In this paper, the author aims to offer an elaboration of simple, yet powerful, mathematical patterns through mathematical games. Mathematical games may serve as colorful instructional tools for teachers and textbooks, and may raise students' motivation and intuition. Patterns are fundamental in mathematics and computer science. In the case of games, they may involve integer properties such as parity, primality, and divisibility, as well as invariant properties of repeated actions. The author presents four simply stated, less familiar mathematical games--Co-primes, Interleaved Parity, Proper Divisors, and Powers of 2--and develops the winning strategy for each of the games. Preliminary novice attempts are mentioned, and then instructive strategy developments are presented, through the illumination of underlying mathematical patterns. The author concludes by reflecting on the illuminated patterns and the way they were reached.