Quantifying over play: Constraining undesirable solutions in puzzle design

Motivated by our ongoing efforts in the development of Refraction 2, a puzzle game targeting mathematics education, we realized that the quality of a puzzle is critically sensitive to the presence of alternative solutions with undesirable properties. Where, in our game, we seek a way to automatically synthesize puzzles that can only be solved if the player demonstrates specific concepts, concern for the possibility of undesirable play touches other interactive design domains. To frame this problem (and our solution to it) in a general context, we formalize the problem of generating solvable puzzles that admit no undesirable solutions as an NPcomplete search problem. By making two design-oriented extensions to answer set programming (a technology that has been recently applied to constrained game content generation problems) we offer a general way to declaratively pose and automatically solve the high-complexity problems coming from this formulation. Applying this technique to Refraction, we demonstrate a qualitative leap in the kind of puzzles we can reliably generate. This work opens up new possibilities for quality-focused content generators that guarantee properties over their entire combinatorial space of play.