The n-Queens problem involves positioning n chess queens on an n×n board without any of them threatening one other. Typical combinatorial issue in computer math. This topic has been studied in graph theory, optimization, and calculation difficulty. Modern combinatorial and discrete arithmetic methods like department-and-bound algorithms, constraint delight approaches, and optimization equipment are better at solving the n-Queens problem, however going backwards has often worked. Sparkling advancements in mathematical frameworks, diagram colouring, and combinatorial designs have added ways to handle huge n. In this study, mathematical methods are tested with computers and compared to the n-Queens problem. The research also covers the problem in higher-dimensional settings and investigates its relevance to aid allocation and timetable construction. The results illuminate combinatorial problem-solving methods in computer mathematics.