Areas of math that can be "gamified"? My old high school math teacher has started up a math club recently. It's become quite popular (to my pleasant surprise), and I've been looking for ways to contribute. To that end I've been looking to introduce games and/or small programs that allow students to investigate different mathematical areas. A key example of what I'm looking to make (and my first project) is a game version of Euclid's Elements. This is an ideal candidate, because we have


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*a few simple rules

*very visual/experiential proof style

*problems are constructions (not "prove this fact is always true") and so students can genuinely learn by progressing through specific examples

*the logical structure of how proofs build on each other is very immediate. This may be emphasized by allowing previous constructions to be used as macros.

*Random guessing will fail. If you're interested in solving the puzzle at all, there are no shortcuts. The intellectual effort needed to solve the problems deliberately is clearly the best way to beat the game, and doesn't just have to serve as 'its own reward' or some such thing.

*doing math is actually the goal of the game. A lot of attempts at gamifying subjects do so by making the subject an obstruction to the real goal that contributes nothing of value from a game design (or game player) standpoint. If this is going to work, the game has to come first. 
The last point is essentially "if I didn't deliberately want to gamify an area of math, but just make a fun game, would this still be worth considering?" I consider the answer here to be yes, for the above reasons. And now we come to the

QUESTION: What areas of math do you think would lend themselves well to games and why? They don't have to be geometric, anything goes (I've seen automata and turing machines used quite effectively, for example). Areas both inside and outside the standard high school curriculum are welcome: I want to expose them to cool new areas while also showing them the value of what they're learning.

BACKGROUND: Some other ideas I had which I rejected as no, for contrast:


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*a shooter game where you were stopped to solve trigonometry and geometry problems that are relevant to programming such a game. i.e, "given your position and the mouse's position, what formula would calculate the direction to shoot bullets in?" and it would shoot them in the way you specified, giving incentive for the right answer. The reason I said no to this is because the math here is just a superficial structure on top of the real game (point 6 fails). Since the problem has nothing to do with the goal, someone who can't immediately solve the problem would have no real incentive to do so. I think it can be tweaked into something more natural, but I'm leaving it alone for now.

*A graph theory game that illustrates various concepts as levels of a puzzle game. "find an optimal colouring of this graph", "find a hamiltonian path", "construct a graph with properties x,y,z", etc. The reason I said no to this is because it's not really a game so much as a collection of exercises with a visual interface. You could progress through the game through brute force and be no better at graph theory than you were when you started. The nature of the levels doesn't 'force insight' to proceed (point 5 fails). It could still be useful for a class, but it's not really a proper game.


Something I'm on the fence about would be "Think like a Topologist": you're given two shapes or everyday objects and told to determine whether or not they're homeomorphic (without using those terms, of course). Gauging correctness/speed or getting the class to play together might make for a quick competitive game. Thoughts?
 A: Well, let me list some ideas from math games (using both words fairly loosely) that I've found interesting.  Hopefully, I'm not a complete outlier here, so others may find them fun too.
Fractals
There are a number of constructions, such as L-systems or iterated function systems that allow one to specify a set of rules and generate interesting fractal shapes based on them.  With a modern computer, one can even allow the user to vary the parameters in real time and see how the shape changes.
Cellular automata
Another fertile area for experimenting with emergent complexity.  There are a number of well known cellular automaton rules (Conway's Game of Life being only the tip of the iceberg) that can produce surprisingly complex (and pretty!) behavior from simple rules and starting conditions, but of course, the real fun is in coming up with your own rules and seeing what they do.
Programming games
The most "game-like" idea in this list: take any traditional game but, instead of the players playing it directly, have them write a set of playing instructions in advance and let the game play out non-interactively according to these instructions.  Then let the players refine their instructions and repeat.
The trick is coming up with games that are sufficiently simple mechanically to make the programming task easy to get started with, yet complex enough strategically to remain interesting.
One possible starting point might be to have the players design strategies for iterated prisoner's dilemma (which is really simple mechanically: on each turn you choose between two alternatives and then find out what your opponent chose) and see if anyone will come up with the tit for tat strategy.  Then change the payoffs or introduce some variations (like random errors, or nearest-neighbor competition and replication on a lattice) and see what happens.

