Play has been discussed by mathematicians and mathematics educators as essential to mathematical progress and has been widely acknowledged to have a role in learning. Yet, play is rarely acknowledged, leveraged, or studied for mathematics learners beyond early childhood. Moreover, there are theoretical and empirical challenges with designing for play in mathematics classrooms: first, play must be defined in a way that can be evaluated through observation; second, students may view play as outside of their frame for mathematics based on their prior experiences in more traditional mathematics classrooms. To begin to address these challenges, I conducted a study that explored how six undergraduate students interact with digital puzzle games and whimsical mathematical problems in a laboratory context. Participants engaged in clinical interviews wherein they played the digital puzzle games Puzzledom and Patrick's Parabox and worked on tasks from the texts Mathematical Puzzles by Peter Winkler and The Riddler by Oliver Roeder. To analyze this data, a theoretical framework for defining and identifying play was developed and utilized as a codebook. In examining the undergraduate participants' interactions with these tasks, characteristics of play were found across all tasks. However, for reasons related to the design of each task and the differences in the participants' backgrounds and relationships with mathematics, these characteristics did not always coalesce into moments of play -- particularly with the mathematical tasks. Reflection on these findings and how they might be interpreted in the context of broader literature led to the development of a series of design conjectures, to be tested in future cycles of design-based research. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]