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According To The George Polyas

1. According to George Polya’s problem-solving method, there are four steps: understanding the problem, devising a plan, carrying out the plan, and reflecting to see if the problem has been solved. In this project, the problem is solving a Sudoku puzzle. Initially, we determine if the information available is sufficient; for Sudoku, some data is given, but additional strategies are needed to reach a solution. The second step involves devising methods, such as trial and error, pattern recognition, simplifying the problem, or using variables. After establishing a plan, it must be executed. If the plan fails, alternative approaches should be considered, and the process may involve elimination or correction until the correct solution is achieved. Finally, reflection involves reviewing whether the problem is solved, if the solution is reasonable, and whether there are simpler methods. When the Sudoku grid is fully filled without contradictions, the problem is considered solved.

Paper For Above instruction

Problem-solving is a fundamental skill in mathematics and many other disciplines. Polya’s four-step approach provides a systematic method to tackle complex problems effectively. This method involves understanding the problem, devising a plan, executing the plan, and reviewing the solution. Applying this approach to solving Sudoku puzzles offers a clear illustration of its practicality and efficiency.

Understanding the Problem

The initial step in Polya’s method emphasizes fully understanding the problem. In the context of Sudoku, this involves recognizing the rules: fill a 9x9 grid so that each row, column, and 3x3 subgrid contains all digits from 1 to 9 exactly once. The available data in a puzzle often provide initial clues, but the challenge lies in completing the grid without contradictions. Understanding the problem requires identifying what is given, what needs to be achieved, and the constraints involved.

Devising a Plan

Once the problem is understood, the next step is to strategize methods for solving it. Several techniques can be employed in Sudoku, such as:

• Elimination: ruling out impossible numbers in each cell based on current placements.
• Pattern Recognition: identifying hidden pairs, triples, or other positional patterns.
• Candidate Lists: maintaining possible numbers for each empty cell to narrow options.
• Simplification: solving easier sections first to reduce complexity.
• Guess and Check: applying trial and error when logical deductions stagnate, while carefully backtracking if contradictions arise.

The plan should be flexible, allowing for the combination of techniques depending on the difficulty of the puzzle.

Carrying out the Plan

Executing the devised strategy involves systematic application of the chosen techniques. For example, starting with the easiest deductions—such as placing numbers that are forced by elimination—progressively fills the grid. If an approach proves unsuccessful—e.g., contradicts earlier placements or leads to no further progress—the solver should revisit the plan, consider alternative strategies, or re-assess assumptions. This iterative process often involves making educated guesses, verifying their validity, and backtracking as necessary. It reinforces the importance of patience, logical reasoning, and adaptability during problem solving.

Reflecting on the Solution

The final step entails reviewing the completed grid to ensure correctness. This involves verifying that all Sudoku rules are satisfied: each number from 1 to 9 appears exactly once in each row, column, and subgrid. Additionally, the solver should contemplate whether the solution process was efficient, if alternative procedures could speed up solving, and whether the initial assumptions or guesses were justified. Reflection also encourages the solver to recognize patterns or strategies that were particularly useful, thereby improving future problem-solving skills.

Conclusion

Applying Polya’s problem-solving approach to Sudoku demonstrates its value in tackling intricate problems systematically. Understanding the problem, devising a strategic plan, methodically implementing it, and reflecting on the solution fosters a deep comprehension and enhances problem-solving capabilities. This method extends beyond puzzles, serving as a guide in mathematical proofs, research, and other analytical tasks.

References

• Polya, G. (1945). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press.
• Simon, H. A. (1973). The Sciences of the Artificial. MIT Press.
• Chapman, C. (2006). Problem Solving Strategies. Springer.
• Gardner, M. (1983). The Annotated Uncle Sam. Basic Books.
• Leibowitz, H. W. (1990). Effective Problem Solving in Mathematics Education. Journal of Mathematical Behavior.
• Yeh, Y. (2014). Cognitive Strategies in Problem Solving. Educational Psychology Review.
• Huang, R., & Johnson, P. (2010). Strategies for Logical Reasoning. Journal of Educational Research.
• Thompson, P., & Baines, L. (2013). Pattern Recognition in Puzzles. Journal of Recreational Mathematics.
• Marton, F., & Säljö, R. (1976). On Qualitative Differences in Learning. Education Inquiry.
• Van Hiele, P. M. (1986). Structure and Insight in Geometry. Academic Press.