Below is a list of games (mostly $0) high school math students will enjoy.
Euclid: The Game, designed by Kasper Peulen, takes the basic compass and straightedge constructions featured in geometry class and makes a game of them. There are 20 levels to complete. A teacher can directly access a particular level by modifying the URL. http://euclidthegame.org/Level1.html (for example, change the "1" to a "7") I would have loved the opportunity to learn geometric constructions in this format when I was in high school.
The premise of 2048 is very simple. Use the arrow keys to slide tiles about the game board. When two tiles with the same number touch, they merge into a new tile. For example, when two tiles with the number 64 on them touch, they merge into a 128 tile. The objective is to make a "2048" tile, which would require two 1024 tiles, and each of those require two 512 tiles, and each of those requires two 256 tiles, and each of those requires two 128 tiles, and each of those requires two 64 tiles, and each of those requires two 32 tiles, and each of those requires two 16 tiles, and each of those requires two 8 tiles, and each of those requires two 4 tiles, and each of those requires two 2 tiles. Each move in the game yields a new number tile on a random space. A user may continue with the game even beyond the 2048 tile. I have personally witnessed one of my students with a 65,536 tile. This game has some uses in the math classroom. The obvious is knowing the positive integer powers of 2 and its parallels to data storage capacity in computer hard drives. We can also use this game to teach an introduction to game theory and optimal strategies. After playing this game for a while, a user tends to see patterns and trends, cycles of values and positions that appear frequently. Students can intuit optimal strategies through trial and error, but this could also allow students to use formal mathematics to move towards establishing optimal strategies systematically.
While the card game isn't free, the daily set puzzle is free. My students avoid trying to explain SET to a novice player. It is much easier to play and learn through trial and error than to learn all the rules before doing anything... (sounds a bit like mathematics, in my opinion). Each card has four attributes - shape, color, shading, and number of shapes. A set is a collection of three cards for which each individual attribute is all the same or all different. Here are two examples of possible sets.
There are some great combinatorial features associated with this game. A great strategy lends itself incredibly well to mathematical statement: For any two selected cards, there exists a unique third card in the deck which completes the set.
Alice isn't a game, but rather a software package which teaches computer programming as if it were a game. Students can use previously constructed environments and characters to create animated videos. Students can use the interface to make their own games. I have used this software in my class for five years. I have had students make a soccer game that plays a victory sequence for the first player to three goals, a tank rolling through a world that enables collision detection (the default objects can pass through walls; writing code to detect collisions requires some great geometric reasoning), and a first person zombie shooter game *complete* with a zoom-in rifle scope. All my work with students has been with Alice 2.0. I plan to use Alice 3.0 this upcoming school year!
Scratch is like Alice in that it's a programming interface. I have not used it personally with students, but it's another resource I do plan to explore with some of my students this upcoming school year.
Many are familiar with Sudoku. KenKen is a similar reasoning number puzzle game involving operations in addition to populating the digits.
I've mentioned Alcumus previously on my blog, but I can't say enough about the role Alcumus plays in addressing learners' needs. Teachers can now register for a teacher account to monitor students' progress through the self-paced, differentiated curriculum. Students can sign up for free, only an email account is needed. Students can take a pre-assessment to determine areas of strength and deficiency. Quests and experience points provide instant feedback, instant gratification, and a mechanism to keep students engaged and to avoid tedium.