Common themes to all the ideas above are indirect control and emergent complexity: instead of having the player control what happens in the game directly in real time, have them specify a fairly simple set of rules in advance and watch complicated behavior arise.  I don't think that's a coincidence.  In a sense, that is exactly what math is about: starting with simple rules and seeing something complex and beautiful emerge.
A: When I saw your question I was immediately reminded of this:
http://logitext.mit.edu/logitext.fcgi/tutorial
See also this related blog post by the author:
http://blog.ezyang.com/2012/05/an-interactive-tutorial-of-the-sequent-calculus/
It's essentially a rudimentary proof assistant for the LK calculus, presented with an interface that makes it feel very game-like. I think it provides strong validation for your idea of gamifying Euclidean geometry, by demonstrating that formal logic in general can be convincingly gamified, even absent the nice visual aids that a geometry game would allow you to provide.
A: Online topology game illustrating a recent theorem
http://www.sci.osaka-cu.ac.jp/math/OCAMI/news/gamehp/etop/gametop.html
A: This is a long comment; hence, community wiki status.
Gamification is not the embedding of educational content into a game. Gamification is the construction of elements typically found in games around a traditional paradigm, whether it be education, training, marketing, etc. This mistaken understanding is a common fallacy and a big reason why there are so many failed attempts at "making learning fun."
Believe it or not, games aren't fun because they're interactive, or have good graphics, or involve knocking things over or shooting things, or anything else like that.
Games are fun because they happen to satisfied a user's intrinsic psychological needs. In fact, most gamers will encounter exceptional frustration at their favorite games. This is certainly not the description of "fun" that most people have!
What games do is they create a framework in which self-actualization is possible. Games synthesize a progression structure in which a user is able to demonstrate competence. Good games also allow a user to dictate their own courses of action, giving them autonomy. And great games allow us to be social with them, creating a sense of relatedness.
Consider any great student: aren't they proud of themselves when they solve a hard problem, even if it frustrated them for hours? Aren't they the ones that like to read three chapters ahead and stay with you after class? Aren't they the ones that like to show their peers the "tricks" they've discovered?
This is what games do. Any mathematical area can be gamified if you do it right. Doing it right is exceptionally hard, which is why most math games are exceptionally boring.
Does adding a colorful graphic make a cosine more fun? No. Does spinning a graph with a mouse make learning about graph theory more fun? No.
None of these things are fun and interesting unless the user is already motivated to see where it goes. To most students, they don't care, so a video game is just a different flavor of worksheet.

If you want to gamify math, don't make the game math. Make the game something else. Make it a story. Give them something to care about, and make mathematics the gateway. Gate rewards behind problems. Scale rewards based on difficulty. The game should have almost nothing to do with mathematics, but rather should motivate students to use the mathematics to uncover the things they're intrigued about.
The star pupil is motivated by learning, and that's great. But the average student doesn't have this motivation, so we need to give it to them somehow. And if you're really clever, you can then tie that story into math, but don't feel like it has to be math. Make it feel like the student can achieve their own goals. Make them want to solve difficult problems and overcome frustration.
If the student hates math, the solution isn't to make the reward more math.
A: Graph theory, particularly for planar graphs, seems a good area for the sort of recreational math puzzles to keep club members interested.
Here are two that I learned about recently:

Planarity The goal is to move vertices
around and obtain a straight-edged embedding without crossings (computer-based)
Sprouts Two players take turns
adding non-crossing edges (which may be curved) to a graph until a maximal planar
graph is reached (pencil-and-paper; normal and misère forms of play).

Better known is the game of Hex,
which I learned of (many years ago) from Martin Gardner's Mathematical Games
column
in Scientific American, doubtless another good source of ideas.
