Terence Tao's Algebra Game

Terence Tao, a famous mathematics professor at the UCLA, blogged about various issues on Gamifying High-School Mathematics in 2012. He also introduced an interesting mathematical puzzle to gamify elementary algebra, i.e., solving a single linear algebra equation with one variable, on his blog.

The idea of Tao's algebra game is to reduce a given linear algebra equation to "x = numerical_solution" through a selection of a finite number of given clues. This algebra game was created using the MIT Scratch software tool and can be played here (requires Java). Let's give an example using a screenshot of the game (taken from Tao's blog) as shown below. Initially, the puzzle is the algebra equation "5x + 3 = x + 11" and the given clues are the three possibilities "Subtract 1", "Divide by 2" and "Subtract x". A player chooses one of the possibilities by clicking on the avatar icon. Say, suppose the player chooses "Subtract 1", the algebra equation changes to "5x + 2 = x + 10" (since both sides of the original equation "5x + 3 = x + 11" get subtracted by one).

Of course, one possible "solution" to the puzzle below is the sequence of "Subtract 1" then "Subtract x" then "Divide by 2" then "Subtract 1"and then finally "Divide by 2" to yield "x = 2". A total of five moves to reach the desired state. What matters is not the final value of x (which can be eye-balled easily), rather, it is the inquisitive problem-solving process while playing.





Tao's algebra game is absolutely fun (and some of the higher levels are fairly challenging, even for an adult like me).  The benefit is obvious: children learn a real subject (e.g., elementary algebra) while playing.  Particularly pleasant is the mathematical depth behind the game. The twelve levels (from easy to hard) in Tao's Scratch mock-up version were cleverly hand-crafted. Yet, there are a few intriguing questions:  First, how to engineer the difficulty level of the game automatically?   Second, how does a computer (not human player) solve a given puzzle efficiently, i.e., with the fewest number of moves?   Third, how to engage the human players in an entertaining manner so that they keep on playing it and, unknowingly, develop a better number sense or mathematical intuition and that such an improvement can be measured?

Another practical question is the means by which these kind of educational games can be readily deployed on existing computing platforms to reach the target audience players (in this case, the target audience is presumably 8 to 12 years old primary school students, and why not adults? I like playing it too). The MIT Scratch software tool is relatively easy (even for 6 years-old) to create games and animations on desktop PC's and laptops, and accessible only through a web browser.  If this were a Facebook game, can it reach K12 students since Facebook requires users to be older than 13?  If this were a mobile app game, can it reach the target audience who may not have personal mobile devices or audience who only have desktop PC's and laptops?