1. The problem is to understand the mathematical logic and grandmaster strategies in chess, focusing on the structure and strategic thinking involved. 2. The chessboard can be represented mathematically as an 8x8 grid, which is a finite set of positions. Each position can be denoted by coordinates $(x,y)$ where $x,y \in \{1,2,\ldots,8\}$. 3. Key concepts include pattern recognition, control of the center, and tactical awareness, which can be analyzed using combinatorial mathematics and probability. 4. Advanced strategies involve positional logic and strategic planning, which can be modeled using decision trees and game theory concepts like minimax optimization. 5. Grandmaster techniques rely heavily on recognizing recurring patterns and anticipating opponent moves, which can be formalized using algorithms and probabilistic models. 6. Psychological and temporal factors such as time pressure and deception add complexity, which can be studied through game theory and behavioral models. 7. Training algorithms for chess intelligence use machine learning and search algorithms to simulate grandmaster-level play. Final answer: Chess strategy can be deeply understood and improved by applying mathematical logic, combinatorial analysis, game theory, and algorithmic training methods.