As understood today, the history of our universe can be described with six parameters. We can constrain these parameters by measuring patterns in the large scale structure of our universe, which are governed by the competition between gravitational collapse and the accelerated expansion of our universe. The most massive collapsed structures are clusters of galaxies, comprised of hundreds to thousands of galaxies. For galaxy clusters, the telltale cosmological pattern is simply their number count as a function of mass and time. In this talk, I will discuss the challenges in using galaxy clusters as a probe for cosmology. We address these challenges through computational methods that explore galaxy formation processes such as energy feedback from active galactic nuclei, synthetic observations of the superheated plasma that permeates galaxy clusters, and methods that automate aspects of pattern-recognition.