A Mathematical Framework for Scalable Tic-Tac-Toe Game Grid Indexing.

This paper presents a mathematical approach for indexing elements in a two-dimensional grid using arithmetic progressions. Each row and column of the grid is represented as an arithmetic sequence, allowing their general terms to be expressed in a structured algebraic form. Formulas are derived for row-wise and column-wise indexing, followed by an analysis of diagonal and anti-diagonal elements. These derivations provide a systematic method for identifying positions within an N × N grid. The proposed framework can be applied to grid-based problems such as Tic-Tac-Toe, matrix indexing, board-game analysis, and computational grid systems.