Permutation sorting, one of the fundamental steps in pre-processing data for the efficient application of other algorithms, has a long history in mathematical research literature and has numerous applications. Two special-purpose sorting operations are considered in this paper: context directed swap, (cds) and context directed reversal, (cdr). These are special cases of sorting operations that were studied in prior work on permutation sorting. Moreover, cds and cdr have been postulated to model molecular sorting events that occur in the genome maintenance program of certain species of single-celled organisms called ciliates. This paper investigates mathematical aspects of these two sorting operations. The main result of this paper is a generalization of previously discovered characterizations of cds-sortability of a permutation. The combinatorial structure underlying this generalization suggests natural combinatorial two-player games. These games are the main mathematical innovation of this paper.

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Permutation sorting, one of the fundamental steps in pre-processing data for the efficient application of other algorithms, has a long history in mathematical research literature and has numerous applications. Two special-purpose sorting operations are considered in this paper: context directed swap, abbreviated cds, and context directed reversal, abbreviated cdr. These are special cases of sorting operations that were studied in prior work on permutation sorting. Moreover, cds and cdr have been postulated to model molecular sorting events that occur in the genome maintenance program of certain species of singlecelled organisms called ciliates.This paper investigates mathematical aspects of these two sorting operations. The main result of this paper is a generalization of previously discovered characterizations of cdssortability of a permutation. The combinatorial structure underlying this generalization suggests natural combinatorial two-player games. These games are the main mathematical innovation of this paper.

Certain video games show promise as tools for training spatial skills, one of the strongest predictors of future success in STEM. However, little is known about the gaming preferences of those who would benefit the most from such interventions: low spatial skill students. To provide guidance on how to design training games for this population, we conducted a survey of 350 participants from three populations: online college-age, students from a low SES high school, and students from a high SES high school. Participants took a timed test of spatial skills and then answered questions about their demographics, gameplay habits, preferences, and motivations. The only predictors of spatial skill were gender and population: female participants from online and low SES high school populations had the lowest spatial skill. In light of these findings, we provide design recommendations for game-based spatial skill interventions targeting low spatial skill students.

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