In this doctoral dissertation, we address the counting of Hamiltonian cycles in 2-tiled graphs. These graphs are a generalization of the construction of large 2-crossing-critical graphs. We also address the study of the learning process involved in solving unsolved mathematical problems, integrating psychological theories of optimal experience (flow) and deliberate practice into a mathematical framework called the dense challenge domain. The introduction presents fundamental graph theory concepts and an overview of ordinal numbers essential for understanding the core of the dissertation. In the second chapter, known results from relevant related fields are introduced, along with the contributions of the doctoral dissertation. In the third chapter, we address the problem of counting Hamiltonian cycles in 2-tiled graphs. First, we introduce basic concepts such as tile, 2-tile, tiled graphs, and 2-tiled graphs. This is followed by results leading to the characterization of Hamiltonian cycle types in 2-tiled graphs and then the introduction of algorithms that count each type of Hamiltonian cycles. We also demonstrate that if the family of 2-tiles used to construct 2-tiled graphs is finite, the algorithms are efficient. Further, we place large 2-crossing-critical graphs in the context of 2-tiled graphs and adapt the previously introduced algorithms to efficiently count all types of Hamiltonian cycles. To describe 2-crossing-critical graphs, we introduce an alphabet and show that 2-traversing and flanking Hamiltonian cycles can be counted by simply counting the occurrences of certain letters from the introduced alphabet. At the end of the chapter, we extend the counting of traversing Hamiltonian cycles to tiled graphs. We attempt to formally capture the experience of this research work in the fourth chapter. We propose a formal mathematical framework for solving a common challenge in mathematical education: how to effectively use limited time to motivate students for research work. We formalize a theoretical mathematical structure called the dense challenge domain and introduce a structured decision process based on Csikszentmihalyi's theory of flow, Duckworth's concept of grit, Snowden's Cynefin framework from decision theory, and Bokal-Steinbacher's time usage optimization model. Unlike traditional educational research, which focuses on primary and secondary education, our approach emphasizes fostering mathematical thinking at the research level through optimal psychological experience. We formalize an algorithm for evolving the dense challenge domain, ensuring a balance between perceived skill and levels of challenge. We further prove that large 2-crossing-critical graphs satisfy the conditions of the dense challenge domain, providing a solid mathematical foundation for this methodology. Through three pilot studies, we demonstrate that this approach not only maintains student engagement but also leads to publishable research outcomes.