video even though you found it somewhat confusing. If you have more specific questions, let us know. And you might enjoy reading some of the other comments on the post if you haven’t already.
Cheers!
Justin

6. This video really got me thinking. It’s really interesting how the lines move. It took me a while to realize it, but the lines swirl because they are beating each other non-stop! Does that mean that it would go on forever? I noticed that towards the middle of the video, there was a big blue gap, and I’m guessing that was because there was no red around it to beat it. Also, in the beginning, the person makes an asterisk-type shape with the lines. If they did it any other way, like let’s say they just made three straight lines through the circles, would the outcome be different?

7. This video was confusing to me but I enjoyed the colors and how they swirl together. I don’t under stand what this video was for? I enjoyed the video though.

8. I didm’t really like the game “Z-Rox”. It just didn’t seem fun to me. The shapes were very vague and later in the game I got confused. I tried drawing the images out, but it still wasn’t clicking. Maybe I’ll try again later, but for now it just confuses me.

9. I thought the video was really cool. I’m dizzy now but it was still interesting how the colors could do that. When the video first started it was almost like the colors were eating away the other color but it got bigger from just those 3 circles so that was really interesting..

10. The game ” Z-Rox ” did not make a whole lot of sense when I started playing it. It was a really challenging game that I got really mad at. It was very interesting how you had to figure out what the line was making. I got to level 18 where i tried everything but could not figure it out. I honestly liked some parts of the game but not others.

11. I totally zoned out while watching the swirly video. It confused me a lot… How did those big circles come up? It was both awesome and weird. I’m kinda dizzy!

12. The video about the swirls was really cool. It looked like the swirls just kept producing more and more. It was funny too because out of nowhere comes the huge glob of blue that looks like it just keeps growing but eventually it got smaller and smaller.

13. this made me really nausea because the lines keep moving around pluse it was really cool

14. the video was cool because the lines kept swerving but I saw a huge blue glob in the middle and I figured out that the globs are just going on top of another and that’s how it gets bigger

15. the game z rox was a cool game but it did not make a lot of sense because on level 12 I could not pass I tried and tried and im still trying so ill figure it out

16. I thought loops of zen was very hard but mind tingling game. Why is it called loops of zen I always wondered that while i played. Over all it was very fun!

• Hi Nico!
Thanks for your comment and your question—and for the inspiration for my new post! (http://wp.me/p1WoPj-1yo) Now you can ask the creator of Loops of Zen about his game yourself!
Cheers,
Justin